An Asymptotic Approximation Scheme for Multigraph Edge Coloring
An Asymptotic Approximation Scheme for Multigraph Edge Coloring Peter Sanders, Universit¨at Karlsruhe David Steurer, Universit¨at des Saarlandes
g
INFORMATIK
Consider an edge coloring of a graph G = (V, E) total coloring: each edge has a color proper col.: adjacent edges have different colors q-coloring: proper total coloring using at most q colors
Chromatic Index χ0 := min{ q | ∃ q-coloring } Edge Coloring Problem: given a graph, find a χ0 -coloring |E(H)| lower bounds ∆ := max deg(v) and Γ := max v∈V H⊆V,|H|>1 b|H|/2c
Classical Results simple graphs: (∆ + 1)-coloring in O(nm) [Vizing ’64 ], χ0 -coloring NP-hard to obtain [Holyer ’81] multigraphs: (µ: max. edge multiplicity) (∆ + µ)-coloring in O((n + ∆)m), ( 43 − ε)χ0 -coloring NP-hard to obtain, Fractional Chromatic Index χ e0 (linear relaxation) χ e0 = max(∆, Γ) [Edmonds ’65 ] Conjecture: χ0 ≤ χ e0 + 1
[Goldberg,Andersen,Seymour ]
In fact their conjectures are much stronger.
(Integer) Linear Programm min
X
xM
X
xM χM = χ E ,
M ∈M
s.t.
M ∈M
x ≥ 0, (x ∈ {0, 1})
alternating path:
Approximation Algorithms In O((n + ∆)m)
3 ∆-coloring 2
7 0 χ e 6
9 0 χ e 8
[Shannon ’49 ]
+ 1-coloring [Hochbaum,Nishizeki,Shmoys ’86 ] + 1-coloring [Goldberg ’84 ]
11 0 χ e 10
+ 1-coloring [Nishizeki,Kashiwagi ’90 ]
√ χ e + O n log n -coloring in polynomial time [Plantholt ’98 ] 0
alternating path:
Approximation Algorithms In O((n + ∆)m)
3 ∆-coloring 2
7 0 χ e 6
9 0 χ e 8
[Shannon ’49 ]
+ 1-coloring [Hochbaum,Nishizeki,Shmoys ’86 ] + 1-coloring [Goldberg ’84 ]
11 0 χ e 10
+ 1-coloring [Nishizeki,Kashiwagi ’90 ]
√ χ e + O n log n -coloring in polynomial time [Plantholt ’98 ] 0
Recent Results on the Conjecture χ0 /e χ0 → 1 for χ0 → ∞ [Kahn ’96 ] χ0 ≤ χ e0 + O(log min(n, χ e0 )) [Plantholt ’03]
but both do not yield polynomial algorithms
We show: q 0 1 + 4.5 -coloring in time polynomial in n and log µ χ e 0 χ e
Outline of Algorithm Relax on totality
partial coloring
G0 = (V, E0 ) induced by uncolored edges ∆0 maximum degree in G0
Start with a naive partial coloring Extend the coloring (hard part) Once G0 is simple and ∆0 is small color G0 with Vizing’s algorithm Combine partial coloring and total coloring of G0
Extending a partial coloring Bad Edges E0(>) : uncolored edges parallel to another uncolored edge Potential of a partial coloring Φ := |E0 | + |E0(>) | Extending the coloring means decreasing its potential
Color Orbit: (moving missing colors) set of nodes connected by uncolored edges weak: contains two nodes with a common missing color (otherwise: hard) Proposition. If there is a weak color orbit, we can decrease Φ.
If q ≥ (1 + ε)∆ then hard color orbits contain at most
1+ε ε
nodes.
Color Orbit: (moving missing colors) set of nodes connected by uncolored edges weak: contains two nodes with a common missing color (otherwise: hard) Proposition. If there is a weak color orbit, we can decrease Φ.
If q ≥ (1 + ε)∆ then hard color orbits contain at most
1+ε ε
nodes.
Edge Orbit: (moving uncolored edges) hierarchy of alternating paths of disjoint colors rooted at a bad edge marked colors: used in an alternating path weak: contains edge parallel only to colored edges (otherwise: hard) Proposition. If there is a weak edge orbit, we can decrease Φ.
Observation. hard edge orbits are color orbits
Edge Orbit: (moving uncolored edges) hierarchy of alternating paths of disjoint colors rooted at a bad edge marked colors: used in an alternating path weak: contains edge parallel only to colored edges (otherwise: hard) Proposition. If there is a weak edge orbit, we can decrease Φ.
Observation. hard edge orbits are color orbits
Edge Orbit: (moving uncolored edges) hierarchy of alternating paths of disjoint colors rooted at a bad edge marked colors: used in an alternating path weak: contains edge parallel only to colored edges (otherwise: hard) Proposition. If there is a weak edge orbit, we can decrease Φ.
Observation. hard edge orbits are color orbits
Edge Orbit: (moving uncolored edges) hierarchy of alternating paths of disjoint colors rooted at a bad edge marked colors: used in an alternating path weak: contains edge parallel only to colored edges (otherwise: hard) Proposition. If there is a weak edge orbit, we can decrease Φ.
