arXiv subjects

Saket Saurabh

Publications and source records attributed to Saket Saurabh.

At least 181 records · Page 10Linked to original sources

arXiv2013

(Meta) Kernelization

In a parameterized problem, every instance I comes with a positive integer k. The problem is said to admit a polynomial kernel if, in polynomial time, one can reduce the size of the instance I to a polynomial in k, while preserving the answer. In this work we give two meta-theorems on kernelzation. The first theorem says that all problems expressible in Counting Monadic Second Order Logic and satisfying a coverability property admit a polynomial kernel on graphs of bounded genus. Our second result is that all problems that have finite integer index and satisfy a weaker coverability property admit a linear kernel on graphs of bounded genus. These theorems unify and extend all previously known kernelization results for planar graph problems.

cs.DM↗

arXiv2013

Linear Time Parameterized Algorithms via Skew-Symmetric Multicuts

A skew-symmetric graph $(D=(V,A),σ)$ is a directed graph $D$ with an involution $σ$ on the set of vertices and arcs. In this paper, we introduce a separation problem, $d$-Skew-Symmetric Multicut, where we are given a skew-symmetric graph $D$, a family of $\cal T$ of $d$-sized subsets of vertices and an integer $k$. The objective is to decide if there is a set $X\subseteq A$ of $k$ arcs such that every set $J$ in the family has a vertex $v$ such that $v$ and $σ(v)$ are in different connected components of $D'=(V,A\setminus (X\cup σ(X))$. In this paper, we give an algorithm for this problem which runs in time $O((4d)^{k}(m+n+\ell))$, where $m$ is the number of arcs in the graph, $n$ the number of vertices and $\ell$ the length of the family given in the input. Using our algorithm, we show that Almost 2-SAT has an algorithm with running time $O(4^kk^4\ell)$ and we obtain algorithms for {\sc Odd Cycle Transversal} and {\sc Edge Bipartization} which run in time $O(4^kk^4(m+n))$ and $O(4^kk^5(m+n))$ 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 also show that Deletion q-Horn Backdoor Set Detection is a special case of 3-Skew-Symmetric Multicut, giving us an algorithm for Deletion q-Horn Backdoor Set Detection which runs in time $O(12^kk^5\ell)$. 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 $O(12^kk^5\ell)$ where $k$ is the size of the smallest q-Horn deletion backdoor set, with $\ell$ being the length of the input formula.

cs.DS↗

arXiv2012

Parameterized Complexity of Directed Steiner Tree on Sparse Graphs

We study the parameterized complexity of the directed variant of the classical {\sc Steiner Tree} problem on various classes of directed sparse graphs. While the parameterized complexity of {\sc Steiner Tree} parameterized by the number of terminals is well understood, not much is known about the parameterization by the number of non-terminals in the solution tree. All that is known for this parameterization is that both the directed and the undirected versions are W[2]-hard on general graphs, and hence unlikely to be fixed parameter tractable FPT. The undirected {\sc Steiner Tree} problem becomes FPT when restricted to sparse classes of graphs such as planar graphs, but the techniques used to show this result break down on directed planar graphs. In this article we precisely chart the tractability border for {\sc Directed Steiner Tree} (DST) on sparse graphs parameterized by the number of non-terminals in the solution tree. Specifically, we show that the problem is fixed parameter tractable on graphs excluding a topological minor, but becomes W[2]-hard on graphs of degeneracy 2. On the other hand we show that if the subgraph induced by the terminals is required to be acyclic then the problem becomes FPT on graphs of bounded degeneracy. We further show that our algorithm achieves the best possible running time dependence on the solution size and degeneracy of the input graph, under standard complexity theoretic assumptions. Using the ideas developed for DST, we also obtain improved algorithms for {\sc Dominating Set} on sparse undirected graphs. These algorithms are asymptotically optimal.

cs.DS↗

arXiv2012

Don't Be Strict in Local Search!

