Derandomization through Approximation: An NC ... - Semantic Scholar
Derandomization An
NC
through
Algorithm
David
Approximation:
for
Minimum
R. Karger*
Rajeev
Department
of Computer
Stanford Stanford, {karger,motwani}
Cuts
Motwanit
Science
University CA
94305
@!cs. stanf
Abstract
ord.
edu
Introduction
1,
Some of the central We show that
the minimum
lems in weighted
undirected
JVC. We do so by giving dently interesting results. sor JVC algorithm
cut
and multi-cut
graphs
algorithms minimum
prob-
can be solved
in
ings, and depth-first
three separate and indepenThe first is an m2/n proces-
for a (2 + c)-approximation
rithms
tion involves a natural combinatorial Safe Sets Problem that can be solved easily in %?JVC. Our third result is of this 7?JVC solution that requires of two widely used tools: pairwise
derandomization
any constant
k and for finding
step towards
resolving
ing the first
MC algorithm
imum
multi-cuts
The
rein-cut
for the rein-cut
by presentproblem
in
cuts.
problem
value within
a sparse k-connectivity bounded values of k.
minimum
for polylogarithmic
these open problems
minimal
we wish to minimize
was only known
There are 77AfC algo-
[19, 23, 1]. The problem
graphs. Our results extend to minand to the problem of enumerating all
any constant factor of the minimum. An additional byproduct of our techniques is an JVC algorithm for finding certificate Previously,
for s-t match-
is defined
as follows:
given
a
Multigraph with n vertices and m (possibly weighted) edges, we wish to partition the vertices into two nonthe number of empty sets S and T so as to minimize edges crossing from S to T (if the graph is weighted,
problems: we k-way cuts for
all cuts with
undirected
approximately
problems.
Our techniques extend to two related give JVC algorithms for finding minimum
search trees.
for all these problems
weighted
independence and random walks on expanders. We believe that the safe sets approach will prove useful in other
in the area of parallel
cuts belongs to this category of finding global minimum of unsolved derandomization problems, and it is representative in that obtaining an AfC algorithm for the case of directed graphs would resolve the other derandomization questions [17]. We take a (possibly small)
to the
minimum cut. The second is a randomized reduction of the minimum cut problem to the problem of obtaining a (2+ e)-approximation to the minimum cut. This reduc-
a derandomization a novel combination
open problems
are those of devising JVC algorithms cuts and maximum flows, maximum
We distinguish
for all polynomially an JVC construction
lem
values of k.
we
the total
the minimum
cut problem. require
that
weight
of crossing edges).
cut problem
from
In the s-t minimum
two specified
vertices
the s-t
cut probs and t be
cut probon opposite sides of the cut; in the minimum Our work deals only lem there is no such restriction.
with minimum cuts, and we assume that the graph is connected, since otherwise the problem is trivial. The value of a minimum cut in an unweighed graph is also called the graph’s edge connectivity.
*Supported by a National Science Foundation Graduate Fellowship, by NSF Grant CCR-901O517, and grants from Mitsubishi and OTL. tSupported by NSF Grant CCR-9010517, NSF Young Investigator Award CCR-9357849, with matching funds from IBM, Schlumberger Foundation, Shell Foundation and Xerox Corporation.
The
rein-cut
problem
many
fields.
tivity
of a network
network
has numerous
The problem arises
frequently
design and network
Queyranne [26] survey many minimum cuts. In information
Permission to co y without fee all or part of this material is granted provide J’ that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Association of Computing Machinery. To copy otherwise, or to republish, requires a fee and/or specific permission.
applications
of determining reliability
in
the connec-
in the
study
[9]. Picard
of and
of weighted applications retrieval,, minimum cuts
have been used to identify clusters of topically related documents in hypertext systems [5]. Padberg and Rlnaldi [25] discovered that the solution c~fminimum cut problems was the computational bottleneck in cutting-
STOC 94- 5/94 Montreal, Quebec, Canada Q 1994 ACM 0-89791 -663-8/94/0005..$3.50
497
plane based algorithms lem and many
other
for the traveling combinatorial
salesman
problems
the minimum
prob-
minimum
whose so-
cut in the contracted
cut in the original
graph is equal to the
graph.
lutions induce connected graphs. Applegate [3] also observed that a faster algorithm for finding all minimum cuts might accelerate the solution of traveling salesman problems. The first known minimum cut algorithm used the du-
tracting non-rein-cut edges until the graph has been reduced to two vertices. These two vertices define a cut in the original graph. If no rein-cut edge is contracted,
alit y between s-t rein-cuts and max-flows [10, 11]. An s-t max-flow algorithm can be used to find an s-t mini-
then the corresponding cut must be a minimum cut. The edges connecting the two vertices correspond to the
mum
cut edges.
cut,
sible
and
choices
Until
by taking
of s and
recently,
the
minimum
over
t,a minimum
the best sequential
all
(~)
Several contraction-based minimum cut algorithms have recently been developed. They all work by con-
pos-
cut may be found. algorithms
Nagamochi
for find-
to develop
and Ibaraki
an O(mn
[24] used graph
+ nz log2 n)-time
contraction
algorithm
for the
ing minimum cuts used this approach [14]. Parallel solutions to the rein-cut problem have also been studied.
rein-cut
Goldschlager, Shaw and Staples [13] showed that the s-t rein-cut problem on weighted directed graphs is Pcomplete. A simple reduction [17] shows that the (unrestricted) rein-cut problem is also P-complete in such graphs.
rein-cut edges but excludes some edge of the graph. This edge can then be contracted without affecting the minimum cut. Matula [22] used the Nagamochi-Ibaraki cer-
For
unweighted
graphs,
the
I?AfC
matching
algo-
randomizing which that
maximum
bipartite
has long been open. the global
rein-cut
is equivalent matching—a
A reduction
problem
for
Karger
[17] observed
contains
find
all the
that
a large number
of
simultaneously. a randomly
to be in the minimum
selected graph cut; it followed of graph This led
cost of the Contraction Algorithm, as well as its sequential running time, to d(n2); this is presently the most
problem
efficient Luby,
is
known
rein-cut
Naor
and Naor
algorithm
for weighted
[21] observed
that
graphs.
in the Con-
traction Algorithm it is not necessary to choose edges randomly one at a time. Instead, given that the mincut size is c, they randomly
Contraction
they
to the Contraction Algorithm, the first I?JVC algorithm for the weighted rein-cut problem, which used mn2 processors. Karger and Stein [18] improved the processor
also equivalent.
1.1
time
that repeated random selection and contraction edges could be used to find a minimum cut.
to de-
graphs
which
to identify
can be contracted
edge is unlikely
in [17] shows
directed
+ n log n)
certificate
use the sparse certificate edges which
the rnin-cut problem in I?AfC. The processor bounds are quite large, and the technique does not extend to graphs with large edge weights. No deterministic parallel algorithm is known. Indeed, derandomizing max-flow graphs
O(rn
tificate algorithm in a linear time (2+ e)-approximation algorithm for finding minimum cuts—the change is to
to yield 7i!hfC algorithms for s-t minimum cuts. By performing n of these computations in parallel, we can solve
undirected
In
a sparse connectivity
rithms of Karp, Upfal and Wigderson [19] or Mulmuley, Vazirani and Vazirani [23] can be combined with a well-known reduction of s-t max-flows to matching [19]
on unweighted,
problem.
Based
Algorithms
mark
each edge with
prob-
on contracting graph edges. Given a graph G and an edge {u, v}, contracting {u, v} means replacing u and v
ability 1/c, and contract all the marked edges. With constant probability, no rein-cut edge is marked but the number of graph vertices is reduced by a constant factor. Thus after O(log n) phases of contraction the graph is reduced to two vertices which define a cut. Since
with
the number
of phases is O (log n) and there
stant
of missing the minimum probability that is an n ‘o(l)
Recently, a new paradigm has emerged for finding minimum cuts in undirected graphs. This approach is based
a new vertex
w and transforming
each edge {x, u}
or {s, v} into a new edge {z, w}. Any {u, v} edge turns into a self loop on w and can be discarded.
phase,
there
is a con-
cut in each no rein-cut
edge is ever contracted; if this happens then the cut determined at the end is the minimum cut. Observing that pairwise-independent marking can be used to achieve the desired behavior, they show that O (log n) random bits suffice to run a phase. Thus, O (log2 ~) bits suffice to run this modified Contraction Algorithm through its O(log n) phases.
