On percolation and ‐hardness

Abstract

The edge-percolation and vertex-percolation random graph models start with an arbitrary graph G, and randomly delete edges or vertices of G with some fixed probability. We study the computational complexity of problems whose inputs are obtained by applying percolation to worst-case instances. Specifically, we show that a number of classical N P-hard problems on graphs remain essentially as hard on percolated instances as they are in the worst-case (assuming N P BPP). We also prove hardness results for other N P-hard problems such as Constraint Satisfaction Problems and Subset-Sum, with suitable definitions of random deletions. Along the way, we establish that for any given graph G the independence number α(G) and the chromatic number χ(G) are robust to percolation in the following sense. Given a graph G, let G � be the graph obtained by randomly deleting edges of G with some probability p ∈ (0, 1). We show that if α(G) is small, then α(G � ) remains small with probability at least 0.99. Similarly, we show that if χ(G) is large, then χ(G � ) remains large with probability at least 0.99. We believe these results are of independent interest.chromatic number, hardness of approximation, independence number, percolation, random subgraphs Random Struct Alg. 2018;00:1-30.wileyonlinelibrary.com/journal/rsa

Phase Transition of Degeneracy in Minor-Closed Families

“…Therefore, s = 3r − 4. By Statement 3 of Lemma 6.6, 3) ). Recall that either H is not a minor of K r−2 ∨ tK 3 for any positive integer t, or r ≥ 4 and H is not a minor of L t of any positive integer t. So p χ r M(H) = O(n −1/(3r−3) ) by Statements 2(a) and 3 of Corollary 3.6.…”

“…Besides the well-studied model G(n, p), rich theories have developed, including the percolation problem, modeling the spread of infectious disease in social network science, and the resilience problem to study the robustness of properties (see e.g. [22,3,10,44]. )…”

“…The aim of this paper is to determine the order of p P G . 3 In many natural random structures, it has been observed that phase transitions appear to influence the computational complexity. It has connections to the boundary between easy and hard approximation problems, such as approximate counting problems for the number of independent sets, random SAT problems, vertex cover problem or colorability in random graphs (see [12,32,42,43,49] for examples).…”

“…Note that M(K 3,3 ) contains the set of planar graphs, denoted by G planar . Since every planar graph on n vertices contains O(n) triangles (by Lemma 4.1) and every triangle-free planar graph is properly 3-colorable by Grőtzsch's theorem, we know p χ 3 G planar = Ω(n −1/3 ). However, we are able to provide the following better estimation for P χ 3 G planar by using Theorem 1.2 and Proposition 1.4.…”

Expected Chromatic Number of Random Subgraphs

“…The chromatic number is one of the most important parameters of a graph, and many problems in computer science-e.g., register allocation, pattern matching, and scheduling problems-can be reduced to finding the chromatic number of a given graph. In the probabilistic setting, the distribution of χ(G p ) is studied in statistical mechanics, where physicists use random subgraphs to model molecular interactions, and properties of the resulting graph colorings are predictive of various macroscopic features [4].…”