878-Approximation Algorithms for MAX CUT and ... - Semantic Scholar
.878-Approximation
Algorithms
Michel
X.
for MAX
Goernans*
David
M.I.T.
always
solutions
of expected
.87856 times the optimal a simple
and elegant
and MAX
2SAT
Williamson University
Introduction Given an undirected graph G = (V, 1?) and nonnegative weights wij = wli on the edges (i, j) c E, the
randomized approximation algorithms CUT and MAX 2SAT problems that
deliver
P.
Cornell
Abstract
We present for the MAX
CUT
value.
value
at least
These algorithms
technique
that
maximum
randomly
cut problem
the set of vertices
(MAX
S that
edges in the cut (S, ~); that
use
with
rounds
one endpoint
CUT)
is that
maximizes
of finding
the weight
is, the weight
in S and the other
of the
of the edges
in $.
For sim-
the solution to a nonlinear programming relaxation. This relaxation can be interpreted both as a semidefi-
plicity, we usually set wij = O for (i, j) @ E and denote wij. the weight of a cut (S, ~) by w(S, ~) = Ei=s,j$s
nite program and as an eigenvalue minimization problem. We then show how to derandomize the algorithm to obtain approximation algorithms with the same performance guarantee of .87856. The previ-
The MAX CUT problem is one of the Karp’s original NP-complete problems [19], and has long been known to be NP-complete even if the problem is unweighed; that is, if w$j = 1 for all (i, j) E E [9]. The MAX
ous best-known approximation algorithms for these problems had performance guarantees of ~ for MAX
CUT problem is solvable in polynomial time for some special classes of graphs (e.g. if the graph is planar [29, 14]). Besides its theoretical importance, the MAX
CUT
and ~ for MAX
analysis
leads
2SAT.
A slight
extension
to a .79607-approximation
of our
CUT problem
algorithm
has applications
in circuit
a &
and statistical
physics
approximation algorithm was the previous best-known algorithm. Our algorithm gives the first substantial progress in approximating MAX CUT in nearly twenty years, and, to the best of our knowledge, represents the first use of semidefinite programming in the design of approximation algorithms.
comprehensive
survey of the MAX
for the maximum
directed
cut problem,
where
reader is referred
*Address: MA
supported contract 1799.
Dept. 02139. in part
14853.
Email: by
NSF
F49620-92-J-0125
tAddress: ~ineering,
of Mathematics,
School 237
Email:
ETC
Room
goemans@nath. contract and
13uildin~,
dpw@cs. cornell.
mit.
contract
Research
Cornell
edu.
an NSF Postdoctoral Fellowship. This while the author was visiting MIT.
Research research
rithm.
Ithaca,
delivers
In 1976, Sahni
lem. Their
For a
problem,
the
and Tuza [38].
a solution
and Gonzales
algorithm
algorithm
decides whether
En-
of value
at least
iterates
[40] presented
for the MAX through
CUT
the vertices
or not to assign vertex
a
proband
i to S based
on which placement maximizes the weight of the cut of vertices 1 to i. This algorithm is essentially equivalent to the randomized algorithm that flips an unbiased coin for each vertex to decide which vertices
NY
supported
that
&approximation
Force
NOOO14-92-J-
and Industrial
University,
CUT
p times the optimal value. The constant p is sometimes called the performance guarantee of the algo-
Research Air
design
approach to solving such a problem is to find a papproximation algorithm; that is, a polynomial-time
M. I. T., Camedu.
9302476-CCR,
DARPA
of Operations
2-372,
to Poljak
layout
et al. [3]).
Because it is unlikely that there exist efficient algorithms for NP-hard maximization problems, a typical
algorithm
bridge,
(Barahona
by
was conducted
Permission to copy without fee all or part of this material is granted provided 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.
are assigned to the set S. Since 1976, a number of researchers have presented approximation algorithms for the unweighed MAX CUT problem with performance; guarantees of ~ + & [42], $ + ~ [37], and [17] (where n = IVI and m = IEI), but no *+= progress was made in improving the constant in the
STOC 94- 5/94 Montreal, Quebec, Canada @ 1994 ACM 0-89791 -663-8/94/0005..$3.50
422
i
performance guarantee zales’s straightforward We
present
a
approximation
beyond that of Sahni and Gonalgorithm.
simple,
algorithm
randomized
(a
for the maximum
–
input, so the running time dependence on e is polynomial in log $). This can be done through the ellipsoid algorithm
e)-
(Grotschel
time algorithms
cut prob-
aa well as interior-point
lem where
mirovskii
~=
20 —
min
time,
rithm, for any e >0. The algorithm for MAX CUT also leads directly to randomized and deterministic (a – e)-approximation 2-satisfiability best known
algorithms
problem algorithm
guarantee
(MAX
for the
2SAT).
maximum
The previously
for this problem
haa a perfor-
of ~ and is due to Yannakakis
(see also Goemans
and Williamson
[10]).
[44]
Coppersmith
and Shmoys have independently observed that the improved 2SAT algorithm leads to a slightly better than ~-approximation algorithm for the overall MAX SAT problem. Finally, a slight extension of our analysis yields a .79607-approximation imum vious
directed
cut problem
best known
performance
algorithm
guarantee
algorithm (MAX
for MAX
The pre-
DICUT
has a
of ~ [32].