A key fact is that contracting vertices cannot decrease the minimum cut. The reason is that any cut in the contracted graph corresponds to a cut of exactly the same size in the original graph—if u and v were con(A, B) in the contracted to w, then a vertex partition w E A corresponds to a partition tracted graph with (AU {u, w} – {w}, B) in the original graph which the same edges. Let us fix a particular minimum
probability
cuts cut,
which from now on we will refer to as the minimum cut (there may be as many as (~)). The power of con-
Unfortunately, this algorithm cannot be fully derandomized. It is indeed possible to try all possible random seeds for a phase and be sure that one of the polynomi-
tractions contract
a~ly many outcomes is good; however, there is no way to which outcome is good. In the next phase is determine
comes from their interaction with cuts. If we an edge which is not in the minimum cut, then
498
thus necessary
to try all possible
of the polynomially
many
random
outcomes
Overview
of
Naor and Naor technique.
In
such an outcome
Results
to contain Our main result is an NC algorithm for the rein-cut and minimum multi-cut problems. Our algorithm is not a derandomization
of the Contraction
stead a new contraction-based we take G to be a Multigraph
Algorithm
If we mark
each edge
with probability @(l/c), then with constant probability we mark no rein-cut edge while marking edges in a constant fraction of the other small cuts. Pairwise independence in the marking of edges is sufficient to make
phase,
squaring the number of outcomes after two phases, all, Cl(nlOg’) combinations of seeds must be tried.
1.2
Luby,
seeds on each
of the first
likely.
However,
this approach
the same flaw as before:
fl(log
seems
n) phases of
selection are needed to mark edges in all the small cuts, and thus fl(log2 n) random bits are needed.
but is in-
This
algorithm. Throughout, with n vertices, m edges
leads to our third
The problem
of finding
can be formulated
building
blc)ck (Section
5).
a good set of edges to contract
abstractly
as the Saj’e Sets Problem:
and rein-cut value c. Most of the paper discusses unweighed graphs; in Section 6 we reduce from weighted graphs to the unweighed graph problem.
given an unknown collection of sets over a known universe, with one of the unknown sets declared %afe,” find a collection of elements which intersects every set
Our algorithm blocks. The first
depends building
except for the safe one. After giving a simple randomized solution, we show that this problem can by solved
an NC
that
algorithm
(2+ ~)-approximation Matula’s sequential
upon block
three major building (Sections 2 and 3) is
uses m2/n
processors
to find
a
in NC
by combining
the techniques
of pairwise
inde-
to the minimum cut. Recall that algorithm [22] was based on the se-
pendence [7, 20] with the technique of random walks on expanders [2]. This is the first time th~ese two impor-
quential sparse certificate algorithm of Nagamochi and Ibaraki [24]. It repeatedly finds a sparse certificate containing all rein-cut edges and then contracts the edges not in the certificate, terminating after a small num-
tant techniques have been combined in a derandomization, although similar ideas have been used earlier to save random bits in the work of Bellare, Goldreich and Goldwasser [4]. We feel that the combination should have further application in derandomizing other algo-
ber of phases.
Our NC
algorithm
uses a new parallel
sparse certificate algorithm to parallelize Matula’s rithm. A parallel sparse k-connectivity certificate rithm
with
running
Kao, and Thurimella an algorithm
that
time
6(k)
algoalgo-
rithms. It should
was given by Cheriyan,
[6]; we improve
this by presenting
runs in NC for all k = no(l).
Our next building block (Section 4) uses a result obtained from the analysis of the Contraction Algorithm. Karger [17] proved that there are only polynomially many cuts whose size is within a constant factor of the If we find a collection of edges which minimum cut. contains minimum
one edge from cut, then
every
contracting
such cut except
relatively
As was the
easy to solve if the solution
goal is to destroy
all but one solution
the problem is unique,
and Naor
hits every vertex
to finding
technique,
An
2 In any (2+
is
this
t) times
Definition
and
in G.
then to easily find the unique solution. Randomization yields a simple solution to this problem: contract each Because the number edge with probability y Cl(log n/c).
building
with the Luby,
it can find
a set of edges cut.
Un-
such an edge set need only halve the number (e.g., if the edge set is a perfect matching),
in
Section
minimum
still
be necessary—the
6 we apply cuts,
the
minimum
that
2.1
A maximal
of the
above
and to enumerating
we describe c > 0, finds
an NC
and
approx-
algorithm
whose
minimum
k-jungle
results
Algorithm
a cut
A k-jungle
same
nnulti-cuts,
Approximation
section,
constant
this third
directly
but not the minimum
weighted minimum cuts; imately minimum cuts.
so the
to the problem
Naor,
Finally,
for the
this set of edges yields
case there,
by itself,
Combined
so fl(log n) phases would flaw as before.
just mentioned. Since the minimum cut will be the only contracted graph cut within the approximation bounds, it will be found by the approximation algorithm. One can view this approach as a variant on the Isolating Lemma approach used to solve the perfect matching [23].
that
which
fortunately, of vertices
a graph with no small cut except for the minimum cut. We can then apply the NC approximation algorithm
problem
be noted
block is not sufficient.
value
that,
for
is less than
cut.
is a set of k disjoint is a k-jungle
forests
such that
no
other edge in G can be added to any one of the jungle’s forests without creating a cycle in that forest.
of small cuts is polynomially bounded, there is a sufficient probability that no edge from the minimum cut is contracted but one edge from every other small cut is contracted. Of course, our goal is to do away with randomization. A step towards this approach is a modification of the
Lemma 2.1 ([24]) A maximal k-jungle the edges in any cut of k or fewer edges.
contains
all
Proofi Consider a maximal k-jungle .7, and suppose it cent ains fewer than k edges of some cut. Some forest in J must
499
have no edge from
this
cut.
Any
cut edge
not in J could be added to this forest a cycle, so all cut edges must already
Procedure
Approx-Min-Cut(
Figure
degree of G.
maximal
[22].
G by contracting
Approximation
3
)).
Algorithm
We give it as an algorithm
all minimum
cut edges and then
contract
all
edges not in the jungle. The algorithm works quickly because so long as we do not have a good approximation to the minimum cut at hand, we can guarantee that many edges are contracted each time. Lemma
2.2
Given
approximation (2+ E)c.
It remains
k-jungle
only to show how to construct
a
in NC.
a graph
algorithm
with
returns
minimum
Finding
cut c, the
a value between c and
Maximal
Jungles
The notation needed to describe this construction is somewhat complex, so first we give some intuition. To construct a maximal jungle, we begin with an empty jungle and repeatedly augment it by adding additional edges from possible.
to approximate
the cut value; it is easily modified to find a cut with the returned value. The basic idea is to find a jungle that contains
above 2 is needed to en-
edges.
The approximation algorithm is described in Figure 1. It is a parallel version of Matula’s approximation algorithm
e factor
time of this algorithm is thus O(T(m, n) . polylog (m)) where T(m, n) is the time needed to construct a maximal jungle.
min(b, Approx-Min-Cut(G’
1: The
the extra
sure a significant reduction in the number of edges at each stage and thus keep the recursion depth small. The depth of recursion is in fact (3(6-1 log m). Each step of this algorithm, except for Step 3, can be implemented in AfC using m processors. The running
G)
k-jungle.
G’ from
all non-jungle
5. Return
■
c).
3. Find a maximal 4. Construct
Note that
creating
be in J.
Multigraph
1. Let 6 be the minimum
2. Let k = 6/(2+
without
the graph
Consider
non-jungle
edges that
out creating
until
no further
one of the forests
augmentation in the jungle.
may be added to that
is The
forest with-
a cycle are just the edges that cross between
two different
trees of that
forest.
We let each tree claim
some such edge incident upon it. Hopefully, each forest will claim and receive a large number of edges, thus significantly increasing the number of edges in the jungle. Two problems arise. The first is that several trees may claim a particular edge. However, the arbitration of these claims can be transformed into a maximal matching problem
and solved in JVC. Another
problem
is that
since each tree is claiming an edge, a cycle might be formed when the claimed edges are added to the forest (for example, two trees may each claim an edge con-
Proofi Clearly the value is at least c because it corresponds to some cut the algorithm encounters. For the upper bound, we use induction on the size of G. We consider two cases. If 6 < (2 + c) c, then since we return a value of at most 6, the algorithm is correct. On the other hand, if 6 > (2 + C)C, then k > c. It follows
of a k-jungle J = Definition 3.1 An augmentation A = {El,..., E~} of k disF~} is a collection {Fl,...,
from
joint
Lemma
2.1 that
all the rein-cut cut is contracted cut c. returns
the jungle
edges. while
By the inductive a value between
we construct
contains
those two trees).
sets sets
2.3
approximation
The
number
algorithm
of levels of recursion
added
■
in the
is O (log m).
