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Fomin F.V.,University of Bergen | Lokshtanov D.,University of California | Misra N.,Institute of Mathematical science | Saurabh S.,Institute of Mathematical science
Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS | Year: 2012

Let F be a finite set of graphs. In the F-deletion problem, we are given an n-vertex graph G and an integer k as input, and asked whether at most k vertices can be deleted from G such that the resulting graph does not contain a graph from F as a minor. F-deletion is a generic problem and by selecting different sets of forbidden minors F, one can obtain various fundamental problems such as Vertex Cover, Feedback Vertex Set or Tree width k-deletion. In this paper we obtain a number of generic algorithmic results about F-deletion, when F contains at least one planar graph. The highlights of our work are: 1. A constant factor approximation algorithm for the optimization version of Planar F-deletion, 2. A linear time and single exponential parameterized algorithm, that is, an algorithm running in time O(exp(2, O(k))n), for the parameterized version of Planar F-deletion where all graphs in Planar F are connected, 3. A polynomial kernel for parameterized F-deletion. These algorithms unify, generalize, and improve a multitude of results in the literature. Our main results have several direct applications, but also the methods we develop on the way have applicability beyond the scope of this paper. Our results - constant factor approximation, polynomial kernelization and FPT algorithms - are stringed together by a common theme of polynomial time preprocessing. © 2012 IEEE.


Cygan M.,University of Lugano | Dell H.,University of Wisconsin - Madison | Lokshtanov D.,University of California | Marx D.,Hungarian Academy of Sciences | And 5 more authors.
Proceedings of the Annual IEEE Conference on Computational Complexity | Year: 2012

The field of exact exponential time algorithms for NP-hard problems has thrived over the last decade. While exhaustive search remains asymptotically the fastest known algorithm for some basic problems, difficult and non-trivial exponential time algorithms have been found for a myriad of problems, including Graph Coloring, Hamiltonian Path, Dominating Set and 3-CNF-Sat. In some instances, improving these algorithms further seems to be out of reach. The CNF-Sat problem is the canonical example of a problem for which the trivial exhaustive search algorithm runs in time O(2 n), where n is the number of variables in the input formula. While there exist non-trivial algorithms for CNF-Sat that run in time o(2 n), no algorithm was able to improve the growth rate 2 to a smaller constant, and hence it is natural to conjecture that 2 is the optimal growth rate. The strong exponential time hypothesis (SETH) by Impagliazzo and Paturi [JCSS 2001] goes a little bit further and asserts that, for every epsilon © 2012 IEEE.


Fomin F.V.,University of Bergen | Lokshtanov D.,University of Bergen | Saurabh S.,University of Bergen | Saurabh S.,Institute of Mathematical science
Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms | Year: 2014

Let M = (E,I) be a matroid and let S = {S1,..., St} be a family of subsets of E of size p. A subfamily Ŝ ⊆ S is q-representative for S if for every set Y ⊆ E of size at most q, if there is a set X ∈ S disjoint from Y with X U Y ∈ X, then there is a set X̂ ∈ Ŝ disjoint from Y with X U Y ∈ I. By the classical result of Bollobás, in a uniform matroid, every family of sets of size p has a q-representative family with at most (p+q p)sets. In his famous "two families theorem" from 1977, Lovász proved that the same bound also holds for any matroid representable over a field F. As observed by Marx, Lovász's proof is constructive. In this paper we show how Lovasz's proof can be turned into an algorithm constructing a q-representative family of size at most (p+q p) in time bounded by a polynomial in (p+q p), t, and the time required for field operations. Copyright © 2014 by the Society for Industrial and Applied Mathematics.


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.


Lei Z.,Fudan University | Lin F.-H.,New York University | Lin F.-H.,Institute of Mathematical science | Zhou Y.,Fudan University
Archive for Rational Mechanics and Analysis | Year: 2015

In this paper we derive a new energy identity for the three-dimensional incompressible Navier–Stokes equations by a special structure of helicity. The new energy functional is critical with respect to the natural scalings of the Navier–Stokes equations. Moreover, it is conditionally coercive. As an application we construct a family of finite energy smooth solutions to the Navier–Stokes equations whose critical norms can be arbitrarily large. © 2015 Springer-Verlag Berlin Heidelberg


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.


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.


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.


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.


Cygan M.,University of Warsaw | Lokshtanov D.,University of Bergen | Pilipczuk M.,University of Bergen | Saurabh S.,University of Bergen | Saurabh S.,Institute of Mathematical science
Proceedings of the Annual ACM Symposium on Theory of Computing | Year: 2014

In the classic Minimum Bisection problem we are given as input a graph G and an integer κ. The task is to determine whether there is a partition of V (G) into two parts A and B such that ∥A| - |B∥ < 1 and there are at most k edges with one endpoint in A and the other in B. In this paper we give an algorithm for Minimum Bisection with running time O(2O (κ3)n3 log3 n). This is the first fixed parameter tractable algorithm for Minimum Bisection. At the core of our algorithm lies a new decomposition theorem that states that every graph G can be decomposed by small separators into parts where each part is "highly connected" in the following sense: any cut of bounded size can separate only a limited number of vertices from each part of the decomposition. Our techniques generalize to the weighted setting, where we seek for a bisection of minimum weight among solutions that contain at most k edges. © 2014 ACM.

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