Local Search is one of the fundamental approaches to combinatorial optimization and it is used throughout AI. Several local search algorithms are based on searching the k-exchange neighborhood. This is the set of solutions that can be obtained from the current solution by exchanging at most k elements. As a rule of thumb, the larger k is, the better are the chances of finding an improved solution. However, for inputs of size n, a naïve brute-force search of the k-exchange neighborhood requires n to the power of O(k) time, which is not practical even for very small values of k. Fellows et al. (IJCAI 2009) studied whether this brute-force search is avoidable and gave positive and negative answers for several combinatorial problems. They used the notion of local search in a strict sense. That is, an improved solution needs to be found in the k-exchange neighborhood even if a global optimum can be found efficiently. In this paper we consider a natural relaxation of local search, called permissive local search (Marx and Schlotter, IWPEC 2009) and investigate whether it enhances the domain of tractable inputs. We exemplify this approach on a fundamental combinatorial problem, Vertex Cover. More precisely, we show that for a class of inputs, finding an optimum is hard, strict local search is hard, but permissive local search is tractable. We carry out this investigation in the framework of parameterized complexity.

cs.DS↗

arXiv2012

Beyond Max-Cut: λ-Extendible Properties Parameterized Above the Poljak-Turzík Bound

Poljak and Turzík (Discrete Math. 1986) introduced the notion of λ-extendible properties of graphs as a generalization of the property of being bipartite. They showed that for any 0<λ<1 and λ-extendible property Π, any connected graph G on n vertices and m edges contains a subgraph H \in Π with at least λm+ (1-λ)/2 (n-1) edges. The property of being bipartite is 1/2-extendible, and thus this bound generalizes the Edwards-Erdős bound for Max-Cut. We define a variant, namely strong λ-extendibility, to which the bound applies. For a strongly λ-extendible graph property Π, we define the parameterized Above Poljak- Turzík (APT) (Π) problem as follows: Given a connected graph G on n vertices and m edges and an integer parameter k, does there exist a spanning subgraph H of G such that H \in Π and H has at least λm + (1-λ)/2 (n - 1) + k edges? The parameter is k, the surplus over the number of edges guaranteed by the Poljak-Turzík bound. We consider properties Π for which APT (Π) is fixed- parameter tractable (FPT) on graphs which are O(k) vertices away from being a graph in which each block is a clique. We show that for all such properties, APT (Π) is FPT for all 0<λ<1. Our results hold for properties of oriented graphs and graphs with edge labels. Our results generalize the result of Crowston et al. (ICALP 2012) on Max-Cut parameterized above the Edwards-Erdős bound, and yield FPT algorithms for several graph problems parameterized above lower bounds, e.g., Max q-Colorable Subgraph problem. Our results also imply that the parameterized above-guarantee Oriented Max Acyclic Digraph problem is FPT, thus solving an open question of Raman and Saurabh (Theor. Comput. Sci. 2006).

cs.DS↗

arXiv2012

A Polynomial kernel for Proper Interval Vertex Deletion

It is known that the problem of deleting at most k vertices to obtain a proper interval graph (Proper Interval Vertex Deletion) is fixed parameter tractable. However, whether the problem admits a polynomial kernel or not was open. Here, we answers this question in affirmative by obtaining a polynomial kernel for Proper Interval Vertex Deletion. This resolves an open question of van Bevern, Komusiewicz, Moser, and Niedermeier.

