2,017 research outputs found

    Packing odd TT-joins with at most two terminals

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    Take a graph GG, an edge subset ΣE(G)\Sigma\subseteq E(G), and a set of terminals TV(G)T\subseteq V(G) where T|T| is even. The triple (G,Σ,T)(G,\Sigma,T) is called a signed graft. A TT-join is odd if it contains an odd number of edges from Σ\Sigma. Let ν\nu be the maximum number of edge-disjoint odd TT-joins. A signature is a set of the form Σδ(U)\Sigma\triangle \delta(U) where UV(G)U\subseteq V(G) and UT)|U\cap T) is even. Let τ\tau be the minimum cardinality a TT-cut or a signature can achieve. Then ντ\nu\leq \tau and we say that (G,Σ,T)(G,\Sigma,T) packs if equality holds here. We prove that (G,Σ,T)(G,\Sigma,T) packs if the signed graft is Eulerian and it excludes two special non-packing minors. Our result confirms the Cycling Conjecture for the class of clutters of odd TT-joins with at most two terminals. Corollaries of this result include, the characterizations of weakly and evenly bipartite graphs, packing two-commodity paths, packing TT-joins with at most four terminals, and a new result on covering edges with cuts.Comment: extended abstract appeared in IPCO 2014 (under the different title "the cycling property for the clutter of odd st-walks"

    Distributed Connectivity Decomposition

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    We present time-efficient distributed algorithms for decomposing graphs with large edge or vertex connectivity into multiple spanning or dominating trees, respectively. As their primary applications, these decompositions allow us to achieve information flow with size close to the connectivity by parallelizing it along the trees. More specifically, our distributed decomposition algorithms are as follows: (I) A decomposition of each undirected graph with vertex-connectivity kk into (fractionally) vertex-disjoint weighted dominating trees with total weight Ω(klogn)\Omega(\frac{k}{\log n}), in O~(D+n)\widetilde{O}(D+\sqrt{n}) rounds. (II) A decomposition of each undirected graph with edge-connectivity λ\lambda into (fractionally) edge-disjoint weighted spanning trees with total weight λ12(1ε)\lceil\frac{\lambda-1}{2}\rceil(1-\varepsilon), in O~(D+nλ)\widetilde{O}(D+\sqrt{n\lambda}) rounds. We also show round complexity lower bounds of Ω~(D+nk)\tilde{\Omega}(D+\sqrt{\frac{n}{k}}) and Ω~(D+nλ)\tilde{\Omega}(D+\sqrt{\frac{n}{\lambda}}) for the above two decompositions, using techniques of [Das Sarma et al., STOC'11]. Moreover, our vertex-connectivity decomposition extends to centralized algorithms and improves the time complexity of [Censor-Hillel et al., SODA'14] from O(n3)O(n^3) to near-optimal O~(m)\tilde{O}(m). As corollaries, we also get distributed oblivious routing broadcast with O(1)O(1)-competitive edge-congestion and O(logn)O(\log n)-competitive vertex-congestion. Furthermore, the vertex connectivity decomposition leads to near-time-optimal O(logn)O(\log n)-approximation of vertex connectivity: centralized O~(m)\widetilde{O}(m) and distributed O~(D+n)\tilde{O}(D+\sqrt{n}). The former moves toward the 1974 conjecture of Aho, Hopcroft, and Ullman postulating an O(m)O(m) centralized exact algorithm while the latter is the first distributed vertex connectivity approximation

    On Strong Diameter Padded Decompositions

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    Given a weighted graph G=(V,E,w), a partition of V is Delta-bounded if the diameter of each cluster is bounded by Delta. A distribution over Delta-bounded partitions is a beta-padded decomposition if every ball of radius gamma Delta is contained in a single cluster with probability at least e^{-beta * gamma}. The weak diameter of a cluster C is measured w.r.t. distances in G, while the strong diameter is measured w.r.t. distances in the induced graph G[C]. The decomposition is weak/strong according to the diameter guarantee. Formerly, it was proven that K_r free graphs admit weak decompositions with padding parameter O(r), while for strong decompositions only O(r^2) padding parameter was known. Furthermore, for the case of a graph G, for which the induced shortest path metric d_G has doubling dimension ddim, a weak O(ddim)-padded decomposition was constructed, which is also known to be tight. For the case of strong diameter, nothing was known. We construct strong O(r)-padded decompositions for K_r free graphs, matching the state of the art for weak decompositions. Similarly, for graphs with doubling dimension ddim we construct a strong O(ddim)-padded decomposition, which is also tight. We use this decomposition to construct (O(ddim),O~(ddim))-sparse cover scheme for such graphs. Our new decompositions and cover have implications to approximating unique games, the construction of light and sparse spanners, and for path reporting distance oracles

    Relaxed Voronoi: A Simple Framework for Terminal-Clustering Problems

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    We reprove three known algorithmic bounds for terminal-clustering problems, using a single framework that leads to simpler proofs. In this genre of problems, the input is a metric space (X,d) (possibly arising from a graph) and a subset of terminals K subset X, and the goal is to partition the points X such that each part, called a cluster, contains exactly one terminal (possibly with connectivity requirements) so as to minimize some objective. The three bounds we reprove are for Steiner Point Removal on trees [Gupta, SODA 2001], for Metric 0-Extension in bounded doubling dimension [Lee and Naor, unpublished 2003], and for Connected Metric 0-Extension [Englert et al., SICOMP 2014]. A natural approach is to cluster each point with its closest terminal, which would partition X into so-called Voronoi cells, but this approach can fail miserably due to its stringent cluster boundaries. A now-standard fix, which we call the Relaxed-Voronoi framework, is to use enlarged Voronoi cells, but to obtain disjoint clusters, the cells are computed greedily according to some order. This method, first proposed by Calinescu, Karloff and Rabani [SICOMP 2004], was employed successfully to provide state-of-the-art results for terminal-clustering problems on general metrics. However, for restricted families of metrics, e.g., trees and doubling metrics, only more complicated, ad-hoc algorithms are known. Our main contribution is to demonstrate that the Relaxed-Voronoi algorithm is applicable to restricted metrics, and actually leads to relatively simple algorithms and analyses

    Min-max results in combinatorial optimization

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    Convexity in partial cubes: the hull number

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    We prove that the combinatorial optimization problem of determining the hull number of a partial cube is NP-complete. This makes partial cubes the minimal graph class for which NP-completeness of this problem is known and improves some earlier results in the literature. On the other hand we provide a polynomial-time algorithm to determine the hull number of planar partial cube quadrangulations. Instances of the hull number problem for partial cubes described include poset dimension and hitting sets for interiors of curves in the plane. To obtain the above results, we investigate convexity in partial cubes and characterize these graphs in terms of their lattice of convex subgraphs, improving a theorem of Handa. Furthermore we provide a topological representation theorem for planar partial cubes, generalizing a result of Fukuda and Handa about rank three oriented matroids.Comment: 19 pages, 4 figure
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