Our algorithm depends on a means of randomly rounding a solution to a nonlinear relaxation. This prorelaxation can either be seen as a semzdej%ate minimization problem. To gram or aa an eigenvalue our knowledge,
this is the first
time
that
solution;
al. [13] and Section
[41])
and Ne-
To terminate implicitly
in
assume
space or on the size
for details
3.3 of Alizadeh
see Grotschel
et
[1].
The importance of semidefinite programming is that it leads to tighter relaxations than the classical linear programming relaxations for mi>ny graph and combinatorial problems. A beautiful application of semidefinite programming is the work of Lov&z [22] on the Shannon with
the
capacity
of a graph.
In conjunction
solvability
of semidefinite
polynomial-time
programs, time
this
leads to the only
algorithm
the largest
for finding
clique)
known
the largest
in a perfect
graph
polynomialstable
set (or
(Grotschel
et al.
[12]). More recently, there has been increased interest in semidefinite programming [24, 25, 1, 33, 35, 8, 23]. This started with the work of Lov&z and Schrijver [24, 25], who developed a machinery tc) define tighter and tighter
for the max-
DICUT).
[1]).
on the feasible
of the optimum
(Vaidya
(Nesterov
these algorithms
some requirement
and e is any positive scalar. The algorithm represents the first substantial progress in approximating the MAX CUT problem in nearly twenty years. We also show how to derandomize the algorithm to obtain a deterministic (a – c)-approximation algo-
methods
[27, 28] and Alizadeh
polynomial
>0.87856,
n’ 1 – Cose
0
–e.
(P) is also equivalent to an eigenvalue upper bound on the value of the maximum cut ZfiC introduced by Delorme and Poljak [5, 4]. To describe the bound, we first introduce some notation. The Laplacian matrix L = (iij ) is defined by lij = –wiJ
for i # j and lii = ~~=1
wik.
Given a vec-
mum eigenvalue
A is denoted
of a matrix
with entries The maxiby Am..(A).
Tr(LY)
that
For the optimum
The relaxation
tor u, diag(u) denotes the diagonal matrix u~fori= l,..., n on the main diagonal.
Noticing
is precisely
objective function of (SD), showing that Z; s Z&IG. mum
matrix
gramming
Y,
correcting
strong
guarantees
and the
for semidefinite
the duality
the
weak duality,
vector
duality
that
four times
one derives
optipro-
gap is equal to
= O, where M is defined as O and, thus, Tr(MY) above. Since both M and Y are positive semidefinite, we derive that MY = O (see [21, p. 218, ex. 14]). From the definition of M and the fact that MY = O, implying that we derive that ~Y – LY = dzag(u)Y, for all i. Since ~ = ~Z~lG = ~Z~, u%= ~ — ~j lijytj we observe that we can easily deduce the optimum
Lemma
1.4 [[5]] Let u E Rn satisfy UI +...
+ u~ = O.
recting vector from the optimum
corY for (SD).
matrix
Then (L+
9(U) = ;Am.z
1.4
dzag(u))
Theorem
is an upper bound on Z~c. The
proof
is simple.
to the integer
yTLy
=
(~~~.(~)
&.z(L
Let
quadratic
4Z~C.
By
g be an optimal program
using
= maxll~ll=l
~ — —
= proving
(Q).
Notice
Rayleigh
Corollary
solution
‘T(L
zfi~ z;
For the 5-cycle, that Z~c/.Z~IG n
1 — n
yTLy ( 4z&~
+ ~ y;UZ 2=1 )
pressed
n’
~~=1 ui = O
The bound
proposed
Delorme
and Poljak
[5] is to optimize
g(u)
by
over all
that
~ a >0.87856.
Delorme =
25~;& —
and Poljak =
[5] have shown
0.88445 ...,
implying
as vi
=
(cos(&),
sin(%))
for 2 =
1,...,5
Z& C/Z$IG
> -A
holds for special subclasses
of graphs, such as planar graphs or line graphs. However, they were unable to prove a bound better than
g(u)
0.5 in the absolute worst-case. Although the worst-case value of Z&C/Z~ is not completely settled, we have constructed instances for
n
to:
CUT,
corresponding to Z; = ~ (1 + cos ~) = w. Since ZfiC = 4 for the 5-cycle, this yields the bound of Delorme and Poljak. Delorme and Poljak have shown
vectors:
subject
corollary:
that our worst-case analysis is almost tight. One can obtain this bound from the relaxation (F’) by observing that for the 5-cycle 1 – 2 – 3 – 4 – 5 – 1, the optimal vectors lie in a 2-dimensional subspace and can be ex-
u satisfying
(EIG)
the following
For any instance of MAX —
‘V~’(U))y
vector.
Inf
1.5
Relaxation
we obtain
A vector
=
1.1 implies
that
a correcting
-%IG
of the
principle
is called correcting
the lemma.
the
ZTMZ),
+ diag(u))
Quality
~ ui = O. $=1
426
which Zf/E[ W] < 0.8786, showing of our algorithm is practically tight.
that
Poljak and Rendl [34, 36] (see also Delorme Poljak [6]) report computational results showing the bound
Z*EIG is typically less than worse than 870 away from Z&c. We have implemented using a code supplied
experiments
have shown that cuts within
see the full paper
randomized
programs.