cs.DS↗

arXiv2012

Faster Parameterized Algorithms using Linear Programming

We investigate the parameterized complexity of Vertex Cover parameterized by the difference between the size of the optimal solution and the value of the linear programming (LP) relaxation of the problem. By carefully analyzing the change in the LP value in the branching steps, we argue that combining previously known preprocessing rules with the most straightforward branching algorithm yields an $O^*((2.618)^k)$ algorithm for the problem. Here $k$ is the excess of the vertex cover size over the LP optimum, and we write $O^*(f(k))$ for a time complexity of the form $O(f(k)n^{O(1)})$, where $f (k)$ grows exponentially with $k$. We proceed to show that a more sophisticated branching algorithm achieves a runtime of $O^*(2.3146^k)$. Following this, using known and new reductions, we give $O^*(2.3146^k)$ algorithms for the parameterized versions of Above Guarantee Vertex Cover, Odd Cycle Transversal, Split Vertex Deletion and Almost 2-SAT, and an $O^*(1.5214^k)$ algorithm for Konig Vertex Deletion, Vertex Cover Param by OCT and Vertex Cover Param by KVD. These algorithms significantly improve the best known bounds for these problems. The most notable improvement is the new bound for Odd Cycle Transversal - this is the first algorithm which beats the dependence on $k$ of the seminal $O^*(3^k)$ algorithm of Reed, Smith and Vetta. Finally, using our algorithm, we obtain a kernel for the standard parameterization of Vertex Cover with at most $2k - c \log k$ vertices. Our kernel is simpler than previously known kernels achieving the same size bound.

cs.DS↗

arXiv2012

A Linear Vertex Kernel for Maximum Internal Spanning Tree

We present a polynomial time algorithm that for any graph G and integer k >= 0, either finds a spanning tree with at least k internal vertices, or outputs a new graph G' on at most 3k vertices and an integer k' such that G has a spanning tree with at least k internal vertices if and only if G' has a spanning tree with at least k' internal vertices. In other words, we show that the Maximum Internal Spanning Tree problem parameterized by the number of internal vertices k, has a 3k-vertex kernel. Our result is based on an innovative application of a classical min-max result about hypertrees in hypergraphs which states that "a hypergraph H contains a hypertree if and only if H is partition connected."

cs.DS↗

arXiv2011

Parameterized Complexity of MaxSat Above Average

In MaxSat, we are given a CNF formula $F$ with $n$ variables and $m$ clauses and asked to find a truth assignment satisfying the maximum number of clauses. Let $r_1,..., r_m$ be the number of literals in the clauses of $F$. Then $asat(F)=\sum_{i=1}^m (1-2^{-r_i})$ is the expected number of clauses satisfied by a random truth assignment (the truth values to the variables are distributed uniformly and independently). It is well-known that, in polynomial time, one can find a truth assignment satisfying at least $asat(F)$ clauses. In the parameterized problem MaxSat-AA, we are to decide whether there is a truth assignment satisfying at least $asat(F)+k$ clauses, where $k$ is the parameter. We prove that MaxSat-AA is para-NP-complete and, thus, MaxSat-AA is not fixed-parameter tractable unless P$=$NP. This is in sharp contrast to MaxLin2-AA which was recently proved to be fixed-parameter tractable by Crowston et al. (arXiv:1104.1135v3). In fact, we consider a more refined version of {\sc MaxSat-AA}, {\sc Max-$r(n)$-Sat-AA}, where $r_j\le r(n)$ for each $j$. Alon {\em et al.} (SODA 2010) proved that if $r=r(n)$ is a constant, then {\sc Max-$r$-Sat-AA} is fixed-parameter tractable. We prove that {\sc Max-$r(n)$-Sat-AA} is para-NP-complete for $r(n)=\lceil \log n\rceil.$ We also prove that assuming the exponential time hypothesis, {\sc Max-$r(n)$-Sat-AA} is not in XP already for any $r(n)\ge \log \log n +ϕ(n)$, where $ϕ(n)$ is any unbounded strictly increasing function. This lower bound on $r(n)$ cannot be decreased much further as we prove that {\sc Max-$r(n)$-Sat-AA} is (i) in XP for any $r(n)\le \log \log n - \log \log \log n$ and (ii) fixed-parameter tractable for any $r(n)\le \log \log n - \log \log \log n - ϕ(n)$, where $ϕ(n)$ is any unbounded strictly increasing function. The proof uses some results on {\sc MaxLin2-AA}.

