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
2153584512 2007385254 2568310503
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. We describe random sampling techniques for approximately solving problems that involve cuts and flows in graphs. We give a near-linear-time randomized combinatorial construction that transforms any graph on @math vertices into an @math -edge graph on the same vertices whose cuts have approximately the same value as the original graph's. In this new graph, for example, we can run the @math -time maximum flow algorithm of Goldberg and Rao to find an @math - @math minimum cut in @math time. This corresponds to a @math -times minimum @math - @math cut in the original graph. A related approach leads to a randomized divide-and-conquer algorithm producing an approximately maximum flow in @math time. Our algorithm can also be used to improve the running time of sparsest cut approximation algorithms from @math to @math and to accelerate several other recent cut and flow algorithms. Our algorithms are based on a general theorem analyzing the concent... The binary symmetric stochastic block model deals with a random graph of @math vertices partitioned into two equal-sized clusters, such that each pair of vertices is connected independently with probability @math within clusters and @math across clusters. In the asymptotic regime of @math and @math for fixed @math and @math , we show that the semidefinite programming relaxation of the maximum likelihood estimator achieves the optimal threshold for exactly recovering the partition from the graph with probability tending to one, resolving a conjecture of Abbe14 . Furthermore, we show that the semidefinite programming relaxation also achieves the optimal recovery threshold in the planted dense subgraph model containing a single cluster of size proportional to @math . We present algorithmic and hardness results for network design problems with degree or order constraints. The first problem we consider is the Survivable Network Design problem with degree constraints on vertices. The objective is to find a minimum cost subgraph which satisfies connectivity requirements between vertices and also degree upper bounds @math on the vertices. This includes the well-studied Minimum Bounded Degree Spanning Tree problem as a special case. Our main result is a @math -approximation algorithm for the edge-connectivity Survivable Network Design problem with degree constraints, where the cost of the returned solution is at most twice the cost of an optimum solution (satisfying the degree bounds) and the degree of each vertex @math is at most @math . This implies the first constant factor (bicriteria) approximation algorithms for many degree constrained network design problems, including the Minimum Bounded Degree Steiner Forest problem. Our results also extend to directed graphs and provide the first constant factor (bicriteria) approximation algorithms for the Minimum Bounded Degree Arborescence problem and the Minimum Bounded Degree Strongly @math -Edge-Connected Subgraph problem. In contrast, we show that the vertex-connectivity Survivable Network Design problem with degree constraints is hard to approximate, even when the cost of every edge is zero. A striking aspect of our algorithmic result is its simplicity. It is based on the iterative relaxation method, which is an extension of Jain's iterative rounding method. This provides an elegant and unifying algorithmic framework for a broad range of network design problems. We also study the problem of finding a minimum cost @math -edge-connected subgraph with at least @math vertices, which we call the @math -subgraph problem. This generalizes some well-studied classical problems such as the @math -MST and the minimum cost @math -edge-connected subgraph problems. We give a polylogarithmic approximation for the @math -subgraph problem. However, by relating it to the Densest @math -Subgraph problem, we provide evidence that the @math -subgraph problem might be hard to approximate for arbitrary @math . Distributed voting is a fundamental topic in distributed computing. In the standard model of pull voting, at each step every vertex chooses a neighbour uniformly at random and adopts its opinion. The voting is completed when all vertices hold the same opinion. In the simplest case, each vertex initially holds one of two different opinions. This partitions the vertices into arbitrary sets A and B. For many graphs, including regular graphs and irrespective of their expansion properties, if both A and B are sufficiently large sets, then pull voting requires @math expected steps, where n is the number of vertices of the graph. In this paper we consider a related class of voting processes based on sampling two opinions. In the simplest case, every vertex v chooses two random neighbours at each step. If both these neighbours have the same opinion, then v adopts this opinion. Otherwise, v keeps its own opinion. Let G be a connected graph with n vertices and m edges. Let P be the transition matrix of a simple random walk on G with second largest eigenvalue @math . We show that if the initial imbalance in degree between the two opinions satisfies @math , then with high probability voting completes in @math steps, and the opinion with the larger initial degree wins. The condition that @math , or only a bound on the conductance of the graph is known, the sampling process can be modified so that voting still provably completes in @math steps with high probability. The modification uses two sampling based on probing to a fixed depth @math from any vertex. In its most general form our voting process allows vertices to bias their sampling of opinions among their neighbours to achieve a desired outcome. This is done by allocating weights to edges. We prove two results regarding cycles in properly edge-colored graphs. First, we make a small improvement to the recent breakthrough work of Alon, Pokrovskiy and Sudakov who showed that every properly edge-colored complete graph @math on @math vertices has a rainbow cycle on at least @math vertices, by showing that @math has a rainbow cycle on at least @math vertices. Second, by modifying the argument of Hatami and Shor which gives a lower bound for the length of a partial transversal in a Latin Square, we prove that every properly colored complete graph has a Hamilton cycle in which at least @math different colors appear. For large @math , this is an improvement of the previous best known lower bound of @math of Andersen. We use random sampling as a tool for solving