Observation. hard edge orbits are color orbits
Problem: (by observation: only remaining case) Hard Orbit: both hard edge orbit and hard color orbit But: For q ≥ (1 + )∆ at most
1+ε ε
nodes and ≤
2 ε
marked colors
Idea: Grow hard orbit until it eventually gets weak or witness appears: e node with no unmarked missing color (∆): j k e subgraph H containing V (H) edges of each unmarked color (Γ): 2 Proposition. Either we can grow the hard orbit or there is a witness
If witness appears, then q < χ e0 +
2 ε
decrease Φ by adding a color
Problem: (by observation: only remaining case) Hard Orbit: both hard edge orbit and hard color orbit But: For q ≥ (1 + )∆ at most
1+ε ε
nodes and ≤
2 ε
marked colors
Idea: Grow hard orbit until it eventually gets weak or witness appears: e node with no unmarked missing color (∆): j k e subgraph H containing V (H) edges of each unmarked color (Γ): 2 Proposition. Either we can grow the hard orbit or there is a witness
If witness appears, then q < χ e0 +
2 ε
decrease Φ by adding a color
Putting everything together Start with q = q0 = (1 + ε)∆ Decrease Φ by maximal amount χ0 + 2/ε, q0 ) no edge orbit, no weak color orbit, q ≤ max(e G0 is simple with component size ≤
1+ε ε
Color G0 using Vizings algorithm with ∆0 + 1 ≤ Combine colorings
max(e χ0 + 2/ε, q0 ) +
1+ε ε
colors
1+ε -coloring ε
Results: ∀ε > 0. max((1 + ε)∆, Γ)+3/ε-coloring in O((n + ∆)m · poly(1/ε)) p χ0 -coloring in poly(n, m) Adding colors more adaptively: χ e0 + 4.5e
Polynomial Algorithm If we tolerate up to n parallel uncolored edges, a witness implies q < χ e0 . Split graph into two even parts and simple remainder: G = 2Gq + Gr Color Gq recursively until max. multiplicity of uncolored edges ≤ n Double the coloring and insert edges of Gr
. . . ≤ 2n + 1
Extend coloring until max. multiplicity of uncolored edges ≤ n In the end: #uncolored edges ≤ n Now: apply first algorithm χ e0 +
n 2 p
and q ≤ χ e0 4.5e χ0 -coloring in time poly(n, log µ)
Conclusion first polynomial algorithm with asymptotic approximation ratio 1. algorithms to use for multigraph edge coloring:
Open Problems: χ e0 + log min(e χ0 , n)-coloring in polynomial time Conjecture χ0 ≤ χ e0 + 1 is still open...
Thank you!
g
INFORMATIK
Consider an edge coloring of a graph G = (V, E) total coloring: each edge has a color proper col.: adjacent edges have different colors q-coloring: proper total coloring using at most q colors
Chromatic Index χ0 := min{ q | ∃ q-coloring } Edge Coloring Problem: given a graph, find a χ0 -coloring |E(H)| lower bounds ∆ := max deg(v) and Γ := max v∈V H⊆V,|H|>1 b|H|/2c
Classical Results simple graphs: (∆ + 1)-coloring in O(nm) [Vizing ’64 ], χ0 -coloring NP-hard to obtain [Holyer ’81] multigraphs: (µ: max. edge multiplicity) (∆ + µ)-coloring in O((n + ∆)m), ( 43 − ε)χ0 -coloring NP-hard to obtain, Fractional Chromatic Index χ e0 (linear relaxation) χ e0 = max(∆, Γ) [Edmonds ’65 ] Conjecture: χ0 ≤ χ e0 + 1
[Goldberg,Andersen,Seymour ]
In fact their conjectures are much stronger.
(Integer) Linear Programm min
X
xM
X
xM χM = χ E ,
M ∈M
s.t.
M ∈M
x ≥ 0, (x ∈ {0, 1})
alternating path:
Approximation Algorithms In O((n + ∆)m)
3 ∆-coloring 2
7 0 χ e 6
9 0 χ e 8
[Shannon ’49 ]
+ 1-coloring [Hochbaum,Nishizeki,Shmoys ’86 ] + 1-coloring [Goldberg ’84 ]
11 0 χ e 10
+ 1-coloring [Nishizeki,Kashiwagi ’90 ]
√ χ e + O n log n -coloring in polynomial time [Plantholt ’98 ] 0
alternating path:
Approximation Algorithms In O((n + ∆)m)
3 ∆-coloring 2
7 0 χ e 6
9 0 χ e 8
[Shannon ’49 ]
+ 1-coloring [Hochbaum,Nishizeki,Shmoys ’86 ] + 1-coloring [Goldberg ’84 ]
11 0 χ e 10
+ 1-coloring [Nishizeki,Kashiwagi ’90 ]
√ χ e + O n log n -coloring in polynomial time [Plantholt ’98 ] 0
Recent Results on the Conjecture χ0 /e χ0 → 1 for χ0 → ∞ [Kahn ’96 ] χ0 ≤ χ e0 + O(log min(n, χ e0 )) [Plantholt ’03]
but both do not yield polynomial algorithms
We show: q 0 1 + 4.5 -coloring in time polynomial in n and log µ χ e 0 χ e
Outline of Algorithm Relax on totality
partial coloring
G0 = (V, E0 ) induced by uncolored edges ∆0 maximum degree in G0
Start with a naive partial coloring Extend the coloring (hard part) Once G0 is simple and ∆0 is small color G0 with Vizing’s algorithm Combine partial coloring and total coloring of G0
Extending a partial coloring Bad Edges E0(>) : uncolored edges parallel to another uncolored edge Potential of a partial coloring Φ := |E0 | + |E0(>) | Extending the coloring means decreasing its potential
Color Orbit: (moving missing colors) set of nodes connected by uncolored edges weak: contains two nodes with a common missing color (otherwise: hard) Proposition. If there is a weak color orbit, we can decrease Φ.
