Semi-random Models Research Articles

On the tractability of coloring semirandom graphs

As part of the efforts to understand the intricacies of the k-colorability problem, different distributions over k-colorable graphs have been analyzed. While the problem is notoriously hard (not even reasonably approximable) in the worst case, the average case (with respect to such distributions) often turns out to be “easy”. Semi-random models mediate between these two extremes and are more suitable to imitate “real-life” instances than purely random models. In this work we consider semi-random variants of the planted k-colorability distribution. This continues a line of research pursued by Coja-Oghlan, and by Krivelevich and Vilenchik. Our aim is to study a more general semi-random framework than those suggested so far. On the one hand we show that previous algorithmic techniques extend to our more general semi-random setting; on the other hand we give a hardness result, proving that a closely related semi-random model is intractable. Thus we provide some indication about which properties of the input distribution make the k-colorability problem hard.

Finding large cliques in sparse semi-random graphs by simple randomized search heuristics

Surprisingly, general heuristics often solve some instances of hard combinatorial problems quite sufficiently, although they do not outperform specialized algorithms. Here, the behavior of simple randomized optimizers on the maximum clique problem is investigated. We focus on semi-random models for sparse graphs, in which an adversary is even allowed to insert a limited number of edges (and not only to remove them). In the course of these investigations the approximation behavior on general graphs and the optimization behavior for sparse graphs and further semi-random graph models are also considered. With regard to the optimizers, particular interest is given to the influences of the population size and the search operator.

Minimum Coloring k-Colorable Graphs in Polynomial Average Time

We present algorithms for minimum (χ(G)-) coloring k-colorable graphs drawn from random and semi-random models. In both models, an adversary initially splits the vertices into k color classes V1,…,Vk of sizes Ω(n) each. In the first model, each potential edge joining vertices in different classes is included with independent probability p. In the latter model, the adversary fixes an ordering of all potential edges and then for each i, the ith edge (according to this ordering) is included with probability p(ei,S)≥p,S⊆{1,…,i−1}. Thus, the probability of inclusion for the ith edge depends on the outcomes on previous i−1 edges. Semi-random models were introduced as a way of striking a balance between random graphs and worst-case adversaries. Our algorithms run in polynomial time on average. χ(G)-coloring is harder than k-coloring because even a “short certificate” is not presently known for the optimality of a coloring. Our algorithms work as long as p≥n−α(k)+ϵ for the semi-random model and p≥n−γ(k)+ϵ for the random model where α(k)=(2k)/((k−1)(k+2)), γ(k)=(2k)/(k2−k+2), and ϵ is any positive constant.