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ConstantTime Approximation Algorithms via Local Improvements
"... We present a technique for transforming classical approximation algorithms into constanttime algorithms that approximate the size of the optimal solution. Our technique is applicable to a certain subclass of algorithms that compute a solution in a constant number of phases. The technique is based o ..."
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Cited by 42 (3 self)
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We present a technique for transforming classical approximation algorithms into constanttime algorithms that approximate the size of the optimal solution. Our technique is applicable to a certain subclass of algorithms that compute a solution in a constant number of phases. The technique is based on greedily considering local improvements in random order. The problems amenable to our technique include
Almost stable matchings by truncating the Gale–Shapley algorithm
 Algorithmica
, 2010
"... We show that the ratio of matched individuals to blocking pairs grows linearly with the number of propose–accept rounds executed by the Gale–Shapley algorithm for the stable marriage problem. Consequently, the participants can arrive at an almost stable matching even without full information about ..."
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We show that the ratio of matched individuals to blocking pairs grows linearly with the number of propose–accept rounds executed by the Gale–Shapley algorithm for the stable marriage problem. Consequently, the participants can arrive at an almost stable matching even without full information about the problem instance; for each participant, knowing only its local neighbourhood is enough. In distributedsystems parlance, this means that if each person has only a constant number of acceptable partners, an almost stable matching emerges after a constant number of synchronous communication rounds. We apply our results to give a distributed (2 + )approximation algorithm for maximumweight matching in bicoloured graphs and a centralised randomised constanttime approximation scheme for estimating the size of a stable matching. 1
Local Computation: Lower and Upper Bounds
"... The question of what can be computed, and how efficiently, are at the core of computer science. Not surprisingly, in distributed systems and networking research, an equally fundamental question is what can be computed in a distributed fashion. More precisely, if nodes of a network must base their de ..."
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The question of what can be computed, and how efficiently, are at the core of computer science. Not surprisingly, in distributed systems and networking research, an equally fundamental question is what can be computed in a distributed fashion. More precisely, if nodes of a network must base their decision on information in their local neighborhood only, how well can they compute or approximate a global (optimization) problem? In this paper we give the first substantial lower bound on such local computation for (optimization) problems including minimum vertex cover, minimum (connected) dominating set, maximum matching, maximal independent set, and maximal matching. In addition we present a new distributed algorithm for solving general covering and packing linear programs. For some problems this algorithm is tight with the lower bounds, for others it is a distributed approximation scheme. Together, our lower and upper bounds establish the local computability and approximability of a large class of problems, characterizing how much local information is required to solve these tasks.
Distance approximation in boundeddegree and general sparse graphs
 In Proceedings of the Tenth International Workshop on Randomization and Computation (RANDOM
, 2006
"... We address the problem of approximating the distance of bounded degree and general sparse graphs from having some predetermined graph property P. Namely, we are interested in sublinear algorithms for estimating the fraction of edges that should be added to / removed from a graph so that it obtains P ..."
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Cited by 15 (5 self)
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We address the problem of approximating the distance of bounded degree and general sparse graphs from having some predetermined graph property P. Namely, we are interested in sublinear algorithms for estimating the fraction of edges that should be added to / removed from a graph so that it obtains P. This fraction is taken with respect to a given upper bound m on the number of edges. In particular, for graphs with degree bound d over n vertices, m = dn. To perform such an approximation the algorithm may ask for the degree of any vertex of its choice, and may ask for the neighbors of any vertex. The problem of estimating the distance to having a property was first explicitly addressed by Parnas et. al. (ECCC 2004). In the context of graphs this problem was studied by Fischer and Newman (FOCS 2005) in the densegraphs model. In this model the fraction of edge modifications is taken with respect to n 2, and the algorithm may ask for the existence of an edge between any pair of vertices of its choice. Fischer and Newman showed that every graph property that has a testing algorithm in this model with query complexity that is independent of the size of the graph, also has a distanceapproximation algorithm with query complexity that is independent of the size of the graph. In this work we focus on boundeddegree and general sparse graphs, and give algorithms for all properties that were shown to have efficient testing algorithms by Goldreich and Ron (Algorithmica, 2002). Specifically, these properties are kedge connectivity, subgraphfreeness (for constant size subgraphs), being a Eulerian graph, and cyclefreeness. A variant of our subgraphfreeness algorithm approximates the size of a minimum vertex cover of a graph in sublinear time. This approximation improves on a recent result of Parnas and Ron (ECCC 2005).
Local Graph Partitions for Approximation and Testing
"... We introduce a new tool for approximation and testing algorithms called partitioning oracles. We develop methods for constructing them for any class of boundeddegree graphs with an excluded minor, and in general, for any hyperfinite class of boundeddegree graphs. These oracles utilize only local ..."