If q ≥ (1 + ε)∆ then hard color orbits contain at most
1+ε ε
nodes.
Color Orbit: (moving missing colors) set of nodes connected by uncolored edges weak: contains two nodes with a common missing color (otherwise: hard) Proposition. If there is a weak color orbit, we can decrease Φ.
If q ≥ (1 + ε)∆ then hard color orbits contain at most
1+ε ε
nodes.
Edge Orbit: (moving uncolored edges) hierarchy of alternating paths of disjoint colors rooted at a bad edge marked colors: used in an alternating path weak: contains edge parallel only to colored edges (otherwise: hard) Proposition. If there is a weak edge orbit, we can decrease Φ.
Observation. hard edge orbits are color orbits
Edge Orbit: (moving uncolored edges) hierarchy of alternating paths of disjoint colors rooted at a bad edge marked colors: used in an alternating path weak: contains edge parallel only to colored edges (otherwise: hard) Proposition. If there is a weak edge orbit, we can decrease Φ.
Observation. hard edge orbits are color orbits
Edge Orbit: (moving uncolored edges) hierarchy of alternating paths of disjoint colors rooted at a bad edge marked colors: used in an alternating path weak: contains edge parallel only to colored edges (otherwise: hard) Proposition. If there is a weak edge orbit, we can decrease Φ.
Observation. hard edge orbits are color orbits
Edge Orbit: (moving uncolored edges) hierarchy of alternating paths of disjoint colors rooted at a bad edge marked colors: used in an alternating path weak: contains edge parallel only to colored edges (otherwise: hard) Proposition. If there is a weak edge orbit, we can decrease Φ.
Observation. hard edge orbits are color orbits
Problem: (by observation: only remaining case) Hard Orbit: both hard edge orbit and hard color orbit But: For q ≥ (1 + )∆ at most
1+ε ε
nodes and ≤
2 ε
marked colors
Idea: Grow hard orbit until it eventually gets weak or witness appears: e node with no unmarked missing color (∆): j k e subgraph H containing V (H) edges of each unmarked color (Γ): 2 Proposition. Either we can grow the hard orbit or there is a witness
If witness appears, then q < χ e0 +
2 ε
decrease Φ by adding a color
Problem: (by observation: only remaining case) Hard Orbit: both hard edge orbit and hard color orbit But: For q ≥ (1 + )∆ at most
1+ε ε
nodes and ≤
2 ε
marked colors
Idea: Grow hard orbit until it eventually gets weak or witness appears: e node with no unmarked missing color (∆): j k e subgraph H containing V (H) edges of each unmarked color (Γ): 2 Proposition. Either we can grow the hard orbit or there is a witness
If witness appears, then q < χ e0 +
2 ε
decrease Φ by adding a color
Putting everything together Start with q = q0 = (1 + ε)∆ Decrease Φ by maximal amount χ0 + 2/ε, q0 ) no edge orbit, no weak color orbit, q ≤ max(e G0 is simple with component size ≤
1+ε ε
Color G0 using Vizings algorithm with ∆0 + 1 ≤ Combine colorings
max(e χ0 + 2/ε, q0 ) +
1+ε ε
colors
1+ε -coloring ε
Results: ∀ε > 0. max((1 + ε)∆, Γ)+3/ε-coloring in O((n + ∆)m · poly(1/ε)) p χ0 -coloring in poly(n, m) Adding colors more adaptively: χ e0 + 4.5e
Polynomial Algorithm If we tolerate up to n parallel uncolored edges, a witness implies q < χ e0 . Split graph into two even parts and simple remainder: G = 2Gq + Gr Color Gq recursively until max. multiplicity of uncolored edges ≤ n Double the coloring and insert edges of Gr
. . . ≤ 2n + 1
Extend coloring until max. multiplicity of uncolored edges ≤ n In the end: #uncolored edges ≤ n Now: apply first algorithm χ e0 +
n 2 p
and q ≤ χ e0 4.5e χ0 -coloring in time poly(n, log µ)
Conclusion first polynomial algorithm with asymptotic approximation ratio 1. algorithms to use for multigraph edge coloring:
Open Problems: χ e0 + log min(e χ0 , n)-coloring in polynomial time Conjecture χ0 ≤ χ e0 + 1 is still open...
Thank you!