MAX
2SAT
We consider
and that
the integer
Maximize (Q’)
[39] for a spe-
subject
to:
vi {: v,
y~ G {–1,1}
where aij and bij are non-negative. function of (Q’) is thus a nonnegative
Very preliminary
the algorithm
program
Z<j
algorithm
typically
4-5% of the semidefinite
quadratic
[aii(l - YzYj) + b~j(l+ Y~Y~)l
~
2-570 and never
by R. Vanderbei
cial class of semidefinite generates
the
2.1
the analysis
1 + y,yj.
We relax
(Q’)
The objective linear form in
to:
bound;
for details.
Maximize
~[aij(l
-vivj)-
tbij(l+vtv
j)]
t0.87856. Lemmla
1.3 by using
– y and nc)ting
that
n –
H
We now show that the maximum 2-satisfiability problem can be modelled by (Q’). An instance of the maximum satisfiability problem (MAX SAT) is defined by a collection C of boolean clauses, where each
is
We merely need to add the following conVI . Vj = 1 for (z, j) c E+ and to (P):
clause is a disjunction of literals drawn from a set of is either a varivariables {xl, X2, . . . , Zm}. A literal
–1 for (z, j) E E-, where E+ (resp. .E-) vi .V3= corresponds to the pair of vertices forced to be on the same side (resp. different sides) of the cut. Using the
able z or its negation
algorithm
An optimal
above and setting
by using
(except
for (Q’); we only need the following
Lemma
probalgo-
problem
(Q’)
as before
vi c v.
Practically the same analysis shows that algorithm is a randomized (a – e)-approxirnation
where ~ >0.79607. to the MAX
Sn
(P’)).
sides of the algorithm.
MAX 2SAT For the maximum 2-satisfiability lem, we also have an (a – e)-approximation rit hm.
v%e
to:
We approximate
problems.
MAX RES CUT For the variation of MAX CUT in which pairs of vertices are forced to be on either
The extension
subject
i.
In addition,
Cj E C there is an associated
yi = 1 if r . v% ~ O and
assignment
yi = – 1 otherwise gives a feasible solution to MAX RES CUT, assuming that a feasible solution exists. Indeed, it is easy to see that if v, . v~ = 1, then the algorithm will produce a solution such that y~yj = 1. If vi . vj = – 1 then the only case in which the algorithm produces a solution such that Yiyj # – 1 is when v%. r = v~ . r = O, an event that happens with probability O. The analysis of the expected value of the cut is unchanged and, therefore, the resulting algorithm is a randomized (a – c)-approximation algorithm. In the next section, we show that the algorithm for MAX CUT can also be used to approximate a more general integer quadratic program, and that MAX 2SAT can be modelled as such. In Section 2.2, we show how to approximate another general integer quadratic program, which can be used to model the MAX DI-
that
solution of truth
maximizes
clauses.
MAX
for each clause
non-negative
to a MAX
SAT
values to the variables
the sum of the weight 2SAT
consists
of MAX
weight
instance
W3. is an
xl, . . . . Zn
of the satisfied SAT instances
in which each clause contains no more than two literals. MAX 2SAT is NP-complete [9]; the best approximation algorithm known previously has a performance guarantee Goemans
of ~ and is due to Yannakakis [44] (see also and Williamson [10]). As with the MAX
CUT problem, MAX 2SAT is known to be MAX SNPhard [32]; thus there exists some constant c 0.79607.
i,~, k
where the v (Cj ) are non-negative
be shown to yield
the expected
guarantee
namely
of any clause with at most two literals per clause can be expressed in the form required in (Q’). As a result,
a MAX
“vk)]
+YOY, - Y;Y2Y,)
larly expressed; for instance, if z~ is negated one only needs to replace yi by –yi. Therefore, the value v(C)
Given
“vk +vj
Vi C V.
vi c Sn
2
4
algorithm
‘vk)
k
; (3+YoYi
=
(SAT)
as
in 1 + Yiyj.
that
A3j)
2 =
yiyj
and, t%
can be interpreted
–Yj = –Yk and o otherwise while 1 +Ytyj +Yiyk +Yjyk is 4 if yz = Y2 = yk and is O otherwise. We relax (Q”) to:
Max V(zj
of (Q”)
restricted
Moreover, 1 –
and
C, we de-
(1– YiYj)(l + YjY~)),
CUT. The previous best known approximation algorithm for MAX DICUT has a performance guarantee of ~ [32].
that
as (1 –
428
We can model the MAX
DICUT
problem
using the Vi for each
distributed is not needed to derive the inequality. We first describe a set of vectors U1(6),. . . . u~ (8) c
2SAT program, we i E V, and, as with the MAX introduce a variable y. that will denote the S’ side of the cut. Thus z E S iff g, = yo. Then arc (z, j)
and, thus, for a fixed value of 9, fe can easily be evaluated. We will then describe a discrete set (3 of size
contributes
polynomial
program
(Q”).
We introduce
weight
a variable
~waj (1 + giyo) ( 1 – yjyo)
Sn_l
to the cut.
such that
approximation
algorithm graph
for MAX
DICUT.
