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.
Randomly subsampling subgraphs by including each edge independently in the sample with probability @math has been studied extensively in works concerned with cuts and flows (e.g., @cite_16 ). More recently, sampling subgraphs has been used to find independent sets @cite_4 (the main sampling technique used, e.g., the layers model is not independent-it introduces dependencies between sampled vertices). The effect of subsampling variables in mathematical relaxations of constraint satisfaction problems on the value of these relaxations was studied in @cite_19 . Edge-percolated graphs have been also used to construct hard-instance for specific algorithms. For example, @cite_1 proved that the well known greedy coloring algorithm performs poorly on the complete @math -partite graph in which every edge is removed independently with probability @math and @math for @math . Namely, for this graph @math , even if vertices are considered in a random order by the greedy algorithm, with high probability @math colors are used to color the percolated graph whereas @math .
@cite_1 @cite_19 @cite_16 @cite_4
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Let G3n,p,3 be a random 3-colorable graph on a set of 3n vertices generated as follows. First, split the vertices arbitrarily into three equal color classes, and then choose every pair of vertices of distinct color classes, randomly and independently, to be edges with probability p. We describe a polynomial-time algorithm that finds a proper 3-coloring of G3n,p,3 with high probability, whenever p @math c n, where c is a sufficiently large absolute constant. This settles a problem of Blum and Spencer, who asked if an algorithm can be designed that works almost surely for p @math polylog(n) n [J. Algorithms, 19 (1995), pp. 204--234]. The algorithm can be extended to produce optimal k-colorings of random k-colorable graphs in a similar model as well as in various related models. Implementation results show that the algorithm performs very well in practice even for moderate values of c. Consider an Erdos---Renyi random graph in which each edge is present independently with probability @math 1 2, except for a subset @math CN of the vertices that form a clique (a completely connected subgraph). We consider the problem of identifying the clique, given a realization of such a random graph. The algorithm of (ANALCO. SIAM, pp 67---75, 2011) provably identifies the clique @math CN in linear time, provided @math |CN|?1.261N. Spectral methods can be shown to fail on cliques smaller than @math N. In this paper we describe a nearly linear-time algorithm that succeeds with high probability for @math |CN|?(1+?)N e for any @math ?>0. This is the first algorithm that provably improves over spectral methods. We further generalize the hidden clique problem to other background graphs (the standard case corresponding to the complete graph on @math N vertices). For large-girth regular graphs of degree @math (Δ+1) we prove that so-called local algorithms succeed if @math |CN|?(1+?)N eΔ and fail if @math |CN|≤(1-?)N eΔ. In the subgraph-freeness problem, we are given a constant-size graph @math , and wish to determine whether the network contains @math as a subgraph or not. The relaxation of the problem only requires us to distinguish graphs that are @math -free from graphs that are @math -far from @math -free, meaning an @math -fraction of their edges must be removed to obtain an @math -free graph. Recently, Censor-Hillel et. al. and showed that in the property-testing regime it is possible to test @math -freeness for any graph @math of size 4 in constant time, @math rounds, regardless of the network size. However, Fraigniaud et. al. also showed that their techniques for graphs @math of size 4 cannot test @math -cycle-freeness in constant time. In this paper we revisit the subgraph-freeness problem and show that @math -cycle-freeness, and indeed @math -freeness for many other graphs @math comprising more than 4 vertices, can be tested in constant time. We show that @math -freeness can be tested in @math rounds for any cycle @math , improving on the running time of @math of the previous algorithms for triangle-freeness and @math -freeness. In the special case of triangles, we show that triangle-freeness can be solved in @math rounds independently of @math , when @math is not too small with respect to the number of nodes and edges. We also show that @math -freeness for any constant-size tree @math can be tested in @math rounds, even without the property-testing relaxation. Building on these results, we define a general class of graphs for which we can test subgraph-freeness in @math rounds. This class includes all graphs over 5 vertices except the 5-clique, @math . For cliques @math over @math nodes, we show that @math -freeness can be tested in @math rounds, where @math is the number of edges. Betweenness centrality is a fundamental measure in social network analysis, expressing the importance or influence of individual vertices (or edges) in a network in terms of the fraction of shortest paths that pass through them. Since exact computation in large networks is prohibitively expensive, we present two efficient randomized algorithms for betweenness estimation. The algorithms are based on random sampling of shortest paths and offer probabilistic guarantees on the quality of the approximation. The first algorithm estimates the betweenness of all vertices (or edges): all approximate values are within an additive factor @math ??(0,1) from the real values, with probability at least @math 1-?. The second algorithm focuses on the top-K vertices (or edges) with highest betweenness and estimate their betweenness value to within a multiplicative factor @math ?, with probability at least @math 1-?. This is the first algorithm that can compute such approximation for the top-K vertices (or edges). By proving upper and lower bounds to the VC-dimension of a range set associated with the problem at hand, we can bound the sample size needed to achieve the desired approximations. We obtain sample sizes that are independent from the number of vertices in the network and only depend on a characteristic quantity that we call the vertex-diameter, that is the maximum number of vertices in a shortest path. In some cases, the sample size is completely independent from any quantitative property of the graph. An extensive experimental evaluation on real and artificial networks shows that our algorithms are significantly faster and much more scalable as the number of vertices grows than other algorithms with similar approximation guarantees.