undirected graph problems. We show that the sparse graph, or skeleton, that arises when we randomly sample a graph's edges will accurately approximate the value of all cuts in the original graph with high probability. This makes sampling effective for problems involving cuts in graphs. We present fast randomized (Monte Carlo and Las Vegas) algorithms for approximating and exactly finding minimum cuts and maximum flows in unweighted, undirected graphs. Our cut-approximation algorithms extend unchanged to weighted graphs while our weighted-graph flow algorithms are somewhat slower. Our approach gives a general paradigm with potential applications to any packing problem. It has since been used in a near-linear time algorithm for finding minimum cuts, as well as faster cut and flow algorithms. Our sampling theorems also yield faster algorithms for several other cut-based problems, including approximating the best balanced cut of a graph, finding a k-connected orientation of a 2k-connected graph, and finding integral multicommodity flows in graphs with a great deal of excess capacity. Our methods also improve the efficiency of some parallel cut and flow algorithms. Our methods also apply to the network design problem, where we wish to build a network satisfying certain connectivity requirements between vertices. We can purchase edges of various costs and wish to satisfy the requirements at minimum total cost. Since our sampling theorems apply even when the sampling probabilities are different for different edges, we can apply randomized rounding to solve network design problems. This gives approximation algorithms that guarantee much better approximations than previous algorithms whenever the minimum connectivity requirement is large. As a particular example, we improve the best approximation bound for the minimum k-connected subgraph problem from 1.85 to [math not displayed]. We study a random graph model named the "block model" in statistics and the "planted partition model" in theoretical computer science. In its simplest form, this is a random graph with two equal-sized clusters, with a between-class edge probability of @math and a within-class edge probability of @math . A striking conjecture of Decelle, Krzkala, Moore and Zdeborov 'a based on deep, non-rigorous ideas from statistical physics, gave a precise prediction for the algorithmic threshold of clustering in the sparse planted partition model. In particular, if @math and @math , @math and @math then conjectured that it is possible to efficiently cluster in a way correlated with the true partition if @math and impossible if @math for some sufficiently large @math . In a previous work, we proved that indeed it is information theoretically impossible to to cluster if @math . A different independent proof of the same result was recently obtained by Laurent Massoulie. Determining whether the vertices of a graph can be colored using @math different colors so that no two adjacent vertices receive the same color is a well-known NP-complete problem. Graph coloring is also of practical interest (for example, in estimating sparse Jacobians and in scheduling), and many heuristic algorithms have been developed. We present a heuristic algorithm based on the eigenvalue decomposition of the adjacency matrix of a graph. Eigenvectors point out bipartite-looking'''' subgraphs that are used to refine the coloring to a valid coloring. The algorithm optimally colors complete @math -partite graphs and certain other classes of graphs with regular structure. In this paper we show a new way of constructing deterministic polynomial-time approximation algorithms for computing complex-valued evaluations of a large class of graph polynomials on bounded degree graphs. In particular, our approach works for the Tutte polynomial and independence polynomial, as well as partition functions of complex-valued spin and edge-coloring models. More specifically, we define a large class of graph polynomials @math and show that if @math and there is a disk @math centered at zero in the complex plane such that @math does not vanish on @math for all bounded degree graphs @math , then for each @math in the interior of @math there exists a deterministic polynomial-time approximation algorithm for evaluating @math at @math . This gives an explicit connection between absence of zeros of graph polynomials and the existence of efficient approximation algorithms, allowing us to show new relationships between well-known conjectures. Our work builds on a recent line of work initiated b... Given two graphs @math and @math , the Subgraph Isomorphism problem asks if @math is isomorphic to a subgraph of @math . While NP-hard in general, algorithms exist for various parameterized versions of the problem: for example, the problem can be solved (1) in time @math using the color-coding technique of Alon, Yuster, and Zwick; (2) in time @math using Courcelle's Theorem; (3) in time @math using a result on first-order model checking by Frick and Grohe; or (4) in time @math for connected @math using the algorithm of Matou s ek and Thomas. Already this small sample of results shows that the way an algorithm can depend on the parameters is highly nontrivial and subtle. We develop a framework involving 10 relevant parameters for each of @math and @math (such as treewidth, pathwidth, genus, maximum degree, number of vertices, number of components, etc.), and ask if an algorithm with running time [ f_1(p_1,p_2,..., p_ ) n^ f_2(p_ +1 ,..., p_k) ] exist, where each of @math is one of the 10 parameters depending only on @math or @math . We show that all the questions arising in this framework are answered by a set of 11 maximal positive results (algorithms) and a set of 17 maximal negative results (hardness proofs); some of these results already appear in the literature, while others are new in this paper. On the algorithmic side, our study reveals for example that an unexpected combination of bounded degree, genus, and feedback vertex set number of @math gives rise to a highly nontrivial algorithm for Subgraph Isomorphism. On the hardness side, we present W[1]-hardness proofs under extremely restricted conditions, such as when @math is a bounded-degree tree of constant pathwidth and @math is a planar graph of bounded pathwidth. Consider the following probabilistic one-player game: The board is a graph with @math vertices, which initially contains no edges. In each step, a new edge is drawn uniformly at random from all nonedges and is presented to the player, henceforth called Painter. Painter must assign one of @math available colors to each edge immediately, where @math is a fixed integer. The game is over as soon as a monochromatic copy of some fixed graph @math has been created, and Painter's goal is to “survive” for as many steps as possible before this happens. We present a new technique for deriving upper bounds on the threshold of this game, i.e., on the typical number of steps Painter will survive with an optimal strategy. More specifically, we consider a deterministic two-player variant of the game where the edges are chosen not randomly, but by a second player Builder. However, Builder has to adhere to the restriction that, for some real number @math , the ratio of edges to vertices in all subgraphs of the evolving board ne... Bootstrap percolation on an arbitrary graph has a random initial configuration, where each vertex is occupied with probability @math , independently of each other, and a deterministic spreading rule with a fixed parameter @math : if a vacant site has at least @math occupied neighbours at a certain time step, then it becomes occupied in the next step. This process is well studied on @math ; here we investigate it on regular and general infinite trees and on non-amenable Cayley graphs. The critical probability is the infimum of those values of @math for which the process achieves complete occupation with positive probability. On trees we find the following discontinuity: if the branching number of a tree is strictly smaller than @math , then the critical probability is 1, while it is @math on the @math -ary tree. A related result is that in any rooted tree @math there is a way of erasing @math children of the root, together with all their descendants, and repeating this for all remaining children, and so on, such that the remaining tree @math has branching number @math . We also prove that on any @math -regular non-amenable graph, the critical probability for the @math -rule is strictly positive. This paper addresses the cornerstone family of in distributed computing, and investigates the curious gap between randomized and deterministic solutions under bandwidth restrictions. Our main contribution is in providing tools for derandomizing solutions to local problems, when the @math nodes can only send @math -bit messages in each round of communication. We combine bounded independence, which we show to be sufficient for some algorithms, with the method of conditional expectations and with additional machinery, to obtain the following results. Our techniques give a deterministic maximal independent set (MIS) algorithm in the CONGEST model, where the communication graph is identical to the input graph, in @math rounds, where @math is the diameter of the graph. The best known running time in terms of @math alone is @math , which is super-polylogarithmic, and requires large messages. For the CONGEST model, the only known previous solution is a coloring-based @math -round algorithm, where @math is the maximal degree in the graph. On the way to obtaining the above, we show that in the model, which allows all-to-all communication, there is a deterministic MIS algorithm that runs in @math rounds. , where @math is the maximum degree. When @math , the bound improves to @math and holds also for @math -coloring. In addition, we deterministically construct a @math -spanner with @math edges in @math rounds. For comparison, in the more stringent CONGEST model, the best deterministic algorithm for constructing a @math -spanner with @math edges runs in @math rounds. A growing family of random graphs is called robust if it retains a giant component after percolation with arbitrary positive retention probability. We study robustness for graphs, in which new vertices are given a spatial position on the @math -dimensional torus and are connected to existing vertices with a probability favouring short spatial distances and high degrees. In this model of a scale-free network with clustering we can independently tune the power law exponent @math of the degree distribution and the rate @math at which the connection probability decreases with the distance of two vertices. We show that the network is robust if @math . In the case of one-dimensional space we also show that the network is not robust if @math . This implies that robustness of a scale-free network depends not only on its power-law exponent but also on its clustering features. Other than the classical models of scale-free networks our model is not locally tree-like, and hence we need to develop novel methods for its study, including, for example, a surprising application of the BK-inequality. We present an algorithm for testing the k-vertex-connectivity of graphs with the given maximum degree. The time complexity of the algorithm is independent of the number of vertices and edges of graphs. Fixed degree bound d, a graph G with n vertices and a maximum degree at most d is called e-far from k-vertex-connectivity when at least @math edges must be added to or removed from G to obtain a k-vertex-connected graph with a maximum degree at most d. The algorithm always accepts every graph that is k-vertex-connected and rejects every graph that is e-far from k-vertex-connectivity with a probability of at least 2 3. The algorithm runs in @math time (c>1 is a constant) for (k−1)-vertex-connected graphs, and in @math time (c>1 is a constant) for general graphs. It is the first constant-time k-vertex-connectivity testing algorithm for general k≥4. Edge and vertex connectivity are fundamental concepts in graph theory. While they have been thoroughly studied in the case of undirected graphs, surprisingly not much has been investigated for directed graphs. In this paper we study @math -edge connectivity problems in directed graphs and, in particular, we consider the computation of the following natural relation: We say that two vertices @math and @math are @math -edge-connected if there are two edge-disjoint paths from @math to @math and two edge-disjoint paths from @math to @math . This relation partitions the vertices into blocks such that all vertices in the same block are @math -edge-connected. Differently from the undirected case, those blocks do not correspond to the @math -edge-connected components of the graph. We show how to compute this relation in linear time so that we can report in constant time if two vertices are @math -edge-connected. We also show how to compute in linear time a sparse certificate for this relation, i.e., a subgraph of the input graph that has @math edges and maintains the same @math -edge-connected blocks as the input graph.