Finally,
has weighted
indegree
of every
max
-$~o
a(l
–
~ = ~ and d = -&. The running
time of the re-
in the performance
guar-
antee. We first
7r/2), im-
show how to define
from
u;(0)
vi and $.
Given a vector .z g Rm, we define, for any scalar a, zla to be the vector in Rm-l having coordinates .zl,’zz, ..., z~–2 and a. Recall that (:c~_l, Zm) = (scosO, ssinf9). We also express the last two coordinates of VI in polar form: (V,,,n-l, vtm) = (t, cos ~,, t,sin ~,), where t,=
on the value of 0. For a given value of 8, let
vn)
. . . ,Vn) – dwtot.
mialit y and to avoid losing
two cox~) =
(Indeed, (Zm-I, Xm) is a bivariate normal variable and its density depends only on S2 = X&_l +x%. ) We shall fo (U , . . . . Vn) = E[W18].
we have scalars
operations are computed with precision polynomial in the input size and log $, in order to guarantee polyno-
plying that s is allowed to be negative. Observe that O is undefined on a set of measure O (when s = O). Although we don’t really need this fact, it is easy to see that 0 is uniformly distributed in the range [– $, ~].
condition
point,
heavily involves square roots and trigonometric functions. As usual, we implicitly assume that all these
Consider a vector r drawn uniformly from S’m. For simplicity of exposition, we shall assume that r is ZI, . . ., z~.
2 f(w,
. . . IYn)
assume
that
sulting deterministic (a – d)-approximat ion algorithm is thus polynomial in ~. Our derandomized algorithm
at a time.
by generating
can
At
c) ZfiC, we see that the derandomized algorithm achieves a performance guarantee of (1 – 2nc$)a(l – e), which can be made to be (a – d) for any d > 0, by
In the deter. ., Vn). we shall decrease the dimension m
variables
such that
f(Yl>...
.,yn)f(tll,l,.
obtained
vn),
f(vl,..., vn ) > Wt~t/2 since this is true for an orthonormal basis of n vectors. Thus
Vj) = E[JV]. In
the existence
to 1.
We
to S’m. Define obtain
7G@@17v2,...,
sion is equal
f(Yl>
and present
to S’l (i.e. scalars taking
fd(vl, vz, . . . ,Wn) – f5WtOt
Y, ● {–1,+1}
the same
a deterministic (a – c)-approximation algorithm for any e >0. Assume we are given a set of unit vectors VI, . . . . Vn m. spanning a linear space of dimension We can thus assume that
. . . . ~n(~))
. . ,Un) ~ f(vl,. . . ,vn)– JWtOt. S~_l such that f(ul,. We can then continue the process until the dimen-
performance guarantee. The method we use is the classical method of conditional expectations. We illustrate the derandomization
= ~(ul(o),
where WtOt = ~z<J wv” Selecting for O the value fe over @, we obtain vectors U1, , . . . Un c maximizing
Derandomization
deterministic
. . ,vn)
in n and 1/6 such that
Summing over all arcs (i, j) c A gives a program of the same form as (Q”). Hence we obtain a .79607if the directed
~e(vl,.
m~~oand
We observe that ?’,C [0,27r).
We know that Xm–lvt,m–l
‘?ql”v]= Ee[E[w’/o]]
+ x~vtm
7r/2 =
/
_=,2
fe(vl,
. . . ,Vn)g(e)de
Setting
=
stl(cos ~, cos (9+ sin ~, sin (3)
=
Sti cos(~t – e).
x’ = X[S and
a.(o) ==W,l[tzCos(-yi – /3)], where g(6) is the probability y density
function.
gued above, g(d) = ~, but the fact that
As ar-
we derive
O is uniformly
ai(0)/
429
that
z . V$ =
x’ . al(6).
[[a,(6) II. The above discussion
Let
u,(6)
=
shows that Sgn(z.
and Schrijver [24, 25], which showed that tighter and tighter relaxations could be obtained through semidefinite programming, it seemed worthwhile to investigate the power of such relaxations from a worst-case perspective. The results of this paper constitute a first step in this direction. As we mentioned in the introduction, further steps have already been made, with improved results for MAX 2SAT and MAX DICUT by Feige and Goemans, and for coloring by Karger, Motwani, and Sudan. We think that the continued investigation of these methods is promising.
vi) = sgn(x’ . w(6) ). Moreover, once we condition on 0, s is normally distributed (and independent of xl, . . . , Xn -2), implying that z’/ IIz’ I[ is uniformly distributed on S~_l. Therefore, fe (v1,..., v~) can be evaluated as .f(ul (0),. . . . un(0)). In order to compute approximately j(zq(f?),..., the interval denote
the maximum
the angle between
the vectors
v,[O and a~(e).
Since v, 10 and az(0) – (vZ10) are orthogonal, that
and from
of
Un((?)) over 6 c [—n/2, 7r/2], we discretize in the following way. For every i, let a,(0)
this that
‘i
al(0)
‘arctan
ranges between
&
‘
As @varies, cq (8) covers the interval
we derive
O and
‘“
[0, mi] twice,
be-
cause the dependence of cq (0) on 8 can be stated terms of cos(~z –O). Therefore, we can choose 2m,/v r/v
values of 0, say 0:)
for j s 2mi/v,
all Oji)
Algorithms
Michel
X.
for MAX
Goernans*
David
M.I.T.
always
solutions
of expected
.87856 times the optimal a simple
and elegant
and MAX
2SAT
Williamson University
Introduction Given an undirected graph G = (V, 1?) and nonnegative weights wij = wli on the edges (i, j) c E, the
randomized approximation algorithms CUT and MAX 2SAT problems that
deliver
P.