Proof: If G has minimum degree cl, it must have at least 6n/2 edges. On the other hand, the graph G’ which we construct contains only jungle edges; since each forest of the jungle contains only n – 1 edges, G’ can have at most k(n – 1) = J(n – 1)/(2 + e) edges. It follows
that
each recursive
remedy
this problem
of non-jungle Ei
to forest
edges from
be non-empty.
must
G. The
At
least
edges
one
of E,
of are
F%.
call Definition does not
Lemma
We will
as well.
the
Thus no edge in the minimum forming G’, so G’ has minimum hypothesis, the recursive c and (2 + E)C.
necting
step reduces the number
of edges in the graph by a constant factor; thus at a recursion depth of O (log m) the problem can be solved trivially. ■
Fact 3.1 mentation. Given
3.2
A valid
augmentation
of J is one that
create any cycles in any of the forests A jungle
a jungle,
is maximal
it is convenient
of J.
iff it has no valid
aug-
to view it in the fol-
lowing fashion. We construct a reduced (Multigraph GF for each forest F. For each tree T in F, the reduced graph contains a reduced vertex VT. For each edge e in G that connects trees T and U, we add an edge eF connecting VT and vu. Since many edges can connect two forests, the reduced graph may have parallel edges. An edge e of G may induce many different edges, one in each forest’s reduced graph.
500
Given any augmentation, F can be mapped to their inducing
the edges added to forest corresponding edges in GF,
an augmentation
assigned to F. The edges of A may induce cycles in GF, which would mean (Fact 3.2) that A does not correspond to a valid augmentation of F. However, if we
subgraph of the reduced graph
GF.
find an acyclic subset of A then the G-edlges correspond-
Fact
3.2
An augmentation
subgraph
it induces
ing to this subset will form
is valid iff the augmentation
in each forest’s
reduced graph
is a
the reduced graph GF. Direct the reduced vertex to which
forest.
vertex
Care should be taken not to confuse the forest F with the forest that
is the augmentation
subgraph
has outdegree
of GF.
in the reduced Multigraph.
of edges which can be added to J by a constant
no cycle obeying
fraction
must contain
each time, and since the maximum jungle size is m, J will have to be maximal after O (log m) phases. mal matching vertex
problem
to a smaller
on a bipartite
graph
H.
tex with
Let one
VT
of GF to Ve if eF is incident we are connecting
edges incident
upon
on VT in GF.
in it in G.
Note
edge in GF is a valid augmenting the size of H, note that 2k incident
Lemma matching
each vertex
Proofi
Consider
a valid
edges of the augmentation follows. For each forest Multigraph augmenting duced
vertex
Thus
the total
of
forest
matching
J
number
G can
numbered hand,
must contain
two, an impossibility.
any
a ver-
It follows
that
gain
at least
half
the edges assigned
so the augmentation
to it
has the desired
induces
a
Each non-root
augmentation
If edge e c G is matched
to reduced
in NC
using
O(km)
k-jungle
of
processors.
the
re-
edge eF
vertex
GF, tentatively assign e to forest F. Consider the set A of edges in GF that correspond to the G-edges
an augmentation.
Let a lbe the size of a
maximum augmentation of J. Lemma 3,,3 shows that H must have a matching of size a. It follows that any max-
of the jungle.
in H between
Given G and k, a maximal
Proof: We begin with an empty jungle and repeatedly augment it. Given the current jungle J, construct the bipartite graph H as was previously described and
imal matching in H must have size at least a/2, since at least one endpoint of each edge in any maximum match-
and the reduced vertices as F in J, consider its reduced
arbitrarily.
3.5
be found
of
in H, a valid augmenLemma 3.4 Given a matching tation of J of size at least half the size of the matching can be constructed in hfC.
Proof:
a larger
On the other
■
T~eorem
on either
leading to its parent. Since edge e is added to F no other forest F’ will use edge eFl, so we can match vT to V.. It follows that every augmentation edge is matched ■ to a unique reduced vertex.
VT E
will
use it to find
augmentation
UT has a unique
from
size.
v. will have at most be incident
GF. Since the augmentation is valid, the edges in GF form a forest (Fact 3.2). Root
each tree in this
vertex.
the edge directions
outdegree
in the matching,
to the
this means each
3.3 A valid augmentation in H of the same size.
We set up a corresponding
an edge directed
edges can form
since such a cycle
edge for F. To bound
edges, because it will
O or 2 trees of each forest. edges in H is O(km).
forest
Equiva-
each tree in the jungle
Its (directed)
the edges of A. form a valid augmentation of F of at least half the size of the matching. If we do this for each forest F in parallel, we get a valid augmentation of the jungle. Furthermore, each
vT in the various
reduced multigraphs, i.e., the trees in the jungle. Let the other vertex set consist of one vertex v. for each non-jungle-edge e in G. Connect each reduced vertex lently,
in
the edges into two
the edge directions,
numbered
cycle disobeying
we solve a maxi-
set of H consist of the vertices
of F.
the vertices
each edge in A away from it was miitched (so each
one), and split
constant fraction of the largest possible valid augmentation. Since we reduce the maximum possible number
augmentation,
number
groups: A. G A are the edges directed from a smaller numbered to a larger numbered vertex, and Al ~ A are the edges directed from a larger numbered to a smaller numbered vertex. One of these sets, say A., contains at least half the edges of A. However, A. creates no cycles
Our construction proceeds in a series of O(log m) phases in which we add edges to the jungle J. In each phase we find a valid augmentation of J whose size is a
To find a large valid
a valid augrnentation
To find this subset, arbitrarily
ing must be matched
in any maximal
matching.
Several
NC algorithms for maximal matching exist—for example, that of Israeli and Shiloach [16]. Lemma 3.4 shows that
after we find a maximal
transform
this
at least a/4.
matching
matching,
into
Since we find
we can (in Nc)
an augmentation
an augmentation
of size of size at
least one fourth the maximum each time, and since the maximum jungle size is m, the number of augmentations
needed to make a J maximal
is O(log m).
Since
each augmentation is found in NC, the maximal jungle can be found in NC. The processor cost of this algorithm is dominated by that of finding the matching in the graph H. The algorithm of Israeli and Shiloach requires a linear number of processors, and is being run on a graph of size O(km). ■
501
Corollary
3.6
be found
in NC
Proofi
A (2+t)
-approximate
using 0(m2 /n)
A graph
with
minimum
It follows
cut can
that
running
the approximation
algorithm
of Section 2 on the contracted graph will find the minimum cut, since the actual minimum cut is the only one
processors.
m edges has a vertex
with
which is small enought to meet the approximation criterion. Our goal is thus to find a collection of edges that hits every target cut but not the minimum cut. This problem can be phrased more abstractly as follows:
de-
cut can therefore be no gree O(m/n); the minimum larger. It follows that our approximation algorithm will ■ construct k-jungles with k = O(m/n).
Over some universe U, an adversary selects a polynomially sized collection of “target” sets of roughly equal 4
Reducing
to
size (the small cuts’ edge sets), together with a disjoint “safe” set of about the same size (the rein-cut edges).
Approximation
We want
In this section, we show how the problem of finding a minimum cut in a graph can be reduced to that of finding a (2 + e)-approximation.
Our technique
is to “kill”
the vertices of G into two sets A and B. graphs induced by A and B.
this idea, partitions
Consider
4.1
The minimum
5
The
Safe
We describe
Proofi Suppose A has a cut into X less than c/2. Only c edges go from X one of X or Y (say X) must have at leading to B. Since X also has less to Y, the cut
that
intersect we do we do
Sets
Problem
a general
form
U of size
of the Safe Sets Problem.
u.
cuts in A and in B have Definition 5.1 A (u, k, c) safe set instance is a collection of at most k target sets of size Q(c), together with a safe set of size O(c) that is disjoint from the target
value at least c/2.
leading
of elements
the Fix a universe
Lemma
a collection
have an upper bound on the number of target sets. We proceed to formalize this problem as the Safe Sets Problem.
all cuts of size less than (2 + ~)c other than the minimum cut itself. The minimum cut is then the only cut of size less than (2+ e) c, and thus must be the output of the approximation algorithm. To implement we focus on a particular minimum cut which
to find
every target set but not the safe set. Note that not know what the target or safe sets are, but
(X, ~)
and Y of value and Y to B, so most c/2 edges than c/2 edges
has value
less than
contradiction.
sets. An isolator for the safe set instance is a set that intersects all the target sets but not the safe set.
c, a ■
Definition
5.2
a collection
of subsets of U that contains
A (u, k, c)-universal
isolating
family
an isolator
is for
any (u, k, c) safe set instance. Theorem
4.2
([17] )
at most Q times
There are O(n2a)
To see that this general formulation captures our cut isolation problem, note that the minimum cut is the safe set in an (m, k, c) safe set instance. The universe is the
cuts of weight
the minimum.
set of edges, of size m; the target
It follows that in each of A and B, every cut has value at least c/2 and there are n ‘(1 J cuts of weight less than (2+ C)C. Call these cuts the target cuts. Lemma
4.3
Let Y be a set containing
edges from
every
target cut but not the minimum
cut. If every edge in Y is
contracted,
graph has a unique
weight
then the contracted
less than
(2 + ~)c,
namely
the original
sets are the small cuts
of the two sides of the minimum cut; k is the number of such small cuts and (by Lemmas 4.2 and 4.3) can be bounded by polynomial in n < m; and c is the cut size. The approximation
algorithm
of Section
2 allows
us to
family
then
estimate c to within a constant factor. If we had an (m, k, c)-universal isolating
cut of
one of the sets in it would be an isolator for the safe sets instance corresponding to the minimum cut. By Lemma 4.3, contracting all the edges in this set would isolate the minimum cut as the only small cut. If the size of the universal family was polynomial in m, k, and c, we could try each set in the universal family in parallel in NC, and be sure that one such set isolates
minimum
cut.
