Efficiently list-edge coloring multigraphs asymptotically optimally

Abstract

We give polynomial time algorithms for the seminal results of Kahn [18,19], who showed that the Goldberg-Seymour and List-Coloring conjectures for (list-)edge coloring multigraphs hold asymptotically. Kahn's arguments are based on the probabilistic method and are non-constructive. Our key insight is to show that the main result of Achlioptas, Iliopoulos and Kolmogorov [2] for analyzing local search algorithms can be used to make constructive applications of a powerful version of the so-called Lopsided Lovász Local Lemma. In particular, we use it to design algorithms that exploit the fact that correlations in the probability spaces on matchings used by Kahn decay with distance. algorithms and techniques are very different from those of [33,35]. Finally, as we will see, we will need to show that the algorithm of Theorem 1.3 is commutative, a notion introduced by Kolmogorov [22]. This fact may be of independent interest since, as shown in [22,14], commutative algorithms have several nice properties: they are typically parallelizable, their output distribution has high entropy, etc.As a final remark, we note that, to the best of our knowledge, Theorem 1.4 is the first result to give an asymptotically optimal polynomial time algorithm for list-edge coloring multigraphs. Technical OverviewThe proofs of Theorems 1.1 and 1.2 are based on a very sophisticated variation of what is known as the semi-random method (also known as the "naive coloring procedure"), which is the main technical tool behind some of the strongest graph coloring results, e.g., [16,17,21,25]. The idea is to gradually color the graph in iterations, until we reach a point where we can finish the coloring using a greedy algorithm. In its most basic form, each iteration consists of the following simple procedure (using vertex coloring as a canonical example): Assign to each vertex a color chosen uniformly at random; then uncolor any vertex which receives the same color as one of its neighbors. Using the Lovász Local Lemma (LLL) [9] and concentration inequalities, one typically shows that, with positive probability, the resulting partial proper coloring has useful properties that allow for the continuation of the argument in the next iteration. For a nice exposition of both the method and the proofs of Theorems 1.1 and 1.2, the reader is referred to [26].The key new ingredient in Kahn's arguments is the method of assigning colors. For each color c, we choose a matching M c from some hard-core distribution on M(G) and assign the color c to the edges in M c . The idea is that by assigning each color exclusively to the edges of one matching, we avoid conflicting color assignments and the resulting uncolorings.The existence of such hard-core distributions is guaranteed by the characterization of the matching polytope due to Edmonds [8] and a result by Lee [23] (also shown independently by Rabinovich et al. [32]). The crucial fact about them is that they are endowed with very useful approximate stochastic independence properties, as was shown by Kahn and Kayll ...

A new notion of commutativity for the algorithmic Lovász Local Lemma

“…The following result of [18] illustrates the connection between this measure and the Lopsided Lovász Local Lemma (LLLL): Theorem 4.1 ([18]). Given a family of flaws F and a measure µ over Ω, then for each set S ⊆ F − Γ(f ) we have µ f | g∈S g ≤ γ f , where the γ f are the charges of the algorithm as defined in (2).…”