cs.CC↗

arXiv2011

Bidimensionality and Geometric Graphs

In this paper we use several of the key ideas from Bidimensionality to give a new generic approach to design EPTASs and subexponential time parameterized algorithms for problems on classes of graphs which are not minor closed, but instead exhibit a geometric structure. In particular we present EPTASs and subexponential time parameterized algorithms for Feedback Vertex Set, Vertex Cover, Connected Vertex Cover, Diamond Hitting Set, on map graphs and unit disk graphs, and for Cycle Packing and Minimum-Vertex Feedback Edge Set on unit disk graphs. Our results are based on the recent decomposition theorems proved by Fomin et al [SODA 2011], and our algorithms work directly on the input graph. Thus it is not necessary to compute the geometric representations of the input graph. To the best of our knowledge, these results are previously unknown, with the exception of the EPTAS and a subexponential time parameterized algorithm on unit disk graphs for Vertex Cover, which were obtained by Marx [ESA 2005] and Alber and Fiala [J. Algorithms 2004], respectively. We proceed to show that our approach can not be extended in its full generality to more general classes of geometric graphs, such as intersection graphs of unit balls in R^d, d >= 3. Specifically we prove that Feedback Vertex Set on unit-ball graphs in R^3 neither admits PTASs unless P=NP, nor subexponential time algorithms unless the Exponential Time Hypothesis fails. Additionally, we show that the decomposition theorems which our approach is based on fail for disk graphs and that therefore any extension of our results to disk graphs would require new algorithmic ideas. On the other hand, we prove that our EPTASs and subexponential time algorithms for Vertex Cover and Connected Vertex Cover carry over both to disk graphs and to unit-ball graphs in R^d for every fixed d.

cs.DS↗

arXiv2011

Bidimensionality and EPTAS

Bidimensionality theory is a powerful framework for the development of metaalgorithmic techniques. It was introduced by Demaine et al. as a tool to obtain sub-exponential time parameterized algorithms for problems on H-minor free graphs. Demaine and Hajiaghayi extended the theory to obtain PTASs for bidimensional problems, and subsequently improved these results to EPTASs. Fomin et. al related the theory to the existence of linear kernels for parameterized problems. In this paper we revisit bidimensionality theory from the perspective of approximation algorithms and redesign the framework for obtaining EPTASs to be more powerful, easier to apply and easier to understand. Two of the most widely used approaches to obtain PTASs on planar graphs are the Lipton-Tarjan separator based approach, and Baker's approach. Demaine and Hajiaghayi strengthened both approaches using bidimensionality and obtained EPTASs for a multitude of problems. We unify the two strenghtened approaches to combine the best of both worlds. At the heart of our framework is a decomposition lemma which states that for "most" bidimensional problems, there is a polynomial time algorithm which given an H-minor-free graph G as input and an e > 0 outputs a vertex set X of size e * OPT such that the treewidth of G n X is f(e). Here, OPT is the objective function value of the problem in question and f is a function depending only on e. This allows us to obtain EPTASs on (apex)-minor-free graphs for all problems covered by the previous framework, as well as for a wide range of packing problems, partial covering problems and problems that are neither closed under taking minors, nor contractions. To the best of our knowledge for many of these problems including cycle packing, vertex-h-packing, maximum leaf spanning tree, and partial r-dominating set no EPTASs on planar graphs were previously known.