Proofi Clearly contracting the edges of Y does not affect the minimum cut. Now suppose this contracted graph had some other cut C of value less than (2+.E)c. It corresponds to some cut of the same value in the original graph. Since it is not the minimum cut, it must induce
the minimum
a cut in either A or have value less than target cut, so one of But this prevents C graph, a contradiction.
can find it. Thus our goal is: given U, a constant factor approximation to c, and an upper bound on k, generate a (u, k, c)-universal isolating family of size polynomial in u, k, and c in NC. We first give an existence proof for universal families of the desired size.
B, and this induced cut must also (2+ C)C. This induced cut is then a its edges will have been contracted. from being a cut in the contracted ■
502
cut so that
the approximation
algorithm
Theorem
5.1
ing family
of size at most uk”~~~.
exists a (u, k, c) -universal
We use a standard
Proofi argument. stance.
There
Fix Suppose
with
probability
form
one member
probabilistic
attention
on a particular
we mark
each element
z and call the set of marked elements For each trial, we have some constant the trial is good for a particular i.
existence
We now observe that
safe set in-
the marking
of the universe
(log k) /c and let the marked of the universal
isolat-
family,
With
analysis
prob-
the probability
of failure
ability
that
them
during
Note vant,
on the instance
we fail
to to generate
all the trials
that
is 2–U~O(’).
all 2U~0(1) safe set instances,
the
since we could
If
the prob-
c is
construct
somewhat
universal
constant
Constructing
Universal
We proceed to derandomize sal isolating
family.
irrele-
we fix our attention
value.
Suppose with bility
we independently
probability
l/c,
obtaining
mark
elements is (1 – l/c)Z. inequalities:
under
We use the following
any marked
1 – t.
a result
due to Ajtai,
~
(~-e-,c)
of T~ is marked.
walks on of a fam-
and a convenient
and Galil
[12].
con-
They
show
The following
5.2
is a minor
Kom16s
presents
adaptation
and Szemer6di
a crucial
of
[2] (see
property
of ran-
G..
([2] ) Let B be a subset
OJF
V(G.)
of size
~ log k on Gn,
all from
the probability B is 0(k–2).
that the vertices
visited are
Notice that performing a random walk of length ~ log k on G. requires O(log n + log k) random bits— choosing a random starting vertex requires log n ranbits
and,
since the degree is constant,
each step
of the walk requires O(1) random bits. We use this random walk result as follows. Each vertex of the ex(e-l
(~-~)
pander corresponds to a seed for the mark set generator f described above; thus, log n = O(log u -!- log c), implying that we need a total of O(log u + log c + log k) random bits for the random walk. Choosing B to be the set of bad seeds for i, i.e. those that generate set families containing no good sets for i, and noting that
)s’c.
Since t~/c = $2(1) and s/c = O(l), there is a constant probability of &,, i.e. that no element of S is marked but some element
of random
at most (1 — ~)n, for some constant ~. Then there exists a constant ~ such that for a random walk of length
dom (l-(l-;)’’)(l-;)s
of Gabber
at most
Theorem
standard
Let &i be the event that T% does contain some and S does not contain any marked elements. Then
=
mark
subsets of that f(s)
construction
degree expanders,
is that
dom walks on the expander
(e-w)”c~ (’-:)’<e-”c Pr[S,]
of the
a, different
the size of the seed needed by only
relies on the behavior
also [15, 8]) which
of U
a mark set. The proba-
that a subset of size x does not contain
iterations
process, each yielding
that for sufficiently large n, there exists a graph Gn on n vertices with the following properties: the graph is 7-regular; it has a constant expansion factor; and, for some constant c, the second eigenvalue of the graph is
sets. Let
each element
inde-
of success to any desired
independent
We need an explicit
ily of bounded struction
safe set instance,
instance
This
expanders.
of a univer-
of target
of complete
The next step is to reduce the probability of failure from a constant 1 – ,6 to an inverse polynomially small
and show that our construction will contain an isolator for that instance. It will follow that our construction contains an isolator for every instance. Let S be the s = IS I and t;= [T.1,for the particular consideration.
in
of being good for i. The instead
seed of O(log u + log c) bits and returns 0(1) U. Randomizing over seeds s, the probability contains at least one good set for i is ~.
the derandomization,
safe set and let {T, } be the collection
independence
is needed to achieve
a constant factor. WJe can think of this pairwise independent marking algorithm as a function f that takes a
Families
on a particular
marking
set. This increases
sets for c =
the construction
To perform
/3 by using O(1)
random
1,2,4,8 . . . . . u and combine them. This would yield a family that was universal for all c. It would increase the number of sets in the family to ukO(l) log u but would increase the total size of the family by only a constant factor.
5.1
probability
of the use of pairwise
We can boost the probability
a safe set for all of ■ 1.
is less than
parameter
is all that
pendence is fairly standard [7, 20]. Such pairwise independence can be achieved using O(log u -tlog c) random bits as a seed to generate pairwise-independent variables for the marking trial. The O(log u) term comes from the need to generate u random variables; the O (log c) term comes from the fact that the denominator in the marking probability is c.
ability k–O(lJ the safe set is not marked but all the target sets are. Thus if we perform kO(lJ trials, we can reduce the probability of not producing an isolator for this instance to 1/2. If we do this ukO(l) times, then we now consider
in fact pairwise
of elements
the desired constant
elements
a, good set for z, probability that
Call this event good for
503
by construction the following Theorem
B has size (1 – ~~, allows
can
A (u, k, c) universal be generated in NC.
family
for U of size
Proofi Use O(log u + log c + log k) random bits in the expander walk to generate @(log k) pseudo-random seeds. Then
use each seed as an input
ber of processors
set
polylogarithmic
generator .f. Let H denote the @(log k) sets generated throughout these trials (we give @(log k) inputs to .f,
can be found
each of which generates that
~ generates
to the mark
O (1 ) sets). Since the probability
a good-for-i
set on a random
input
are
n o(l) targetsets. Thus in NC
we can generate and try all members of a universal isolating family of m ‘(l) sets. One of the sets we try will be an isolator for our problem, hitting all small cuts except for the minimum cut. When we contract the edges in this set, running the approximation algorithm on the contracted graph will find the minimum cut. The num-
theorem. 5.3
(ukc)”t’j
of 2 + e), and there
us to prove
6.1
is
used is me(l),
and the running
time
in m. In other words, the minimum in
cut
Nc.
Extension
to
Multiway
Cuts
~, we can choose constants and apply Theorem 5.2 to ensure that with probability 1 – 1/k2, one of our pseudorandom seeds is such that H contains a good set for Ti. It follows that with probability 1 – l/k, H contains good sets for every one of the Ti. Note that the good sets for
The r-way rein-cut problem is to partition a graph’s vertices into r nonempty groups so as to minimize the number of edges crossing between groups. An I?.JVC algorithm for constant r appears in [17], and a more
different
sets technique
targets
the collection
might
be different.
However,
C of all possible unions of sets in H.
H has O(log k) sets, C’ has size 21HI = kO(l). C’ consists
of the union
efficient
consider
to solve the r-way
r, we can use the safe cut problem
in NC.
Since Lemma
One set in
of all the good-for-some-i
one in [18]. For constant
6.1
value within
sets
([17])
The
number
a of the r-way
of r-way
rein-cut
cuts
is 0(n2a(”–1)
with ).
in H; this set hits every Ti but does not hit the safe set, Lemma 6.2 In an r-way rein-cut (Xl,. ... X,) of value c, each Xi has minimum cut at least 2c/(r – 1) (r + 2).
and is thus an isolator for our instance. We have shown that with O(log u + log c + log k) random bits, we generate a family of kO(l) sets such that there is a nonzero probability that one of the sets isolates the safe sets instance. It follows that if we try all possible O(log u + log c + log k) bit seeds, one of them must
yield
a collection
these seeds together which
contains
an isolator.