Cornell
Abstract
We present for the MAX
CUT
value.
value
at least
These algorithms
technique
that
maximum
randomly
cut problem
the set of vertices
(MAX
S that
edges in the cut (S, ~); that
use
with
rounds
one endpoint
CUT)
is that
maximizes
of finding
the weight
is, the weight
in S and the other
of the
of the edges
in $.
For sim-
the solution to a nonlinear programming relaxation. This relaxation can be interpreted both as a semidefi-
plicity, we usually set wij = O for (i, j) @ E and denote wij. the weight of a cut (S, ~) by w(S, ~) = Ei=s,j$s
nite program and as an eigenvalue minimization problem. We then show how to derandomize the algorithm to obtain approximation algorithms with the same performance guarantee of .87856. The previ-
The MAX CUT problem is one of the Karp’s original NP-complete problems [19], and has long been known to be NP-complete even if the problem is unweighed; that is, if w$j = 1 for all (i, j) E E [9]. The MAX
ous best-known approximation algorithms for these problems had performance guarantees of ~ for MAX
CUT problem is solvable in polynomial time for some special classes of graphs (e.g. if the graph is planar [29, 14]). Besides its theoretical importance, the MAX
CUT
and ~ for MAX
analysis
leads
2SAT.
A slight
extension
to a .79607-approximation
of our
CUT problem
algorithm
has applications
in circuit
a &
and statistical
physics
approximation algorithm was the previous best-known algorithm. Our algorithm gives the first substantial progress in approximating MAX CUT in nearly twenty years, and, to the best of our knowledge, represents the first use of semidefinite programming in the design of approximation algorithms.
comprehensive
survey of the MAX
for the maximum
directed
cut problem,
where
reader is referred
*Address: MA
supported contract 1799.
Dept. 02139. in part
14853.
Email: by
NSF
F49620-92-J-0125
tAddress: ~ineering,
of Mathematics,
School 237
Email:
ETC
Room
goemans@nath. contract and
13uildin~,
dpw@cs. cornell.
mit.
contract
Research
Cornell
edu.
an NSF Postdoctoral Fellowship. This while the author was visiting MIT.
Research research
rithm.
Ithaca,
delivers
In 1976, Sahni
lem. Their
For a
problem,
the
and Tuza [38].
a solution
and Gonzales
algorithm
algorithm
decides whether
En-
of value
at least
iterates
[40] presented
for the MAX through
CUT
the vertices
or not to assign vertex
a
proband
i to S based
on which placement maximizes the weight of the cut of vertices 1 to i. This algorithm is essentially equivalent to the randomized algorithm that flips an unbiased coin for each vertex to decide which vertices
NY
supported
that
&approximation
Force
NOOO14-92-J-
and Industrial
University,
CUT
p times the optimal value. The constant p is sometimes called the performance guarantee of the algo-
Research Air
design
approach to solving such a problem is to find a papproximation algorithm; that is, a polynomial-time
M. I. T., Camedu.
9302476-CCR,
DARPA
of Operations
2-372,
to Poljak
layout
et al. [3]).
Because it is unlikely that there exist efficient algorithms for NP-hard maximization problems, a typical
algorithm
bridge,
(Barahona
by
was conducted
Permission to copy without fee all or part of this material is granted provided 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.
are assigned to the set S. Since 1976, a number of researchers have presented approximation algorithms for the unweighed MAX CUT problem with performance; guarantees of ~ + & [42], $ + ~ [37], and [17] (where n = IVI and m = IEI), but no *+= progress was made in improving the constant in the
STOC 94- 5/94 Montreal, Quebec, Canada @ 1994 ACM 0-89791 -663-8/94/0005..$3.50
422
i
performance guarantee zales’s straightforward We
present
a
approximation
beyond that of Sahni and Gonalgorithm.
simple,
algorithm
randomized
(a
for the maximum
–
input, so the running time dependence on e is polynomial in log $). This can be done through the ellipsoid algorithm
e)-
(Grotschel
time algorithms
cut prob-
aa well as interior-point
lem where
mirovskii
~=
20 —
min
time,
rithm, for any e >0. The algorithm for MAX CUT also leads directly to randomized and deterministic (a – e)-approximation 2-satisfiability best known
algorithms
problem algorithm
guarantee
(MAX
for the
2SAT).
maximum
The previously
for this problem
haa a perfor-
of ~ and is due to Yannakakis
(see also Goemans
and Williamson
[10]).
[44]
Coppersmith
and Shmoys have independently observed that the improved 2SAT algorithm leads to a slightly better than ~-approximation algorithm for the overall MAX SAT problem. Finally, a slight extension of our analysis yields a .79607-approximation imum vious
directed
cut problem
best known
performance
algorithm
guarantee
algorithm (MAX
for MAX
The pre-
DICUT
has a
of ~ [32].
