13,693 research outputs found

    Fast and Deterministic Approximations for k-Cut

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    In an undirected graph, a k-cut is a set of edges whose removal breaks the graph into at least k connected components. The minimum weight k-cut can be computed in n^O(k) time, but when k is treated as part of the input, computing the minimum weight k-cut is NP-Hard [Goldschmidt and Hochbaum, 1994]. For poly(m,n,k)-time algorithms, the best possible approximation factor is essentially 2 under the small set expansion hypothesis [Manurangsi, 2017]. Saran and Vazirani [1995] showed that a (2 - 2/k)-approximately minimum weight k-cut can be computed via O(k) minimum cuts, which implies a O~(km) randomized running time via the nearly linear time randomized min-cut algorithm of Karger [2000]. Nagamochi and Kamidoi [2007] showed that a (2 - 2/k)-approximately minimum weight k-cut can be computed deterministically in O(mn + n^2 log n) time. These results prompt two basic questions. The first concerns the role of randomization. Is there a deterministic algorithm for 2-approximate k-cuts matching the randomized running time of O~(km)? The second question qualitatively compares minimum cut to 2-approximate minimum k-cut. Can 2-approximate k-cuts be computed as fast as the minimum cut - in O~(m) randomized time? We give a deterministic approximation algorithm that computes (2 + eps)-minimum k-cuts in O(m log^3 n / eps^2) time, via a (1 + eps)-approximation for an LP relaxation of k-cut

    Subjectively interesting connecting trees

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    Vertex Sparsifiers: New Results from Old Techniques

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    Given a capacitated graph G=(V,E)G = (V,E) and a set of terminals KāŠ†VK \subseteq V, how should we produce a graph HH only on the terminals KK so that every (multicommodity) flow between the terminals in GG could be supported in HH with low congestion, and vice versa? (Such a graph HH is called a flow-sparsifier for GG.) What if we want HH to be a "simple" graph? What if we allow HH to be a convex combination of simple graphs? Improving on results of Moitra [FOCS 2009] and Leighton and Moitra [STOC 2010], we give efficient algorithms for constructing: (a) a flow-sparsifier HH that maintains congestion up to a factor of O(logā”k/logā”logā”k)O(\log k/\log \log k), where k=āˆ£Kāˆ£k = |K|, (b) a convex combination of trees over the terminals KK that maintains congestion up to a factor of O(logā”k)O(\log k), and (c) for a planar graph GG, a convex combination of planar graphs that maintains congestion up to a constant factor. This requires us to give a new algorithm for the 0-extension problem, the first one in which the preimages of each terminal are connected in GG. Moreover, this result extends to minor-closed families of graphs. Our improved bounds immediately imply improved approximation guarantees for several terminal-based cut and ordering problems.Comment: An extended abstract appears in the 13th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX), 2010. Final version to appear in SIAM J. Computin

    The Minimum Wiener Connector

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    The Wiener index of a graph is the sum of all pairwise shortest-path distances between its vertices. In this paper we study the novel problem of finding a minimum Wiener connector: given a connected graph G=(V,E)G=(V,E) and a set QāŠ†VQ\subseteq V of query vertices, find a subgraph of GG that connects all query vertices and has minimum Wiener index. We show that The Minimum Wiener Connector admits a polynomial-time (albeit impractical) exact algorithm for the special case where the number of query vertices is bounded. We show that in general the problem is NP-hard, and has no PTAS unless P=NP\mathbf{P} = \mathbf{NP}. Our main contribution is a constant-factor approximation algorithm running in time O~(āˆ£Qāˆ£āˆ£Eāˆ£)\widetilde{O}(|Q||E|). A thorough experimentation on a large variety of real-world graphs confirms that our method returns smaller and denser solutions than other methods, and does so by adding to the query set QQ a small number of important vertices (i.e., vertices with high centrality).Comment: Published in Proceedings of the 2015 ACM SIGMOD International Conference on Management of Dat

    Connecting Seed Lists of Mammalian Proteins Using Steiner Trees

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    Multivariate experiments and genomics studies applied to mammalian cells often produce lists of genes or proteins altered under treatment/disease vs. control/normal conditions. Such lists can be identified in known protein-protein interaction networks to produce subnetworks that “connect” the genes or proteins from the lists. Such subnetworks are valuable for biologists since they can suggest regulatory mechanisms that are altered under different conditions. Often such subnetworks are overloaded with links and nodes resulting in connectivity diagrams that are illegible due to edge overlap. In this study, we attempt to address this problem by implementing an approximation to the Steiner Tree problem to connect seed lists of mammalian proteins/genes using literature-based protein-protein interaction networks. To avoid over-representation of hubs in the resultant Steiner Trees we assign a cost to Steiner Vertices based on their connectivity degree. We applied the algorithm to lists of genes commonly mutated in colorectal cancer to demonstrate the usefulness of this approach
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