(uck)O(l)
Suppose
seed, the pairwise
All
two sets X,
value, a contradiction.
gen-
Given a par-
It follows
cut of smaller
the total
number
and merging A with X3 would produce a smaller r-way cut, a contradiction. It follows that the number of edges incident on Xl can be at most 2(r - 1). Combining the previous two arguments, the r-way cut value c must satisfy c
NC
through
Algorithm
David
Approximation:
for
Minimum
R. Karger*
Rajeev
Department
of Computer
Stanford Stanford, {karger,motwani}
Cuts
Motwanit
Science
University CA
94305
@!cs. stanf
Abstract
ord.
edu
Introduction
1,
Some of the central We show that
the minimum
lems in weighted
undirected
JVC. We do so by giving dently interesting results. sor JVC algorithm
cut
and multi-cut
graphs
algorithms minimum
prob-
can be solved
in
ings, and depth-first
three separate and indepenThe first is an m2/n proces-
for a (2 + c)-approximation
rithms
tion involves a natural combinatorial Safe Sets Problem that can be solved easily in %?JVC. Our third result is of this 7?JVC solution that requires of two widely used tools: pairwise
derandomization
any constant
k and for finding
step towards
resolving
ing the first
MC algorithm
imum
multi-cuts
The
rein-cut
for the rein-cut
by presentproblem
in
cuts.
problem
value within
a sparse k-connectivity bounded values of k.
minimum
for polylogarithmic
these open problems
minimal
we wish to minimize
was only known
There are 77AfC algo-
[19, 23, 1]. The problem
graphs. Our results extend to minand to the problem of enumerating all
any constant factor of the minimum. An additional byproduct of our techniques is an JVC algorithm for finding certificate Previously,
for s-t match-
is defined
as follows:
given
a
Multigraph with n vertices and m (possibly weighted) edges, we wish to partition the vertices into two nonthe number of empty sets S and T so as to minimize edges crossing from S to T (if the graph is weighted,
problems: we k-way cuts for
all cuts with
undirected
approximately
problems.
Our techniques extend to two related give JVC algorithms for finding minimum
search trees.
for all these problems
weighted
independence and random walks on expanders. We believe that the safe sets approach will prove useful in other
in the area of parallel
cuts belongs to this category of finding global minimum of unsolved derandomization problems, and it is representative in that obtaining an AfC algorithm for the case of directed graphs would resolve the other derandomization questions [17]. We take a (possibly small)
to the
minimum cut. The second is a randomized reduction of the minimum cut problem to the problem of obtaining a (2+ e)-approximation to the minimum cut. This reduc-
a derandomization a novel combination
open problems
are those of devising JVC algorithms cuts and maximum flows, maximum
We distinguish
for all polynomially an JVC construction
lem
values of k.
we
the total
the minimum
cut problem. require
that
weight
of crossing edges).
cut problem
from
In the s-t minimum
two specified
vertices
the s-t
cut probs and t be
cut probon opposite sides of the cut; in the minimum Our work deals only lem there is no such restriction.
with minimum cuts, and we assume that the graph is connected, since otherwise the problem is trivial. The value of a minimum cut in an unweighed graph is also called the graph’s edge connectivity.
*Supported by a National Science Foundation Graduate Fellowship, by NSF Grant CCR-901O517, and grants from Mitsubishi and OTL. tSupported by NSF Grant CCR-9010517, NSF Young Investigator Award CCR-9357849, with matching funds from IBM, Schlumberger Foundation, Shell Foundation and Xerox Corporation.
The
rein-cut
problem
many
fields.
tivity
of a network
network
has numerous
The problem arises
frequently
design and network
Queyranne [26] survey many minimum cuts. In information
Permission to co y without fee all or part of this material is granted provide J’ that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Association of Computing Machinery. To copy otherwise, or to republish, requires a fee and/or specific permission.
applications
of determining reliability
in
the connec-
in the
study
[9]. Picard
of and
of weighted applications retrieval,, minimum cuts
have been used to identify clusters of topically related documents in hypertext systems [5]. Padberg and Rlnaldi [25] discovered that the solution c~fminimum cut problems was the computational bottleneck in cutting-
STOC 94- 5/94 Montreal, Quebec, Canada Q 1994 ACM 0-89791 -663-8/94/0005..$3.50
497
plane based algorithms lem and many
other
for the traveling combinatorial
salesman
problems
the minimum
prob-
minimum
whose so-
cut in the contracted
cut in the original
graph is equal to the
graph.
lutions induce connected graphs. Applegate [3] also observed that a faster algorithm for finding all minimum cuts might accelerate the solution of traveling salesman problems. The first known minimum cut algorithm used the du-
tracting non-rein-cut edges until the graph has been reduced to two vertices. These two vertices define a cut in the original graph. If no rein-cut edge is contracted,
alit y between s-t rein-cuts and max-flows [10, 11]. An s-t max-flow algorithm can be used to find an s-t mini-
then the corresponding cut must be a minimum cut. The edges connecting the two vertices correspond to the
mum
cut edges.
cut,
sible
and
choices
Until
by taking
of s and
recently,
the
minimum
over
t,a minimum
the best sequential
all
(~)
Several contraction-based minimum cut algorithms have recently been developed. They all work by con-
pos-
cut may be found. algorithms
Nagamochi
for find-
to develop
and Ibaraki
an O(mn
[24] used graph
+ nz log2 n)-time
contraction
algorithm
for the
ing minimum cuts used this approach [14]. Parallel solutions to the rein-cut problem have also been studied.
rein-cut
Goldschlager, Shaw and Staples [13] showed that the s-t rein-cut problem on weighted directed graphs is Pcomplete. A simple reduction [17] shows that the (unrestricted) rein-cut problem is also P-complete in such graphs.
rein-cut edges but excludes some edge of the graph. This edge can then be contracted without affecting the minimum cut. Matula [22] used the Nagamochi-Ibaraki cer-
For
unweighted
graphs,
the
I?AfC
matching
algo-
randomizing which that
maximum
bipartite
has long been open. the global
rein-cut
is equivalent matching—a
A reduction
problem
for
Karger
[17] observed
contains
find
all the
that
a large number
of
simultaneously. a randomly
to be in the minimum
selected graph cut; it followed of graph This led
cost of the Contraction Algorithm, as well as its sequential running time, to d(n2); this is presently the most
problem
efficient Luby,
is
known
rein-cut
Naor
and Naor
algorithm
for weighted
[21] observed
that
graphs.
in the Con-
traction Algorithm it is not necessary to choose edges randomly one at a time. Instead, given that the mincut size is c, they randomly
Contraction
they
to the Contraction Algorithm, the first I?JVC algorithm for the weighted rein-cut problem, which used mn2 processors. Karger and Stein [18] improved the processor
also equivalent.
1.1
time
that repeated random selection and contraction edges could be used to find a minimum cut.
to de-
graphs
which
to identify
can be contracted
edge is unlikely
in [17] shows
directed
+ n log n)
certificate
use the sparse certificate edges which
the rnin-cut problem in I?AfC. The processor bounds are quite large, and the technique does not extend to graphs with large edge weights. No deterministic parallel algorithm is known. Indeed, derandomizing max-flow graphs
O(rn
tificate algorithm in a linear time (2+ e)-approximation algorithm for finding minimum cuts—the change is to
to yield 7i!hfC algorithms for s-t minimum cuts. By performing n of these computations in parallel, we can solve
undirected
In
a sparse connectivity
rithms of Karp, Upfal and Wigderson [19] or Mulmuley, Vazirani and Vazirani [23] can be combined with a well-known reduction of s-t max-flows to matching [19]
on unweighted,
problem.
Based
Algorithms
mark
each edge with
prob-
on contracting graph edges. Given a graph G and an edge {u, v}, contracting {u, v} means replacing u and v
ability 1/c, and contract all the marked edges. With constant probability, no rein-cut edge is marked but the number of graph vertices is reduced by a constant factor. Thus after O(log n) phases of contraction the graph is reduced to two vertices which define a cut. Since
with
the number
of phases is O (log n) and there
stant
of missing the minimum probability that is an n ‘o(l)
Recently, a new paradigm has emerged for finding minimum cuts in undirected graphs. This approach is based
a new vertex
w and transforming
each edge {x, u}
or {s, v} into a new edge {z, w}. Any {u, v} edge turns into a self loop on w and can be discarded.
phase,
there
is a con-
cut in each no rein-cut
edge is ever contracted; if this happens then the cut determined at the end is the minimum cut. Observing that pairwise-independent marking can be used to achieve the desired behavior, they show that O (log n) random bits suffice to run a phase. Thus, O (log2 ~) bits suffice to run this modified Contraction Algorithm through its O(log n) phases.
