We consider the robustness of computational hardness of problems whose input is obtained by applying independent random deletions to worst-case instances. For some classical @math -hard problems on graphs, such as Coloring, Vertex-Cover, and Hamiltonicity, we examine the complexity of these problems when edges (or vertices) of an arbitrary graph are deleted independently with probability @math . We prove that for @math -vertex graphs, these problems remain as hard as in the worst-case, as long as @math for arbitrary @math , unless @math . We also prove hardness results for Constraint Satisfaction Problems, where random deletions are applied to clauses or variables, as well as the Subset-Sum problem, where items of a given instance are deleted at random.
When @math is sufficiently small, algorithms for random graphs and random formulas can be proven to find the optimal solution (with high probability) for percolated instance. For example, for graph coloring, it is known that for @math with some positive constant @math , with high probability @math is @math -degenerate, and hence can be 3-colored in polynomial time @cite_2 . Since the property of being 2-degenerate is monotone, and as @math -colorability can be decided in polynomial time, it follows that for every @math -vertex graph and @math , one can find in polynomial time with high probability a coloring of the edge percolated graph with the minimum number of colors.
@cite_2
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For many random constraint satisfaction problems, by now there exist asymptotically tight estimates of the largest constraint density for which solutions exist. At the same time, for many of these problems, all known polynomial-time algorithms stop finding solutions at much smaller densities. For example, it is well-known that it is easy to color a random graph using twice as many colors as its chromatic number. Indeed, some of the simplest possible coloring algorithms achieve this goal. Given the simplicity of those algorithms, one would expect room for improvement. Yet, to date, no algorithm is known that uses (2 - epsiv)chi colors, in spite of efforts by numerous researchers over the years. In view of the remarkable resilience of this factor of 2 against every algorithm hurled at it, we find it natural to inquire into its origin. We do so by analyzing the evolution of the set of k-colorings of a random graph, viewed as a subset of 1,...,k n, as edges are added. We prove that the factor of 2 corresponds in a precise mathematical sense to a phase transition in the geometry of this set. Roughly speaking, we prove that the set of k-colorings looks like a giant ball for k ges 2chi, but like an error-correcting code for k les (2 - epsiv)chi. We also prove that an analogous phase transition occurs both in random k-SAT and in random hypergraph 2-coloring. And that for each of these three problems, the location of the transition corresponds to the point where all known polynomial-time algorithms fail. To prove our results we develop a general technique that allows us to establish rigorously much of the celebrated 1-step replica-symmetry-breaking hypothesis of statistical physics for random CSPs. Consider an n-vertex graph G = (V, E) of maximum degree Δ, and suppose that each vertex v ∈ V hosts a processor. The processors are allowed to communicate only with their neighbors in G. The communication is synchronous, that is, it proceeds in discrete rounds. In the distributed vertex coloring problem, the objective is to color G with Δ + 1, or slightly more than Δ + 1, colors using as few rounds of communication as possible. (The number of rounds of communication will be henceforth referred to as running time.) Efficient randomized algorithms for this problem are known for more than twenty years [ 1986; Luby 1986]. Specifically, these algorithms produce a (Δ + 1)-coloring within O(log n) time, with high probability. On the other hand, the best known deterministic algorithm that requires polylogarithmic time employs O(Δ2) colors. This algorithm was devised in a seminal FOCS’87 paper by Linial [1987]. Its running time is O(log* n). In the same article, Linial asked whether one can color with significantly less than Δ2 colors in deterministic polylogarithmic time. By now, this question of Linial became one of the most central long-standing open questions in this area. In this article, we answer this question in the affirmative, and devise a deterministic algorithm that employs Δ1+o(1) colors, and runs in polylogarithmic time. Specifically, the running time of our algorithm is O(f(Δ)log Δ log n), for an arbitrarily slow-growing function f(Δ) = ω(1). We can also produce an O(Δ1+η)-coloring in O(log Δ log n)-time, for an arbitrarily small constant η > 0, and an O(Δ)-coloring in O(Δe log n) time, for an arbitrarily small constant e > 0. Our results are, in fact, far more general than this. In particular, for a graph of arboricity a, our algorithm produces an O(a1+η)-coloring, for an arbitrarily small constant η > 0, in time O(log a log n). We study the following one-person game against a random graph process: the Player's goal is to @math -colour a random sequence of edges @math of a complete graph on @math vertices, avoiding a monochromatic triangle for as long as possible. The game is over when a monochromatic triangle is created. The online version of the game requires that the Player should colour each edge as it comes, before looking at the next edge.While it is not hard to prove that the expected length of this game is about @math , the proof of the upper bound suggests the following relaxation: instead of colouring online, the random graph is generated in only two rounds, and the Player colours the edges of the first round before the edges of the second round are thrown in. Given the size of the first round, how many edges can there be in the second round for the Player to be likely to win? In the extreme case, when the first round consists of a random graph with @math edges, where @math is a positive constant, we show that the Player can win with high probability only if constantly many edges are generated in the second round. We study deterministic, distributed algorithms for two weak variants of the standard graph coloring problem. We consider defective colorings, i.e., colorings where nodes of a color class may induce a graph of maximum degree d for some parameter d>0. We also look at colorings where a minimum number of multi-chromatic edges is required. For an integer k>0, we call a coloring k-partially proper if every node v has at least min k,deg(v) neighbors with a different color. We show that for all d∈ 1,...,Δ , it is possible to compute a O(Δ2 d2)-coloring with defect d in time O(log*n) where Δ is the largest degree of the network graph. Similarly, for all k∈ 1,...,Δ , a k-partially proper O(k2)-coloring can be computed in O(log*n) rounds. As an application of our weak defective coloring algorithm, we obtain a faster deterministic algorithm for the standard vertex coloring problem on graphs with moderate degrees. We show that in time O(Δ+log*n), a (Δ+1)-coloring can be computed, a task for which the best previous algorithm required time O(Δ*log(Δ) + log*n). The same result holds for the problem of computing a maximal independent set.