cs.DS↗

arXiv2010

Hitting forbidden minors: Approximation and Kernelization

We study a general class of problems called F-deletion problems. In an F-deletion problem, we are asked whether a subset of at most $k$ vertices can be deleted from a graph $G$ such that the resulting graph does not contain as a minor any graph from the family F of forbidden minors. We obtain a number of algorithmic results on the F-deletion problem when F contains a planar graph. We give (1) a linear vertex kernel on graphs excluding $t$-claw $K_{1,t}$, the star with $t$ leves, as an induced subgraph, where $t$ is a fixed integer. (2) an approximation algorithm achieving an approximation ratio of $O(\log^{3/2} OPT)$, where $OPT$ is the size of an optimal solution on general undirected graphs. Finally, we obtain polynomial kernels for the case when F contains graph $θ_c$ as a minor for a fixed integer $c$. The graph $θ_c$ consists of two vertices connected by $c$ parallel edges. Even though this may appear to be a very restricted class of problems it already encompasses well-studied problems such as {\sc Vertex Cover}, {\sc Feedback Vertex Set} and Diamond Hitting Set. The generic kernelization algorithm is based on a non-trivial application of protrusion techniques, previously used only for problems on topological graph classes.

cs.DS↗

arXiv2010

Known Algorithms on Graphs of Bounded Treewidth are Probably Optimal

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)^{n}m^{O(1)} time, we show that for any e > 0; {\sc Independent Set} cannot be solved in (2-e)^{tw(G)}|V(G)|^{O(1)} time, {\sc Dominating Set} cannot be solved in (3-e)^{tw(G)}|V(G)|^{O(1)} time, {\sc Max Cut} cannot be solved in (2-e)^{tw(G)}|V(G)|^{O(1)} time, {\sc Odd Cycle Transversal} cannot be solved in (3-e)^{tw(G)}|V(G)|^{O(1)} time, For any $q \geq 3$, $q$-{\sc Coloring} cannot be solved in (q-e)^{tw(G)}|V(G)|^{O(1)} time, {\sc Partition Into Triangles} cannot be solved in (2-e)^{tw(G)}|V(G)|^{O(1)} time. Our lower bounds match the running times for the best known algorithms for the problems, up to the e in the base.

cs.DS↗

arXiv2010

Beyond Bidimensionality: Parameterized Subexponential Algorithms on Directed Graphs

We develop two different methods to achieve subexponential time parameterized algorithms for problems on sparse directed graphs. We exemplify our approaches with two well studied problems. For the first problem, {\sc $k$-Leaf Out-Branching}, which is to find an oriented spanning tree with at least $k$ leaves, we obtain an algorithm solving the problem in time $2^{O(\sqrt{k} \log k)} n+ n^{O(1)}$ on directed graphs whose underlying undirected graph excludes some fixed graph $H$ as a minor. For the special case when the input directed graph is planar, the running time can be improved to $2^{O(\sqrt{k})}n + n^{O(1)}$. The second example is a generalization of the {\sc Directed Hamiltonian Path} problem, namely {\sc $k$-Internal Out-Branching}, which is to find an oriented spanning tree with at least $k$ internal vertices. We obtain an algorithm solving the problem in time $2^{O(\sqrt{k} \log k)} + n^{O(1)}$ on directed graphs whose underlying undirected graph excludes some fixed apex graph $H$ as a minor. Finally, we observe that for any $ε>0$, the {\sc $k$-Directed Path} problem is solvable in time $O((1+ε)^k n^{f(ε)})$, where $f$ is some function of $\ve$. Our methods are based on non-trivial combinations of obstruction theorems for undirected graphs, kernelization, problem specific combinatorial structures and a layering technique similar to the one employed by Baker to obtain PTAS for planar graphs.