A key fact is that contracting vertices cannot decrease the minimum cut. The reason is that any cut in the contracted graph corresponds to a cut of exactly the same size in the original graph—if u and v were con(A, B) in the contracted to w, then a vertex partition w E A corresponds to a partition tracted graph with (AU {u, w} – {w}, B) in the original graph which the same edges. Let us fix a particular minimum
probability
cuts cut,
which from now on we will refer to as the minimum cut (there may be as many as (~)). The power of con-
Unfortunately, this algorithm cannot be fully derandomized. It is indeed possible to try all possible random seeds for a phase and be sure that one of the polynomi-
tractions contract
a~ly many outcomes is good; however, there is no way to which outcome is good. In the next phase is determine
comes from their interaction with cuts. If we an edge which is not in the minimum cut, then
498
thus necessary
to try all possible
of the polynomially
many
random
outcomes
Overview
of
Naor and Naor technique.
In
such an outcome
Results
to contain Our main result is an NC algorithm for the rein-cut and minimum multi-cut problems. Our algorithm is not a derandomization
of the Contraction
stead a new contraction-based we take G to be a Multigraph
Algorithm
If we mark
each edge
with probability @(l/c), then with constant probability we mark no rein-cut edge while marking edges in a constant fraction of the other small cuts. Pairwise independence in the marking of edges is sufficient to make
phase,
squaring the number of outcomes after two phases, all, Cl(nlOg’) combinations of seeds must be tried.
1.2
Luby,
seeds on each
of the first
likely.
However,
this approach
the same flaw as before:
fl(log
seems
n) phases of
selection are needed to mark edges in all the small cuts, and thus fl(log2 n) random bits are needed.
but is in-
This
algorithm. Throughout, with n vertices, m edges
leads to our third
The problem
of finding
can be formulated
building
blc)ck (Section
5).
a good set of edges to contract
abstractly
as the Saj’e Sets Problem:
and rein-cut value c. Most of the paper discusses unweighed graphs; in Section 6 we reduce from weighted graphs to the unweighed graph problem.
given an unknown collection of sets over a known universe, with one of the unknown sets declared %afe,” find a collection of elements which intersects every set
Our algorithm blocks. The first
depends building
except for the safe one. After giving a simple randomized solution, we show that this problem can by solved
an NC
that
algorithm
(2+ ~)-approximation Matula’s sequential
upon block
three major building (Sections 2 and 3) is
uses m2/n
processors
to find
a
in NC
by combining
the techniques
of pairwise
inde-
to the minimum cut. Recall that algorithm [22] was based on the se-
pendence [7, 20] with the technique of random walks on expanders [2]. This is the first time th~ese two impor-
quential sparse certificate algorithm of Nagamochi and Ibaraki [24]. It repeatedly finds a sparse certificate containing all rein-cut edges and then contracts the edges not in the certificate, terminating after a small num-
tant techniques have been combined in a derandomization, although similar ideas have been used earlier to save random bits in the work of Bellare, Goldreich and Goldwasser [4]. We feel that the combination should have further application in derandomizing other algo-
ber of phases.
Our NC
algorithm
uses a new parallel
sparse certificate algorithm to parallelize Matula’s rithm. A parallel sparse k-connectivity certificate rithm
with
running
Kao, and Thurimella an algorithm
that
time
6(k)
algoalgo-
rithms. It should
was given by Cheriyan,
[6]; we improve
this by presenting
runs in NC for all k = no(l).
Our next building block (Section 4) uses a result obtained from the analysis of the Contraction Algorithm. Karger [17] proved that there are only polynomially many cuts whose size is within a constant factor of the If we find a collection of edges which minimum cut. contains minimum
one edge from cut, then
every
contracting
such cut except
relatively
As was the
easy to solve if the solution
goal is to destroy
all but one solution
the problem is unique,
and Naor
hits every vertex
to finding
technique,
An
2 In any (2+
is
this
t) times
Definition
and
in G.
then to easily find the unique solution. Randomization yields a simple solution to this problem: contract each Because the number edge with probability y Cl(log n/c).
building
with the Luby,
it can find
a set of edges cut.
Un-
such an edge set need only halve the number (e.g., if the edge set is a perfect matching),
in
Section
minimum
still
be necessary—the
6 we apply cuts,
the
minimum
that
2.1
A maximal
of the
above
and to enumerating
we describe c > 0, finds
an NC
and
approx-
algorithm
whose
minimum
k-jungle
results
Algorithm
a cut
A k-jungle
same
nnulti-cuts,
Approximation
section,
constant
this third
directly
but not the minimum
weighted minimum cuts; imately minimum cuts.
so the
to the problem
Naor,
Finally,
for the
this set of edges yields
case there,
by itself,
Combined
so fl(log n) phases would flaw as before.
just mentioned. Since the minimum cut will be the only contracted graph cut within the approximation bounds, it will be found by the approximation algorithm. One can view this approach as a variant on the Isolating Lemma approach used to solve the perfect matching [23].
that
which
fortunately, of vertices
a graph with no small cut except for the minimum cut. We can then apply the NC approximation algorithm
problem
be noted
block is not sufficient.
value
that,
for
is less than
cut.
is a set of k disjoint is a k-jungle
forests
such that
no
other edge in G can be added to any one of the jungle’s forests without creating a cycle in that forest.
of small cuts is polynomially bounded, there is a sufficient probability that no edge from the minimum cut is contracted but one edge from every other small cut is contracted. Of course, our goal is to do away with randomization. A step towards this approach is a modification of the
Lemma 2.1 ([24]) A maximal k-jungle the edges in any cut of k or fewer edges.
contains
all
Proofi Consider a maximal k-jungle .7, and suppose it cent ains fewer than k edges of some cut. Some forest in J must
499
have no edge from
this
cut.
Any
cut edge
not in J could be added to this forest a cycle, so all cut edges must already
Procedure
Approx-Min-Cut(
Figure
degree of G.
maximal
[22].
G by contracting
Approximation
3
)).
Algorithm
We give it as an algorithm
all minimum
cut edges and then
contract
all
edges not in the jungle. The algorithm works quickly because so long as we do not have a good approximation to the minimum cut at hand, we can guarantee that many edges are contracted each time. Lemma
2.2
Given
approximation (2+ E)c.
It remains
k-jungle
only to show how to construct
a
in NC.
a graph
algorithm
with
returns
minimum
Finding
cut c, the
a value between c and
Maximal
Jungles
The notation needed to describe this construction is somewhat complex, so first we give some intuition. To construct a maximal jungle, we begin with an empty jungle and repeatedly augment it by adding additional edges from possible.
to approximate
the cut value; it is easily modified to find a cut with the returned value. The basic idea is to find a jungle that contains
above 2 is needed to en-
edges.
The approximation algorithm is described in Figure 1. It is a parallel version of Matula’s approximation algorithm
e factor
time of this algorithm is thus O(T(m, n) . polylog (m)) where T(m, n) is the time needed to construct a maximal jungle.
min(b, Approx-Min-Cut(G’
1: The
the extra
sure a significant reduction in the number of edges at each stage and thus keep the recursion depth small. The depth of recursion is in fact (3(6-1 log m). Each step of this algorithm, except for Step 3, can be implemented in AfC using m processors. The running
G)
k-jungle.
G’ from
all non-jungle
5. Return
■
c).
3. Find a maximal 4. Construct
Note that
creating
be in J.
Multigraph
1. Let 6 be the minimum
2. Let k = 6/(2+
without
the graph
Consider
non-jungle
edges that
out creating
until
no further
one of the forests
augmentation in the jungle.
may be added to that
is The
forest with-
a cycle are just the edges that cross between
two different
trees of that
forest.
We let each tree claim
some such edge incident upon it. Hopefully, each forest will claim and receive a large number of edges, thus significantly increasing the number of edges in the jungle. Two problems arise. The first is that several trees may claim a particular edge. However, the arbitration of these claims can be transformed into a maximal matching problem
and solved in JVC. Another
problem
is that
since each tree is claiming an edge, a cycle might be formed when the claimed edges are added to the forest (for example, two trees may each claim an edge con-
Proofi Clearly the value is at least c because it corresponds to some cut the algorithm encounters. For the upper bound, we use induction on the size of G. We consider two cases. If 6 < (2 + c) c, then since we return a value of at most 6, the algorithm is correct. On the other hand, if 6 > (2 + C)C, then k > c. It follows
of a k-jungle J = Definition 3.1 An augmentation A = {El,..., E~} of k disF~} is a collection {Fl,...,
from
joint
Lemma
2.1 that
all the rein-cut cut is contracted cut c. returns
the jungle
edges. while
By the inductive a value between
we construct
contains
those two trees).
sets sets
2.3
approximation
The
number
algorithm
of levels of recursion
added
■
in the
is O (log m).