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We introduce a new tool for approximation and testing algorithms called partitioning oracles. We develop methods for constructing them for any class of boundeddegree graphs with an excluded minor, and in general, for any hyperfinite class of boundeddegree graphs. These oracles utilize only local computation to consistently answer queries about a global partition that breaks the graph into small connected components by removing only a small fraction of the edges. We illustrate the power of this technique by using it to extend and simplify a number of previous approximation and testing results for sparse graphs, as well as to provide new results that were unachievable with existing techniques. For instance: • We give constanttime approximation algorithms for the size of the minimum vertex cover, the minimum dominating set, and the maximum independent set for any class of graphs with an excluded minor. • We show a simple proof that any minorclosed graph property is testable in constant time in the bounded degree model. • We prove that it is possible to approximate the distance to almost any hereditary property in any bounded degree hereditary families of graphs. Hereditary properties of interest include bipartiteness, kcolorability, and perfectness.
Maintaining a Large Matching and a Small Vertex Cover
"... We consider the problem of maintaining a large matching and a small vertex cover in a dynamically changing graph. Each update to the graph is either an edge deletion or an edge insertion. We give the first randomized data structure that simultaneously achieves a constant approximation factor and han ..."
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We consider the problem of maintaining a large matching and a small vertex cover in a dynamically changing graph. Each update to the graph is either an edge deletion or an edge insertion. We give the first randomized data structure that simultaneously achieves a constant approximation factor and handles a sequence of K updates in K · polylog(n) time, where n is the number of vertices in the graph. Previous data structures require a polynomial amount of computation per update.
Approximating maxmin linear programs with local algorithms
 In Proc. 22nd IEEE International Parallel and Distributed Processing Symposium (IPDPS
, 2008
"... Abstract. A local algorithm is a distributed algorithm where each node must operate solely based on the information that was available at system startup within a constantsize neighbourhood of the node. We study the applicability of local algorithms to maxmin LPs where the objective is to maximise ..."
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Abstract. A local algorithm is a distributed algorithm where each node must operate solely based on the information that was available at system startup within a constantsize neighbourhood of the node. We study the applicability of local algorithms to maxmin LPs where the objective is to maximise mink v ckvxv subject to ∑ v aivxv ≤ 1 for each i and xv ≥ 0 for each v. Here ckv ≥ 0, aiv ≥ 0, and the support sets Vi = {v: aiv> 0}, Vk = {v: ckv> 0}, Iv = {i: aiv> 0} and Kv = {k: ckv> 0} have bounded size. In the distributed setting, each agent v is responsible for choosing the value of xv, and the communication network is a hypergraph H where the sets Vk and Vi constitute the hyperedges. We present inapproximability results for a wide range of structural assumptions; for example, even if Vi  and Vk  are bounded by some constants larger than 2, there is no local approximation scheme. To contrast the negative results, we present a local approximation algorithm which achieves good approximation ratios if we can bound the relative growth of the vertex neighbourhoods in H. 1.
Counting Stars and Other Small Subgraphs in Sublinear Time
"... Detecting and counting the number of copies of certain subgraphs (also known as network motifs or graphlets), is motivated by applications in a variety of areas ranging from Biology to the study of the WorldWideWeb. Several polynomialtime algorithms have been suggested for counting or detecting t ..."
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Cited by 10 (3 self)
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Detecting and counting the number of copies of certain subgraphs (also known as network motifs or graphlets), is motivated by applications in a variety of areas ranging from Biology to the study of the WorldWideWeb. Several polynomialtime algorithms have been suggested for counting or detecting the number of occurrences of certain network motifs. However, a need for more efficient algorithms arises when the input graph is very large, as is indeed the case in many applications of motif counting. In this paper we design sublineartime algorithms for approximating the number of copies of certain constantsize subgraphs in a graph G. That is, our algorithms do not read the whole graph, but rather query parts of the graph. Specifically, we consider algorithms that may query the degree of any vertex of their choice and may ask for any neighbor of any vertex of their choice. The main focus of this work is on the basic problem of counting the number of length2 paths and more generally on counting the number of stars of a certain size. Specifically, we design an algorithm that, given an approximation parameter 0 < ɛ < 1 and query access to a graph G, outputs an estimate ˆνs such that with high constant probability, (1−ɛ)νs(G) ≤ ˆνs ≤ (1+ɛ)νs(G), where νs(G) denotes the number of stars of size s + 1 in the graph. The expected query ( complexity and { running time of}) the algorithm are O
Fast Local Computation Algorithms
"... For input x, let F (x) denote the set of outputs that are the “legal ” answers for a computational problem F. Suppose x and members of F (x) are so large that there is not time to read them in their entirety. We propose a model of local computation algorithms which for a given input x, support queri ..."
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Cited by 9 (4 self)
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For input x, let F (x) denote the set of outputs that are the “legal ” answers for a computational problem F. Suppose x and members of F (x) are so large that there is not time to read them in their entirety. We propose a model of local computation algorithms which for a given input x, support queries by a user to values of specified locations yi in a legal output y ∈ F (x). When more than one legal output y exists for a given x, the local computation algorithm should output in a way that is consistent with at least one such y. Local computation algorithms are intended to distill the common features of several concepts that have appeared in various algorithmic subfields, including local distributed computation, local algorithms, locally decodable codes, and local reconstruction. We develop a technique, based on Beck’s analysis in his algorithmic approach to the Lovász Local Lemma, which under certain conditions can be applied to construct polylogarithmic time local computation algorithms. We apply this technique to maximal independent set computations, scheduling radio network broadcasts, hypergraph coloring and satisfying kSAT formulas.