Our algorithm depends on a means of randomly rounding a solution to a nonlinear relaxation. This prorelaxation can either be seen as a semzdej%ate minimization problem. To gram or aa an eigenvalue our knowledge,
this is the first
time
that
solution;
al. [13] and Section
[41])
and Ne-
To terminate implicitly
in
assume
space or on the size
for details
3.3 of Alizadeh
see Grotschel
et
[1].
The importance of semidefinite programming is that it leads to tighter relaxations than the classical linear programming relaxations for mi>ny graph and combinatorial problems. A beautiful application of semidefinite programming is the work of Lov&z [22] on the Shannon with
the
capacity
of a graph.
In conjunction
solvability
of semidefinite
polynomial-time
programs, time
this
leads to the only
algorithm
the largest
for finding
clique)
known
the largest
in a perfect
graph
polynomialstable
set (or
(Grotschel
et al.
[12]). More recently, there has been increased interest in semidefinite programming [24, 25, 1, 33, 35, 8, 23]. This started with the work of Lov&z and Schrijver [24, 25], who developed a machinery tc) define tighter and tighter
for the max-
DICUT).
[1]).
on the feasible
of the optimum
(Vaidya
(Nesterov
these algorithms
some requirement
and e is any positive scalar. The algorithm represents the first substantial progress in approximating the MAX CUT problem in nearly twenty years. We also show how to derandomize the algorithm to obtain a deterministic (a – c)-approximation algo-
methods
[27, 28] and Alizadeh
polynomial
>0.87856,
n’ 1 – Cose
0
–e.
(P) is also equivalent to an eigenvalue upper bound on the value of the maximum cut ZfiC introduced by Delorme and Poljak [5, 4]. To describe the bound, we first introduce some notation. The Laplacian matrix L = (iij ) is defined by lij = –wiJ
for i # j and lii = ~~=1
wik.
Given a vec-
mum eigenvalue
A is denoted
of a matrix
with entries The maxiby Am..(A).
Tr(LY)
that
For the optimum
The relaxation
tor u, diag(u) denotes the diagonal matrix u~fori= l,..., n on the main diagonal.
Noticing
is precisely
objective function of (SD), showing that Z; s Z&IG. mum
matrix
gramming
Y,
correcting
strong
guarantees
and the
for semidefinite
the duality
the
weak duality,
vector
duality
that
four times
one derives
optipro-
gap is equal to
= O, where M is defined as O and, thus, Tr(MY) above. Since both M and Y are positive semidefinite, we derive that MY = O (see [21, p. 218, ex. 14]). From the definition of M and the fact that MY = O, implying that we derive that ~Y – LY = dzag(u)Y, for all i. Since ~ = ~Z~lG = ~Z~, u%= ~ — ~j lijytj we observe that we can easily deduce the optimum
Lemma
1.4 [[5]] Let u E Rn satisfy UI +...
+ u~ = O.
recting vector from the optimum
corY for (SD).
matrix
Then (L+
9(U) = ;Am.z
1.4
dzag(u))
Theorem
is an upper bound on Z~c. The
proof
is simple.
to the integer
yTLy
=
(~~~.(~)
&.z(L
Let
quadratic
4Z~C.
By
g be an optimal program
using
= maxll~ll=l
~ — —
= proving
(Q).
Notice
Rayleigh
Corollary
solution
‘T(L
zfi~ z;
For the 5-cycle, that Z~c/.Z~IG n
1 — n
yTLy ( 4z&~
+ ~ y;UZ 2=1 )
pressed
n’
~~=1 ui = O
The bound
proposed
Delorme
and Poljak
[5] is to optimize
g(u)
by
over all
that
~ a >0.87856.
Delorme =
25~;& —
and Poljak =
[5] have shown
0.88445 ...,
implying
as vi
=
(cos(&),
sin(%))
for 2 =
1,...,5
Z& C/Z$IG
> -A
holds for special subclasses
of graphs, such as planar graphs or line graphs. However, they were unable to prove a bound better than
g(u)
0.5 in the absolute worst-case. Although the worst-case value of Z&C/Z~ is not completely settled, we have constructed instances for
n
to:
CUT,
corresponding to Z; = ~ (1 + cos ~) = w. Since ZfiC = 4 for the 5-cycle, this yields the bound of Delorme and Poljak. Delorme and Poljak have shown
vectors:
subject
corollary:
that our worst-case analysis is almost tight. One can obtain this bound from the relaxation (F’) by observing that for the 5-cycle 1 – 2 – 3 – 4 – 5 – 1, the optimal vectors lie in a 2-dimensional subspace and can be ex-
u satisfying
(EIG)
the following
For any instance of MAX —
‘V~’(U))y
vector.
Inf
1.5
Relaxation
we obtain
A vector
=
1.1 implies
that
a correcting
-%IG
of the
principle
is called correcting
the lemma.
the
ZTMZ),
+ diag(u))
Quality
~ ui = O. $=1
426
which Zf/E[ W] < 0.8786, showing of our algorithm is practically tight.
that
Poljak and Rendl [34, 36] (see also Delorme Poljak [6]) report computational results showing the bound
Z*EIG is typically less than worse than 870 away from Z&c. We have implemented using a code supplied
experiments
have shown that cuts within
see the full paper
randomized
programs.
