24 research outputs found

    Maximum flow is approximable by deterministic constant-time algorithm in sparse networks

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    We show a deterministic constant-time parallel algorithm for finding an almost maximum flow in multisource-multitarget networks with bounded degrees and bounded edge capacities. As a consequence, we show that the value of the maximum flow over the number of nodes is a testable parameter on these networks.Comment: 8 page

    A Quasi-Polynomial Time Partition Oracle for Graphs with an Excluded Minor

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    Motivated by the problem of testing planarity and related properties, we study the problem of designing efficient {\em partition oracles}. A {\em partition oracle} is a procedure that, given access to the incidence lists representation of a bounded-degree graph G=(V,E)G= (V,E) and a parameter \eps, when queried on a vertex v∈Vv\in V, returns the part (subset of vertices) which vv belongs to in a partition of all graph vertices. The partition should be such that all parts are small, each part is connected, and if the graph has certain properties, the total number of edges between parts is at most \eps |V|. In this work we give a partition oracle for graphs with excluded minors whose query complexity is quasi-polynomial in 1/\eps, thus improving on the result of Hassidim et al. ({\em Proceedings of FOCS 2009}) who gave a partition oracle with query complexity exponential in 1/\eps. This improvement implies corresponding improvements in the complexity of testing planarity and other properties that are characterized by excluded minors as well as sublinear-time approximation algorithms that work under the promise that the graph has an excluded minor.Comment: 13 pages, 1 figur

    What does the local structure of a planar graph tell us about its global structure?

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    The local k-neighborhood of a vertex v in an unweighted graph G = (V,E) with vertex set V and edge set E is the subgraph induced by all vertices of distance at most k from v. The rooted k-neighborhood of v is also called a k-disk around vertex v. If a graph has maximum degree bounded by a constant d, and k is also constant, the number of isomorphism classes of k-disks is constant as well. We can describe the local structure of a bounded-degree graph G by counting the number of isomorphic copies in G of each possible k-disk. We can summarize this information in form of a vector that has an entry for each isomorphism class of k-disks. The value of the entry is the number of isomorphic copies of the corresponding k-disk in G. We call this vector frequency vector of k-disks. If we only know this vector, what does it tell us about the structure of G? In this paper we will survey a series of papers in the area of Property Testing that leads to the following result (stated informally): There is a k = k(ε,d) such that for any planar graph G its local structure (described by the frequency vector of k-disks) determines G up to insertion and deletion of at most εd n edges (and relabelling of vertices)

    Random local algorithms

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    Consider the problem when we want to construct some structure on a bounded degree graph, e.g. an almost maximum matching, and we want to decide about each edge depending only on its constant radius neighbourhood. We show that the information about the local statistics of the graph does not help here. Namely, if there exists a random local algorithm which can use any local statistics about the graph, and produces an almost optimal structure, then the same can be achieved by a random local algorithm using no statistics.Comment: 9 page

    A Sublinear Tester for Outerplanarity (and Other Forbidden Minors) With One-Sided Error

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    We consider one-sided error property testing of F\mathcal{F}-minor freeness in bounded-degree graphs for any finite family of graphs F\mathcal{F} that contains a minor of K2,kK_{2,k}, the kk-circus graph, or the (k×2)(k\times 2)-grid for any k∈Nk\in\mathbb{N}. This includes, for instance, testing whether a graph is outerplanar or a cactus graph. The query complexity of our algorithm in terms of the number of vertices in the graph, nn, is O~(n2/3/ϵ5)\tilde{O}(n^{2/3} / \epsilon^5). Czumaj et~al.\ showed that cycle-freeness and CkC_k-minor freeness can be tested with query complexity O~(n)\tilde{O}(\sqrt{n}) by using random walks, and that testing HH-minor freeness for any HH that contains a cycles requires Ω(n)\Omega(\sqrt{n}) queries. In contrast to these results, we analyze the structure of the graph and show that either we can find a subgraph of sublinear size that includes the forbidden minor HH, or we can find a pair of disjoint subsets of vertices whose edge-cut is large, which induces an HH-minor.Comment: extended to testing outerplanarity, full version of ICALP pape

    An Efficient Partitioning Oracle for Bounded-Treewidth Graphs

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    Partitioning oracles were introduced by Hassidim et al. (FOCS 2009) as a generic tool for constant-time algorithms. For any epsilon > 0, a partitioning oracle provides query access to a fixed partition of the input bounded-degree minor-free graph, in which every component has size poly(1/epsilon), and the number of edges removed is at most epsilon*n, where n is the number of vertices in the graph. However, the oracle of Hassidimet al. makes an exponential number of queries to the input graph to answer every query about the partition. In this paper, we construct an efficient partitioning oracle for graphs with constant treewidth. The oracle makes only O(poly(1/epsilon)) queries to the input graph to answer each query about the partition. Examples of bounded-treewidth graph classes include k-outerplanar graphs for fixed k, series-parallel graphs, cactus graphs, and pseudoforests. Our oracle yields poly(1/epsilon)-time property testing algorithms for membership in these classes of graphs. Another application of the oracle is a poly(1/epsilon)-time algorithm that approximates the maximum matching size, the minimum vertex cover size, and the minimum dominating set size up to an additive epsilon*n in graphs with bounded treewidth. Finally, the oracle can be used to test in poly(1/epsilon) time whether the input bounded-treewidth graph is k-colorable or perfect.Comment: Full version of a paper to appear in RANDOM 201

    Every property of hyperfinite graphs is testable

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