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Ramanujan M.S.,Institute of Mathematical science | Saurabh S.,Institute of Mathematical science | Saurabh S.,University of Bergen
Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms | Year: 2014

A skew-symmetric graph (D = (V, A), σ) is a directed graph D with an involution a on the set of vertices and arcs. Flows on skew-symmetric graphs have been used to generalize maximum flow and maximum matching problems on graphs, initially by Tutte [1967], and later by Goldberg and Karzanov [1994, 1995]. In this paper, we introduce a separation problem, d-skew- Symmetric Multicut, where we are given a skew- symmetric graph D, a family of T of d-sized subsets of vertices and an integer k. The objective is to decide if there is a set X ⊆ A of k arcs such that every set J in the family has a vertex v such that v and σ(v) are in different strongly connected components of D' = (V, A \ (X ∪ σ(X)). In this paper, we give an algorithm for d-skew-Symmetric Multicut which runs in time O((4d)k(m+n+ℓ)), where m is the number of arcs in the graph, n the number of vertices and ℓ the length of the family given in the input. This problem, apart from being independently interesting, also abstracts out and captures the main combinatorial obstacles towards solving numerous classical problems. Our algorithm for d-Skew-Symmetric Multicut paves the way for the first linear time parameterized algorithms for several problems. We demonstrate its utility by obtaining the following linear time parameterized algorithms. • We show that Almost 2-SAT is a special case of 1-Skew-Symmetric Multicut, resulting in an algorithm for Almost 2-SAT which runs in time O(4kk 4ℓ) where k is the size of the solution and ℓ is the length of the input formula. Then, using linear time parameter preserving reductions to Almost 2-SAT, we obtain algorithms for Odd Cycle Transversal and Edge Bipartization which run in time O(4Kk4(m+n)) and O(4 kk5(m + n)) respectively where k is size of the solution,m and n are the number of edges and vertices respectively. This resolves an open problem posed by Reed, Smith and Vetta [Operations Research Letters, 2003] and improves upon the earlier almost linear time algorithm of Kawarabayashi and Reed [SODA, 2010]. • We show that Deletion g-Horn Backdoor Set Detection is a special case of 3-Skew- Symmetric Multicut, giving us an algorithm for Deletion g-Horn Backdoor Set Detection which runs in time O(12kk 5ℓ) where k is the size of the solution and £ is the length of the input formula. This gives the first fixed-parameter tractable algorithm for this problem answering a question posed in a paper by a superset of the authors [STACS, 2013]. Using this result, we get an algorithm for Satisfiability which runs in time O(12kk5ℓ) where k is the size of the smallest q-Horn deletion backdoor set, with ℓ being the length of the input formula. Copyright © 2014 by the Society for Industrial and Applied Mathematics. Source


Suematsu K.,Institute of Mathematical science
Colloid and Polymer Science | Year: 2012

The concentration dependence of the excluded volume effects in polymer solutions is investigated. Through thermodynamic arguments for the interpenetration of polymer segments and the free energy change, we show that the disappearance of the excluded volume effects should occur at medium concentration. The result is in accord with the recent experimental observations. © 2012 Springer Science+Business Media, LLC. Source


Lokshtanov D.,University of Bergen | Pilipczuk M.,University of Bergen | Saurabh S.,Institute of Mathematical science
Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS | Year: 2014

We give a fixed-parameter tractable algorithm that, given a parameter k and two graphs G1, G2, either concludes that one of these graphs has treewidth at least k, or determines whether G1 and G2 are isomorphic. The running time of the algorithm on an n-vertex graph is 2O(k5 log k) n5, and this is the first fixed-parameter algorithm for Graph Isomorphism parameterized by treewidth. Our algorithm in fact solves the more general canonization problem. We namely design a procedure working in 2 O(k5 log k) n5 time that, for a given graph G on n vertices, either concludes that the treewidth of G is at least k, or finds an isomorphism-invariant construction term-an algebraic expression that encodes G together with a tree decomposition of G of width O(k4). Hence, a canonical graph isomorphic to G can be constructed by simply evaluating the obtained construction term, while the isomorphism test reduces to verifying whether the computed construction terms for G1 and G2 are equal. © 2014 IEEE. Source


Lokshtanov D.,University of California at San Diego | Marx D.,Humboldt University of Berlin | Saurabh S.,Institute of Mathematical science
Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms | Year: 2011

A central problem in parameterized algorithms is to obtain algorithms with running time f(k) · nO(1) such that f is as slow growing function of the parameter k as possible. In particular, the first natural goal is to make f(k) single-exponential, that is, ck for some constant c. This has led to the development of parameterized algorithms for various problems where f(k) appearing in their running time is of form 2O(k). However there are still plenty of problems where the'"slightly superexponential" f(k) appearing in the best known running time has remained non single-exponential even after a lot of attempts to bring it down. A natural question to ask is whether the f(k) appearing in the running time of the best-known algorithms is optimal for any of these problems. In this paper, we examine parameterized problems where f(k) is kO(k) = 2 O(k log k) in the best known running time and for a number of such problems, we show that the dependence on k in the running time cannot be improved to single exponential. More precisely we prove following tight lower bounds, for three natural problems, arising from three different domains: • The pattern matching problem CLOSEST STRING is known to be solvable in time O(d log d) · nO(1) and 2 O(d log |Σ|) middot; nO(1) We show that there is no 2O(d log d) · nO(1) and 2 O(d log |Σ|) · nO(1) time algorithm, unless Exponential Time Hypothesis (ETH) fails. • The graph embedding problem DISTORTION, that is, deciding whether a graph G has a metric embedding into the integers with distortion at most d can be done in time 2O(d log d) · nO(1). We show that there is no 2O(d log d) · nO(1) time algorithm, unless ETH fails. • The DISJOINT PATHS problem can be solved in time in time 2O(w log w) · nO(1) on graphs of treewidth at most w. We show that there is no 2O(w log w) · nO(1) time algorithm, unless ETH fails. To obtain our result we first prove the lower bound for variants of basic problems: finding cliques, independent sets, and hitting sets. These artificially constrained variants form a good starting point for proving lower bounds on natural problems without any technical restrictions and could be of independent interest. We believe that many further results of this form can be obtained by using the framework of the current paper. Source


Lokshtanov D.,University of California at San Diego | Marx D.,Humboldt University of Berlin | Saurabh S.,Institute of Mathematical science
Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms | Year: 2011

We obtain a number of lower bounds on the running time of algorithms solving problems on graphs of bounded treewidth. We prove the results under the Strong Exponential Time Hypothesis of Impagliazzo and Paturi. In particular, assuming that SAT cannot be solved in (2 - ε)nmO(1) time, we show that for any ε > 0; • INDEPENDENT SET cannot be solved in time (2 - ε)tw(G)|V(G)|O(1), • DOMINATING SET cannot be solved in time (3 - ε)tw(G) |V(G)| O(1), • MAX CUT cannot be solved in time (2 - ε) tw(G)|V(G)|O(1), • ODD CYCLE TRANSVERSAL cannot be solved in time (3 - ε)tw(G)|V(G)|O(1), • For any q > 3, q-COLORING cannot be solved in time (q - ε)tw(G)|V(G) |O(1), • PARTITION INTO TRIANGLES cannot be solved in time (2 - ε)tw(G)|V(G)|O(1). Our lower bounds match the running times for the best known algorithms for the problems, up to the ε in the base. Source

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