MAX
2SAT
We consider
and that
the integer
Maximize (Q’)
[39] for a spe-
subject
to:
vi {: v,
y~ G {–1,1}
where aij and bij are non-negative. function of (Q’) is thus a nonnegative
Very preliminary
the algorithm
program
Z<j
algorithm
typically
4-5% of the semidefinite
quadratic
[aii(l - YzYj) + b~j(l+ Y~Y~)l
~
2-570 and never
by R. Vanderbei
cial class of semidefinite generates
the
2.1
the analysis
1 + y,yj.
We relax
(Q’)
The objective linear form in
to:
bound;
for details.
Maximize
~[aij(l
-vivj)-
tbij(l+vtv
j)]
t0.87856. Lemmla
1.3 by using
– y and nc)ting
that
n –
H
We now show that the maximum 2-satisfiability problem can be modelled by (Q’). An instance of the maximum satisfiability problem (MAX SAT) is defined by a collection C of boolean clauses, where each
is
We merely need to add the following conVI . Vj = 1 for (z, j) c E+ and to (P):
clause is a disjunction of literals drawn from a set of is either a varivariables {xl, X2, . . . , Zm}. A literal
–1 for (z, j) E E-, where E+ (resp. .E-) vi .V3= corresponds to the pair of vertices forced to be on the same side (resp. different sides) of the cut. Using the
able z or its negation
algorithm
An optimal
above and setting
by using
(except
for (Q’); we only need the following
Lemma
probalgo-
problem
(Q’)
as before
vi c v.
Practically the same analysis shows that algorithm is a randomized (a – e)-approxirnation
where ~ >0.79607. to the MAX
Sn
(P’)).
sides of the algorithm.
MAX 2SAT For the maximum 2-satisfiability lem, we also have an (a – e)-approximation rit hm.
v%e
to:
We approximate
problems.
MAX RES CUT For the variation of MAX CUT in which pairs of vertices are forced to be on either
The extension
subject
i.
In addition,
Cj E C there is an associated
yi = 1 if r . v% ~ O and
assignment
yi = – 1 otherwise gives a feasible solution to MAX RES CUT, assuming that a feasible solution exists. Indeed, it is easy to see that if v, . v~ = 1, then the algorithm will produce a solution such that y~yj = 1. If vi . vj = – 1 then the only case in which the algorithm produces a solution such that Yiyj # – 1 is when v%. r = v~ . r = O, an event that happens with probability O. The analysis of the expected value of the cut is unchanged and, therefore, the resulting algorithm is a randomized (a – c)-approximation algorithm. In the next section, we show that the algorithm for MAX CUT can also be used to approximate a more general integer quadratic program, and that MAX 2SAT can be modelled as such. In Section 2.2, we show how to approximate another general integer quadratic program, which can be used to model the MAX DI-
that
solution of truth
maximizes
clauses.
MAX
for each clause
non-negative
to a MAX
SAT
values to the variables
the sum of the weight 2SAT
consists
of MAX
weight
instance
W3. is an
xl, . . . . Zn
of the satisfied SAT instances
in which each clause contains no more than two literals. MAX 2SAT is NP-complete [9]; the best approximation algorithm known previously has a performance guarantee Goemans
of ~ and is due to Yannakakis [44] (see also and Williamson [10]). As with the MAX
CUT problem, MAX 2SAT is known to be MAX SNPhard [32]; thus there exists some constant c 0.79607.
i,~, k
where the v (Cj ) are non-negative
be shown to yield
the expected
guarantee
namely
of any clause with at most two literals per clause can be expressed in the form required in (Q’). As a result,
a MAX
“vk)]
+YOY, - Y;Y2Y,)
larly expressed; for instance, if z~ is negated one only needs to replace yi by –yi. Therefore, the value v(C)
Given
“vk +vj
Vi C V.
vi c Sn
2
4
algorithm
‘vk)
k
; (3+YoYi
=
(SAT)
as
in 1 + Yiyj.
that
A3j)
2 =
yiyj
and, t%
can be interpreted
–Yj = –Yk and o otherwise while 1 +Ytyj +Yiyk +Yjyk is 4 if yz = Y2 = yk and is O otherwise. We relax (Q”) to:
Max V(zj
of (Q”)
restricted
Moreover, 1 –
and
C, we de-
(1– YiYj)(l + YjY~)),
CUT. The previous best known approximation algorithm for MAX DICUT has a performance guarantee of ~ [32].
that
as (1 –
428
We can model the MAX
DICUT
problem
using the Vi for each
distributed is not needed to derive the inequality. We first describe a set of vectors U1(6),. . . . u~ (8) c
2SAT program, we i E V, and, as with the MAX introduce a variable y. that will denote the S’ side of the cut. Thus z E S iff g, = yo. Then arc (z, j)
and, thus, for a fixed value of 9, fe can easily be evaluated. We will then describe a discrete set (3 of size
contributes
polynomial
program
(Q”).
We introduce
weight
a variable
~waj (1 + giyo) ( 1 – yjyo)
Sn_l
to the cut.
such that
approximation
algorithm graph
for MAX
DICUT.
