Finding independent transversals efficiently

Abstract

We give an efficient algorithm that, given a graph G and a partition V1,…,V m of its vertex set, finds either an independent transversal (an independent set {v1,…,v m } in G such that ${v_i} \in {V_i}$ for each i), or a subset ${\cal B}$ of vertex classes such that the subgraph of G induced by $\bigcup\nolimits_{\cal B}$ has a small dominating set. A non-algorithmic proof of this result has been known for a number of years and has been used to solve many other problems. Thus we are able to give algorithmic versions of many of these applications, a few of which we describe explicitly here.

Deterministic algorithms for the Lovász local lemma: Simpler, more general, and more parallel

“…Aharoni et al [1] showed that when b ≥ 2Δ, there exists an IT I with w(I) ≥ w(V)∕b. The work [10], building on [9], gives a nearly-matching randomized algorithm under the condition b ≥ (2 + 𝜀)Δ for constant 𝜀 > 0. Similarly, when w(v) ≥ 0 for all v (we say in this case that w is non-negative) and b ≥ 4Δ, then [19] shows that the randomized MT algorithm directly gives an IT I with w(I) ≥ Ω(w(V)∕b).…”

“…Many combinatorial problems, such as graph list‐coloring, can be formulated in terms of independent transversals; see [8] for a more extensive background. One fundamental problem is to determine sufficient conditions for the existence of an independent transversal in a graph.…”

“…Aharoni et al [1] showed that when $$$ b\ge 2\Delta $$$, there exists an IT $$$ I $$$ with $$$ w(I)\ge w(V)/b $$$. The work [10], building on [9], gives a nearly‐matching randomized algorithm under the condition $$$ b\ge \left(2+\varepsilon \right)\Delta $$$ for constant $$$ \varepsilon >0 $$$. Similarly, when $$$ w(v)\ge 0 $$$ for all $$$ v $$$ (we say in this case that $$$ w $$$ is non‐negative ) and $$$ b\ge 4\Delta $$$, then [19] shows that the randomized MT algorithm directly gives an IT $$$ I $$$ with $$$ w(I)\ge \Omega \left(w(V)/b\right) $$$.…”

Deterministic algorithms for the Lovasz Local Lemma: simpler, more general, and more parallel

“…Many combinatorial problems can be formulated in terms of independent transversals, such as satisfiability and graph list-coloring. See [12] for a more extensive background.…”

Degree criteria and stability for independent transversals

“…When no special information is known about the structure of the graph with respect to the vertex partition, the combinatorial and topological methods appear to work best, giving the best possible results in many cases. One example is the following [14,15], where a partition  is said to be t-thick if each of its blocks has size at least t, and as usual G Δ( ) denotes the maximum degree of G. (This is one of the most frequently applied IT theorems, see, e.g., [12] and the references therein.) Theorem 1 (Haxell [14,15]).…”