cs.DS↗

arXiv2009

Faster Algorithms for Finding and Counting Subgraphs

In this paper we study a natural generalization of both {\sc $k$-Path} and {\sc $k$-Tree} problems, namely, the {\sc Subgraph Isomorphism} problem. In the {\sc Subgraph Isomorphism} problem we are given two graphs $F$ and $G$ on $k$ and $n$ vertices respectively as an input, and the question is whether there exists a subgraph of $G$ isomorphic to $F$. We show that if the treewidth of $F$ is at most $t$, then there is a randomized algorithm for the {\sc Subgraph Isomorphism} problem running in time $\cO^*(2^k n^{2t})$. To do so, we associate a new multivariate {Homomorphism polynomial} of degree at most $k$ with the {\sc Subgraph Isomorphism} problem and construct an arithmetic circuit of size at most $n^{\cO(t)}$ for this polynomial. Using this polynomial, we also give a deterministic algorithm to count the number of homomorphisms from $F$ to $G$ that takes $n^{\cO(t)}$ time and uses polynomial space. For the counting version of the {\sc Subgraph Isomorphism} problem, where the objective is to count the number of distinct subgraphs of $G$ that are isomorphic to $F$, we give a deterministic algorithm running in time and space $\cO^*({n \choose k/2}n^{2p})$ or ${n\choose k/2}n^{\cO(t \log k)}$. We also give an algorithm running in time $\cO^{*}(2^{k}{n \choose k/2}n^{5p})$ and taking space polynomial in $n$. Here $p$ and $t$ denote the pathwidth and the treewidth of $F$, respectively. Thus our work not only improves on known results on {\sc Subgraph Isomorphism} but it also extends and generalize most of the known results on {\sc $k$-Path} and {\sc $k$-Tree}.

cs.DS↗

arXiv2009

Kernels for Feedback Arc Set In Tournaments

A tournament T=(V,A) is a directed graph in which there is exactly one arc between every pair of distinct vertices. Given a digraph on n vertices and an integer parameter k, the Feedback Arc Set problem asks whether the given digraph has a set of k arcs whose removal results in an acyclic digraph. The Feedback Arc Set problem restricted to tournaments is known as the k-Feedback Arc Set in Tournaments (k-FAST) problem. In this paper we obtain a linear vertex kernel for k-FAST. That is, we give a polynomial time algorithm which given an input instance T to k-FAST obtains an equivalent instance T' on O(k) vertices. In fact, given any fixed e>0, the kernelized instance has at most (2+e)k vertices. Our result improves the previous known bound of O(k^2) on the kernel size for k-FAST. Our kernelization algorithm solves the problem on a subclass of tournaments in polynomial time and uses a known polynomial time approximation scheme for k-FAST.

cs.DS↗

arXiv2009

FPT Algorithms for Connected Feedback Vertex Set

We study the recently introduced Connected Feedback Vertex Set (CFVS) problem from the view-point of parameterized algorithms. CFVS is the connected variant of the classical Feedback Vertex Set problem and is defined as follows: given a graph G=(V,E) and an integer k, decide whether there exists a subset F of V, of size at most k, such that G[V F] is a forest and G[F] is connected. We show that Connected Feedback Vertex Set can be solved in time $O(2^{O(k)}n^{O(1)})$ on general graphs and in time $O(2^{O(\sqrt{k}\log k)}n^{O(1)})$ on graphs excluding a fixed graph H as a minor. Our result on general undirected graphs uses as subroutine, a parameterized algorithm for Group Steiner Tree, a well studied variant of Steiner Tree. We find the algorithm for Group Steiner Tree of independent interest and believe that it will be useful for obtaining parameterized algorithms for other connectivity problems.

cs.DS↗

arXiv2009

Algorithm for Finding $k$-Vertex Out-trees and its Application to $k$-Internal Out-branching Problem

An out-tree $T$ is an oriented tree with only one vertex of in-degree zero. A vertex $x$ of $T$ is internal if its out-degree is positive. We design randomized and deterministic algorithms for deciding whether an input digraph contains a given out-tree with $k$ vertices. The algorithms are of runtime $O^*(5.704^k)$ and $O^*(5.704^{k(1+o(1))})$, respectively. We apply the deterministic algorithm to obtain a deterministic algorithm of runtime $O^*(c^k)$, where $c$ is a constant, for deciding whether an input digraph contains a spanning out-tree with at least $k$ internal vertices. This answers in affirmative a question of Gutin, Razgon and Kim (Proc. AAIM'08).

cs.DS↗