Finally,
has weighted
indegree
of every
max
-$~o
a(l
–
~ = ~ and d = -&. The running
time of the re-
in the performance
guar-
antee. We first
7r/2), im-
show how to define
from
u;(0)
vi and $.
Given a vector .z g Rm, we define, for any scalar a, zla to be the vector in Rm-l having coordinates .zl,’zz, ..., z~–2 and a. Recall that (:c~_l, Zm) = (scosO, ssinf9). We also express the last two coordinates of VI in polar form: (V,,,n-l, vtm) = (t, cos ~,, t,sin ~,), where t,=
on the value of 0. For a given value of 8, let
vn)
. . . ,Vn) – dwtot.
mialit y and to avoid losing
two cox~) =
(Indeed, (Zm-I, Xm) is a bivariate normal variable and its density depends only on S2 = X&_l +x%. ) We shall fo (U , . . . . Vn) = E[W18].
we have scalars
operations are computed with precision polynomial in the input size and log $, in order to guarantee polyno-
plying that s is allowed to be negative. Observe that O is undefined on a set of measure O (when s = O). Although we don’t really need this fact, it is easy to see that 0 is uniformly distributed in the range [– $, ~].
condition
point,
heavily involves square roots and trigonometric functions. As usual, we implicitly assume that all these
Consider a vector r drawn uniformly from S’m. For simplicity of exposition, we shall assume that r is ZI, . . ., z~.
2 f(w,
. . . IYn)
assume
that
sulting deterministic (a – d)-approximat ion algorithm is thus polynomial in ~. Our derandomized algorithm
at a time.
by generating
can
At
c) ZfiC, we see that the derandomized algorithm achieves a performance guarantee of (1 – 2nc$)a(l – e), which can be made to be (a – d) for any d > 0, by
In the deter. ., Vn). we shall decrease the dimension m
variables
such that
f(Yl>...
.,yn)f(tll,l,.
obtained
vn),
f(vl,..., vn ) > Wt~t/2 since this is true for an orthonormal basis of n vectors. Thus
Vj) = E[JV]. In
the existence
to 1.
We
to S’m. Define obtain
7G@@17v2,...,
sion is equal
f(Yl>
and present
to S’l (i.e. scalars taking
fd(vl, vz, . . . ,Wn) – f5WtOt
Y, ● {–1,+1}
the same
a deterministic (a – c)-approximation algorithm for any e >0. Assume we are given a set of unit vectors VI, . . . . Vn m. spanning a linear space of dimension We can thus assume that
. . . . ~n(~))
. . ,Un) ~ f(vl,. . . ,vn)– JWtOt. S~_l such that f(ul,. We can then continue the process until the dimen-
performance guarantee. The method we use is the classical method of conditional expectations. We illustrate the derandomization
= ~(ul(o),
where WtOt = ~z<J wv” Selecting for O the value fe over @, we obtain vectors U1, , . . . Un c maximizing
Derandomization
deterministic
. . ,vn)
in n and 1/6 such that
Summing over all arcs (i, j) c A gives a program of the same form as (Q”). Hence we obtain a .79607if the directed
~e(vl,.
m~~oand
We observe that ?’,C [0,27r).
We know that Xm–lvt,m–l
‘?ql”v]= Ee[E[w’/o]]
+ x~vtm
7r/2 =
/
_=,2
fe(vl,
. . . ,Vn)g(e)de
Setting
=
stl(cos ~, cos (9+ sin ~, sin (3)
=
Sti cos(~t – e).
x’ = X[S and
a.(o) ==W,l[tzCos(-yi – /3)], where g(6) is the probability y density
function.
gued above, g(d) = ~, but the fact that
As ar-
we derive
O is uniformly
ai(0)/
429
that
z . V$ =
x’ . al(6).
[[a,(6) II. The above discussion
Let
u,(6)
=
shows that Sgn(z.
and Schrijver [24, 25], which showed that tighter and tighter relaxations could be obtained through semidefinite programming, it seemed worthwhile to investigate the power of such relaxations from a worst-case perspective. The results of this paper constitute a first step in this direction. As we mentioned in the introduction, further steps have already been made, with improved results for MAX 2SAT and MAX DICUT by Feige and Goemans, and for coloring by Karger, Motwani, and Sudan. We think that the continued investigation of these methods is promising.
vi) = sgn(x’ . w(6) ). Moreover, once we condition on 0, s is normally distributed (and independent of xl, . . . , Xn -2), implying that z’/ IIz’ I[ is uniformly distributed on S~_l. Therefore, fe (v1,..., v~) can be evaluated as .f(ul (0),. . . . un(0)). In order to compute approximately j(zq(f?),..., the interval denote
the maximum
the angle between
the vectors
v,[O and a~(e).
Since v, 10 and az(0) – (vZ10) are orthogonal, that
and from
of
Un((?)) over 6 c [—n/2, 7r/2], we discretize in the following way. For every i, let a,(0)
this that
‘i
al(0)
‘arctan
ranges between
&
‘
As @varies, cq (8) covers the interval
we derive
O and
‘“
[0, mi] twice,
be-
cause the dependence of cq (0) on 8 can be stated terms of cos(~z –O). Therefore, we can choose 2m,/v r/v
values of 0, say 0:)
for j s 2mi/v,
all Oji)