Proof: If G has minimum degree cl, it must have at least 6n/2 edges. On the other hand, the graph G’ which we construct contains only jungle edges; since each forest of the jungle contains only n – 1 edges, G’ can have at most k(n – 1) = J(n – 1)/(2 + e) edges. It follows
that
each recursive
remedy
this problem
of non-jungle Ei
to forest
edges from
be non-empty.
must
G. The
At
least
edges
one
of E,
of are
F%.
call Definition does not
Lemma
We will
as well.
the
Thus no edge in the minimum forming G’, so G’ has minimum hypothesis, the recursive c and (2 + E)C.
necting
step reduces the number
of edges in the graph by a constant factor; thus at a recursion depth of O (log m) the problem can be solved trivially. ■
Fact 3.1 mentation. Given
3.2
A valid
augmentation
of J is one that
create any cycles in any of the forests A jungle
a jungle,
is maximal
it is convenient
of J.
iff it has no valid
aug-
to view it in the fol-
lowing fashion. We construct a reduced (Multigraph GF for each forest F. For each tree T in F, the reduced graph contains a reduced vertex VT. For each edge e in G that connects trees T and U, we add an edge eF connecting VT and vu. Since many edges can connect two forests, the reduced graph may have parallel edges. An edge e of G may induce many different edges, one in each forest’s reduced graph.
500
Given any augmentation, F can be mapped to their inducing
the edges added to forest corresponding edges in GF,
an augmentation
assigned to F. The edges of A may induce cycles in GF, which would mean (Fact 3.2) that A does not correspond to a valid augmentation of F. However, if we
subgraph of the reduced graph
GF.
find an acyclic subset of A then the G-edlges correspond-
Fact
3.2
An augmentation
subgraph
it induces
ing to this subset will form
is valid iff the augmentation
in each forest’s
reduced graph
is a
the reduced graph GF. Direct the reduced vertex to which
forest.
vertex
Care should be taken not to confuse the forest F with the forest that
is the augmentation
subgraph
has outdegree
of GF.
in the reduced Multigraph.
of edges which can be added to J by a constant
no cycle obeying
fraction
must contain
each time, and since the maximum jungle size is m, J will have to be maximal after O (log m) phases. mal matching vertex
problem
to a smaller
on a bipartite
graph
H.
tex with
Let one
VT
of GF to Ve if eF is incident we are connecting
edges incident
upon
on VT in GF.
in it in G.
Note
edge in GF is a valid augmenting the size of H, note that 2k incident
Lemma matching
each vertex
Proofi
Consider
a valid
edges of the augmentation follows. For each forest Multigraph augmenting duced
vertex
Thus
the total
of
forest
matching
J
number
G can
numbered hand,
must contain
two, an impossibility.
any
a ver-
It follows
that
gain
at least
half
the edges assigned
so the augmentation
to it
has the desired
induces
a
Each non-root
augmentation
If edge e c G is matched
to reduced
in NC
using
O(km)
k-jungle
of
processors.
the
re-
edge eF
vertex
GF, tentatively assign e to forest F. Consider the set A of edges in GF that correspond to the G-edges
an augmentation.
Let a lbe the size of a
maximum augmentation of J. Lemma 3,,3 shows that H must have a matching of size a. It follows that any max-
of the jungle.
in H between
Given G and k, a maximal
Proof: We begin with an empty jungle and repeatedly augment it. Given the current jungle J, construct the bipartite graph H as was previously described and
imal matching in H must have size at least a/2, since at least one endpoint of each edge in any maximum match-
and the reduced vertices as F in J, consider its reduced
arbitrarily.
3.5
be found
of
in H, a valid augmenLemma 3.4 Given a matching tation of J of size at least half the size of the matching can be constructed in hfC.
Proof:
a larger
On the other
■
T~eorem
on either
leading to its parent. Since edge e is added to F no other forest F’ will use edge eFl, so we can match vT to V.. It follows that every augmentation edge is matched ■ to a unique reduced vertex.
VT E
will
use it to find
augmentation
UT has a unique
from
size.
v. will have at most be incident
GF. Since the augmentation is valid, the edges in GF form a forest (Fact 3.2). Root
each tree in this
vertex.
the edge directions
outdegree
in the matching,
to the
this means each
3.3 A valid augmentation in H of the same size.
We set up a corresponding
an edge directed
edges can form
since such a cycle
edge for F. To bound
edges, because it will
O or 2 trees of each forest. edges in H is O(km).
forest
Equiva-
each tree in the jungle
Its (directed)
the edges of A. form a valid augmentation of F of at least half the size of the matching. If we do this for each forest F in parallel, we get a valid augmentation of the jungle. Furthermore, each
vT in the various
reduced multigraphs, i.e., the trees in the jungle. Let the other vertex set consist of one vertex v. for each non-jungle-edge e in G. Connect each reduced vertex lently,
in
the edges into two
the edge directions,
numbered
cycle disobeying
we solve a maxi-
set of H consist of the vertices
of F.
the vertices
each edge in A away from it was miitched (so each
one), and split
constant fraction of the largest possible valid augmentation. Since we reduce the maximum possible number
augmentation,
number
groups: A. G A are the edges directed from a smaller numbered to a larger numbered vertex, and Al ~ A are the edges directed from a larger numbered to a smaller numbered vertex. One of these sets, say A., contains at least half the edges of A. However, A. creates no cycles
Our construction proceeds in a series of O(log m) phases in which we add edges to the jungle J. In each phase we find a valid augmentation of J whose size is a
To find a large valid
a valid augrnentation
To find this subset, arbitrarily
ing must be matched
in any maximal
matching.
Several
NC algorithms for maximal matching exist—for example, that of Israeli and Shiloach [16]. Lemma 3.4 shows that
after we find a maximal
transform
this
at least a/4.
matching
matching,
into
Since we find
we can (in Nc)
an augmentation
an augmentation
of size of size at
least one fourth the maximum each time, and since the maximum jungle size is m, the number of augmentations
needed to make a J maximal
is O(log m).
Since
each augmentation is found in NC, the maximal jungle can be found in NC. The processor cost of this algorithm is dominated by that of finding the matching in the graph H. The algorithm of Israeli and Shiloach requires a linear number of processors, and is being run on a graph of size O(km). ■
501
Corollary
3.6
be found
in NC
Proofi
A (2+t)
-approximate
using 0(m2 /n)
A graph
with
minimum
It follows
cut can
that
running
the approximation
algorithm
of Section 2 on the contracted graph will find the minimum cut, since the actual minimum cut is the only one
processors.
m edges has a vertex
with
which is small enought to meet the approximation criterion. Our goal is thus to find a collection of edges that hits every target cut but not the minimum cut. This problem can be phrased more abstractly as follows:
de-
cut can therefore be no gree O(m/n); the minimum larger. It follows that our approximation algorithm will ■ construct k-jungles with k = O(m/n).
Over some universe U, an adversary selects a polynomially sized collection of “target” sets of roughly equal 4
Reducing
to
size (the small cuts’ edge sets), together with a disjoint “safe” set of about the same size (the rein-cut edges).
Approximation
We want
In this section, we show how the problem of finding a minimum cut in a graph can be reduced to that of finding a (2 + e)-approximation.
Our technique
is to “kill”
the vertices of G into two sets A and B. graphs induced by A and B.
this idea, partitions
Consider
4.1
The minimum
5
The
Safe
We describe
Proofi Suppose A has a cut into X less than c/2. Only c edges go from X one of X or Y (say X) must have at leading to B. Since X also has less to Y, the cut
that
intersect we do we do
Sets
Problem
a general
form
U of size
of the Safe Sets Problem.
u.
cuts in A and in B have Definition 5.1 A (u, k, c) safe set instance is a collection of at most k target sets of size Q(c), together with a safe set of size O(c) that is disjoint from the target
value at least c/2.
leading
of elements
the Fix a universe
Lemma
a collection
have an upper bound on the number of target sets. We proceed to formalize this problem as the Safe Sets Problem.
all cuts of size less than (2 + ~)c other than the minimum cut itself. The minimum cut is then the only cut of size less than (2+ e) c, and thus must be the output of the approximation algorithm. To implement we focus on a particular minimum cut which
to find
every target set but not the safe set. Note that not know what the target or safe sets are, but
(X, ~)
and Y of value and Y to B, so most c/2 edges than c/2 edges
has value
less than
contradiction.
sets. An isolator for the safe set instance is a set that intersects all the target sets but not the safe set.
c, a ■
Definition
5.2
a collection
of subsets of U that contains
A (u, k, c)-universal
isolating
family
an isolator
is for
any (u, k, c) safe set instance. Theorem
4.2
([17] )
at most Q times
There are O(n2a)
To see that this general formulation captures our cut isolation problem, note that the minimum cut is the safe set in an (m, k, c) safe set instance. The universe is the
cuts of weight
the minimum.
set of edges, of size m; the target
It follows that in each of A and B, every cut has value at least c/2 and there are n ‘(1 J cuts of weight less than (2+ C)C. Call these cuts the target cuts. Lemma
4.3
Let Y be a set containing
edges from
every
target cut but not the minimum
cut. If every edge in Y is
contracted,
graph has a unique
weight
then the contracted
less than
(2 + ~)c,
namely
the original
sets are the small cuts
of the two sides of the minimum cut; k is the number of such small cuts and (by Lemmas 4.2 and 4.3) can be bounded by polynomial in n < m; and c is the cut size. The approximation
algorithm
of Section
2 allows
us to
family
then
estimate c to within a constant factor. If we had an (m, k, c)-universal isolating
cut of
one of the sets in it would be an isolator for the safe sets instance corresponding to the minimum cut. By Lemma 4.3, contracting all the edges in this set would isolate the minimum cut as the only small cut. If the size of the universal family was polynomial in m, k, and c, we could try each set in the universal family in parallel in NC, and be sure that one such set isolates
minimum
cut.
