Approximating Maximum Edge Coloring in Multigraphs

Uriel Feige Eran Ofek Udi WiederWeizmann Institute

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The Problem Max Edge t-Coloring

Given a multigraph G and a number t, edge color as many edges as possible using t colors.

t = 3 Gray edges – uncoloredEvery color class is a matching

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Hardness results

For fixed t ¸ 2: Max Edge t-Coloring is NP-hard. Implicit in [Holyer ‘81, Leven Galil ’83, Papadimitriou ‘80].

Easily extends to hardness of approximation within a factor of 1 - t.

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Greedy Algorithm

Repeatedly take a maximum matching from the graph (and remove it).

The approximation ratio is at least The ratio may be as bad as ¾ (for every even t):

Greedy:

Optimum:

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The chromatic index ’(G): the least number of colors required to edge color all the edges of G.

In a simple graph:

(G) · ’(G) · (G) +1 [Vizing ‘64].

A Related Problem

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Algorithm for Simple Graphs

Find an edge subgraph with degree bounded by t, containing as many edges as possible [Edmonds and Johnson ’70].

Edge color the subgraph using at most t +1 colors (can be done using Vizing’s algorithm).

Take the t largest color classes.

Approximation ratio:

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Input: multigraph G, number of colors t = 4.

What About Multigraphs ?

Maximum subgraph with degree 4:

In each triangle only 4 edges can be colored (using 4 colors).

8 edges are colored.

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What About Multigraphs ?

Input: multigraph G, number of colors t = 4. The optimum solution has 12 edges:

The approximation is not better than (can be generalized for every even t).

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Let S be a subset of vertices of size : A matching over the edges in S has size . At least colors are needed. For odd S: it may be as large as .

Odd density:

What Went Wrong?

The maximum degree is not enough to predict the chromatic index, the chromatic index can be as large as :

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Let G be a multigraph. is the maximum degree of G and is

the odd density of G.

Lower bound: ’(G) ¸ max{, }. Upper bounds:

’(G) · max{1.1 + 0.8, } [Nishizeki, Kashiwagi ’90].

’(G) · (1 + o(1))max{, }, not constructive [Kahn ‘96].

Conjecture: ’(G) · max{ + 1, } [Seymour ‘79]

Bounds for the chromatic index ’(G)

algorithm [NK ’90]

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Modifying the Approach for Multigraphs

Find a maximum edge subgraph with , · t.

Edge color the subgraph using as few colors as possible (can be done using [NK ‘90]).

Take the t largest color classes.

This yields a approximation.

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Difficulty

The first step of finding a largest edge subgraph with , · t is NP-hard for t=2 (also for t > 2 ??).

Solution: Algorithm which “approximates” the first step:

OPT= number of edges in an optimal solution (, · t).

The algorithm outputs an edge subgraph G’ with:

(G’) · t+1, (G’) · t + O(t1/2).

The number of edges in G’ is at least OPT.

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Approximation Using a Linear Program

Every edge has value between 0 and 1.

The solution has degree and density bounded by t, however it contains fractional edges. Fractional edges are denoted by GF, integer edges denoted by GI.

Round the fractional edges of GF (in 3 steps).

Return GI plus Round(GF).

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Step 1 – Parallel Edges in GF

Handling parallel fractional edges:

The solution GF + GI remains valid. GF becomes a simple graph.

GIGF

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Steps 2 and 3: Round(GF)

Round the fractional edges of GF achieving an integral solution with the following properties:

Degrees exceed the original fractional degrees by at most 1.

The sum of weights of edges does not decrease.

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Putting All Together

Theorem: Adding Round(GF) to GI yields: Degrees are bounded by t+1. Odd densities are bounded by t + O(t1/2).

Proof: Consider S µ V : If |S| · t1/2: because GF contributes at most |S| edges

per vertex. If |S| ¸ t1/2: follows from degree bound.

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Find a fractional even length cycle in GF. Shift values from red edges to green edges until one

edge becomes integral (0/1):

The degrees of vertices in GF remain the same. Iterating this step eliminates all fractional even cycles

in GF.

Step 2 – Rounding GF

Case 1: Case 2:

GFGF

0

1

Gray edges: edges which become 0/1.

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Step 3 – Tree like structure of fractional GF

Odd cycles are vertex disjoint (otherwise an even cycle exists).

The fractional edges of GF form a tree like structure. A node of the tree is either a vertex of G or an odd cycle of G:

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LPval Gives a Good Prediction

The value of the LP predicts the optimal solution of Max Edge t-Coloring within a factor of 1+ot(1):

Round(LP) has: , · t+o(t) and value ¸ LPval . Round(LP) be colored with t+o(t) colors (Kahn ‘96). Thus there exist a solution for Max Edge t-Coloring

with value of (1-ot(1))LPval .

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Summary

Red – Greedy algorithm.Blue – LP based algorithm.Green – Special algorithm for t=2

with ratio of 0.77 .

t

ratio