Proofi Clearly contracting the edges of Y does not affect the minimum cut. Now suppose this contracted graph had some other cut C of value less than (2+.E)c. It corresponds to some cut of the same value in the original graph. Since it is not the minimum cut, it must induce
the minimum
a cut in either A or have value less than target cut, so one of But this prevents C graph, a contradiction.
can find it. Thus our goal is: given U, a constant factor approximation to c, and an upper bound on k, generate a (u, k, c)-universal isolating family of size polynomial in u, k, and c in NC. We first give an existence proof for universal families of the desired size.
B, and this induced cut must also (2+ C)C. This induced cut is then a its edges will have been contracted. from being a cut in the contracted ■
502
cut so that
the approximation
algorithm
Theorem
5.1
ing family
of size at most uk”~~~.
exists a (u, k, c) -universal
We use a standard
Proofi argument. stance.
There
Fix Suppose
with
probability
form
one member
probabilistic
attention
on a particular
we mark
each element
z and call the set of marked elements For each trial, we have some constant the trial is good for a particular i.
existence
We now observe that
safe set in-
the marking
of the universe
(log k) /c and let the marked of the universal
isolat-
family,
With
analysis
prob-
the probability
of failure
ability
that
them
during
Note vant,
on the instance
we fail
to to generate
all the trials
that
is 2–U~O(’).
all 2U~0(1) safe set instances,
the
since we could
If
the prob-
c is
construct
somewhat
universal
constant
Constructing
Universal
We proceed to derandomize sal isolating
family.
irrele-
we fix our attention
value.
Suppose with bility
we independently
probability
l/c,
obtaining
mark
elements is (1 – l/c)Z. inequalities:
under
We use the following
any marked
1 – t.
a result
due to Ajtai,
~
(~-e-,c)
of T~ is marked.
walks on of a fam-
and a convenient
and Galil
[12].
con-
They
show
The following
5.2
is a minor
Kom16s
presents
adaptation
and Szemer6di
a crucial
of
[2] (see
property
of ran-
G..
([2] ) Let B be a subset
OJF
V(G.)
of size
~ log k on Gn,
all from
the probability B is 0(k–2).
that the vertices
visited are
Notice that performing a random walk of length ~ log k on G. requires O(log n + log k) random bits— choosing a random starting vertex requires log n ranbits
and,
since the degree is constant,
each step
of the walk requires O(1) random bits. We use this random walk result as follows. Each vertex of the ex(e-l
(~-~)
pander corresponds to a seed for the mark set generator f described above; thus, log n = O(log u -!- log c), implying that we need a total of O(log u + log c + log k) random bits for the random walk. Choosing B to be the set of bad seeds for i, i.e. those that generate set families containing no good sets for i, and noting that
)s’c.
Since t~/c = $2(1) and s/c = O(l), there is a constant probability of &,, i.e. that no element of S is marked but some element
of random
at most (1 — ~)n, for some constant ~. Then there exists a constant ~ such that for a random walk of length
dom (l-(l-;)’’)(l-;)s
of Gabber
at most
Theorem
standard
Let &i be the event that T% does contain some and S does not contain any marked elements. Then
=
mark
subsets of that f(s)
construction
degree expanders,
is that
dom walks on the expander
(e-w)”c~ (’-:)’<e-”c Pr[S,]
of the
a, different
the size of the seed needed by only
relies on the behavior
also [15, 8]) which
of U
a mark set. The proba-
that a subset of size x does not contain
iterations
process, each yielding
that for sufficiently large n, there exists a graph Gn on n vertices with the following properties: the graph is 7-regular; it has a constant expansion factor; and, for some constant c, the second eigenvalue of the graph is
sets. Let
each element
inde-
of success to any desired
independent
We need an explicit
ily of bounded struction
safe set instance,
instance
This
expanders.
of a univer-
of target
of complete
The next step is to reduce the probability of failure from a constant 1 – ,6 to an inverse polynomially small
and show that our construction will contain an isolator for that instance. It will follow that our construction contains an isolator for every instance. Let S be the s = IS I and t;= [T.1,for the particular consideration.
in
of being good for i. The instead
seed of O(log u + log c) bits and returns 0(1) U. Randomizing over seeds s, the probability contains at least one good set for i is ~.
the derandomization,
safe set and let {T, } be the collection
independence
is needed to achieve
a constant factor. WJe can think of this pairwise independent marking algorithm as a function f that takes a
Families
on a particular
marking
set. This increases
sets for c =
the construction
To perform
/3 by using O(1)
random
1,2,4,8 . . . . . u and combine them. This would yield a family that was universal for all c. It would increase the number of sets in the family to ukO(l) log u but would increase the total size of the family by only a constant factor.
5.1
probability
of the use of pairwise
We can boost the probability
a safe set for all of ■ 1.
is less than
parameter
is all that
pendence is fairly standard [7, 20]. Such pairwise independence can be achieved using O(log u -tlog c) random bits as a seed to generate pairwise-independent variables for the marking trial. The O(log u) term comes from the need to generate u random variables; the O (log c) term comes from the fact that the denominator in the marking probability is c.
ability k–O(lJ the safe set is not marked but all the target sets are. Thus if we perform kO(lJ trials, we can reduce the probability of not producing an isolator for this instance to 1/2. If we do this ukO(l) times, then we now consider
in fact pairwise
of elements
the desired constant
elements
a, good set for z, probability that
Call this event good for
503
by construction the following Theorem
B has size (1 – ~~, allows
can
A (u, k, c) universal be generated in NC.
family
for U of size
Proofi Use O(log u + log c + log k) random bits in the expander walk to generate @(log k) pseudo-random seeds. Then
use each seed as an input
ber of processors
set
polylogarithmic
generator .f. Let H denote the @(log k) sets generated throughout these trials (we give @(log k) inputs to .f,
can be found
each of which generates that
~ generates
to the mark
O (1 ) sets). Since the probability
a good-for-i
set on a random
input
are
n o(l) targetsets. Thus in NC
we can generate and try all members of a universal isolating family of m ‘(l) sets. One of the sets we try will be an isolator for our problem, hitting all small cuts except for the minimum cut. When we contract the edges in this set, running the approximation algorithm on the contracted graph will find the minimum cut. The num-
theorem. 5.3
(ukc)”t’j
of 2 + e), and there
us to prove
6.1
is
used is me(l),
and the running
time
in m. In other words, the minimum in
cut
Nc.
Extension
to
Multiway
Cuts
~, we can choose constants and apply Theorem 5.2 to ensure that with probability 1 – 1/k2, one of our pseudorandom seeds is such that H contains a good set for Ti. It follows that with probability 1 – l/k, H contains good sets for every one of the Ti. Note that the good sets for
The r-way rein-cut problem is to partition a graph’s vertices into r nonempty groups so as to minimize the number of edges crossing between groups. An I?.JVC algorithm for constant r appears in [17], and a more
different
sets technique
targets
the collection
might
be different.
However,
C of all possible unions of sets in H.
H has O(log k) sets, C’ has size 21HI = kO(l). C’ consists
of the union
efficient
consider
to solve the r-way
r, we can use the safe cut problem
in NC.
Since Lemma
One set in
of all the good-for-some-i
one in [18]. For constant
6.1
value within
sets
([17])
The
number
a of the r-way
of r-way
rein-cut
cuts
is 0(n2a(”–1)
with ).
in H; this set hits every Ti but does not hit the safe set, Lemma 6.2 In an r-way rein-cut (Xl,. ... X,) of value c, each Xi has minimum cut at least 2c/(r – 1) (r + 2).
and is thus an isolator for our instance. We have shown that with O(log u + log c + log k) random bits, we generate a family of kO(l) sets such that there is a nonzero probability that one of the sets isolates the safe sets instance. It follows that if we try all possible O(log u + log c + log k) bit seeds, one of them must
yield
a collection
these seeds together which
contains
an isolator.
(uck)O(l)
Suppose
seed, the pairwise
All
two sets X,
value, a contradiction.
gen-
Given a par-
It follows
cut of smaller
the total
number
and merging A with X3 would produce a smaller r-way cut, a contradiction. It follows that the number of edges incident on Xl can be at most 2(r - 1). Combining the previous two arguments, the r-way cut value c must satisfy c
