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Incompressibility through Colors and IDs
"... In parameterized complexity each problem instance comes with a parameter k and the parameterized problem is said to admit a polynomial kernel if there are polynomial time preprocessing rules that reduce the input instance down to an instance with size polynomial in k. Many problems have been shown t ..."
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In parameterized complexity each problem instance comes with a parameter k and the parameterized problem is said to admit a polynomial kernel if there are polynomial time preprocessing rules that reduce the input instance down to an instance with size polynomial in k. Many problems have been shown to admit polynomial kernels, but it is only recently that a framework for showing the nonexistence of polynomial kernels for specific problems has been developed by Bodlaender et al. [6] and Fortnow and Santhanam [15]. With few exceptions, all known kernelization lower bounds result have been obtained by directly applying this framework. In this paper we show how to combine these results with combinatorial reductions which use colors and IDs in order to prove kernelization lower bounds for a variety of basic problems. Below we give a summary of our main results. All our results are under the assumption that the polynomial hierarchy does not collapse to the third level. • We show that the Steiner Tree problem parameterized by the number of terminals and solution size, and the Connected Vertex Cover and Capacitated Vertex Cover problems do not admit a polynomial kernel. The two latter results are surprising because the closely related Vertex Cover problem admits a kernel of size 2k.
Capacitated domination and covering: A parameterized perspective
 Proceedings 3rd International Workshop on Parameterized and Exact Computation, IWPEC 2008
"... Capacitated versions of Dominating Set and Vertex Cover have been studied intensively in terms of polynomial time approximation algorithms. Although the problems Dominating Set and Vertex Cover have been subjected to considerable scrutiny in the parameterized complexity world, this is not true for t ..."
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Capacitated versions of Dominating Set and Vertex Cover have been studied intensively in terms of polynomial time approximation algorithms. Although the problems Dominating Set and Vertex Cover have been subjected to considerable scrutiny in the parameterized complexity world, this is not true for the capacitated versions. Here we make an attempt to understand the behavior of the problems Capacitated Dominating Set and Capacitated Vertex Cover from the perspective of parameterized complexity. The original versions of these problems, Vertex Cover and Dominating Set, are known to be fixed parameter tractable when parameterized by a structure of the graph called the treewidth (tw). In this paper we show that the capacitated versions of these problems behave differently. Our results are: • Capacitated Dominating Set is W[1]hard when parameterized by treewidth. In fact, Capacitated Dominating Set is W[1]hard when parameterized by both treewidth and solution size k of the capacitated dominating set. • Capacitated Vertex Cover is W[1]hard when parameterized by treewidth. • Capacitated Vertex Cover can be solved in time 2O(tw log k) nO(1) where tw is the treewidth of the input graph and k is the solution size. As a corollary, we show that the weighted version of Capacitated Vertex Cover in general graphs can be solved in time 2O(k log k) nO(1). This improves the earlier algorithm of Guo et al. [15] running in time O(1.2k2 + n2). We would also like to point out that our W[1]hardness result for Capacitated Vertex Cover, when parameterized by treewidth, makes it (to the best of our knowledge) the first known “subset problem ” which has turned out to be fixed parameter tractable when parameterized by solution size but W[1]hard when parameterized by treewidth. 1
Subexponential Algorithms for Partial Cover Problems
"... Partial Cover problems are optimization versions of fundamental and well studied problems like Vertex Cover and Dominating Set. Here one is interested in covering (or dominating) the maximum number of edges (or vertices) using a given number (k) of vertices, rather than covering all edges (or vertic ..."
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Partial Cover problems are optimization versions of fundamental and well studied problems like Vertex Cover and Dominating Set. Here one is interested in covering (or dominating) the maximum number of edges (or vertices) using a given number (k) of vertices, rather than covering all edges (or vertices). In general graphs, these problems are hard for parameterized complexity classes when parameterized by k. It was recently shown by Amini et. al. [FSTTCS 08] that Partial Vertex Cover and Partial Dominating Set are fixed parameter tractable on large classes of sparse graphs, namely Hminor free graphs, which include planar graphs and graphs of bounded genus. In particular, it was shown that on planar graphs both problems can be solved in time 2 O(k) n O(1). During the last decade there has been an extensive study on parameterized subexponential algorithms. In particular, it was shown that the classical Vertex Cover and Dominating Set problems can be solved in subexponential time on Hminor free graphs. The techniques developed to obtain subexponential algorithms for classical problems do not apply to partial cover problems. It was left as an open problem by Amini et al. [FSTTCS 08] whether there is a subexponential algorithm for Partial Vertex Cover and Partial Dominating Set. In this paper, we answer the question affirmatively by solving both problems in time 2 O( √ k) n O(1) not only on planar graphs but also on much larger classes of graphs, namely, apexminor free graphs. Compared to previously known algorithms for these problems our algorithms are significantly faster and simpler. 1
Implicit Branching and Parameterized Partial Cover Problems
 IN PROC. OF IARCS CONFERENCE ON FOUNDATIONS OF SOFTWARE TECHNOLOGY AND THEORETICAL COMPUTER SCIENCE (FSTTCS), LEIBNIZ INTERNATIONAL PROCEEDINGS IN INFORMATICS, SCHLOSS DAGSTUHL–LEIBNIZZENTRUM FUER INFORMATIK
"... Covering problems are fundamental classical problems in optimization, computer science and complexity theory. Typically an input to these problems is a family of sets over a finite universe and the goal is to cover the elements of the universe with as few sets of the family as possible. The variatio ..."
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Cited by 8 (1 self)
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Covering problems are fundamental classical problems in optimization, computer science and complexity theory. Typically an input to these problems is a family of sets over a finite universe and the goal is to cover the elements of the universe with as few sets of the family as possible. The variations of covering problems include well known problems like Set Cover, Vertex Cover, Dominating Set and Facility Location to name a few. Recently there has been a lot of study on partial covering problems, a natural generalization of covering problems. Here, the goal is not to cover all the elements but to cover the specified number of elements with the minimum number of sets. In this paper we study partial covering problems in graphs in the realm of parameterized complexity. Classical (nonpartial) version of all these problems have been intensively studied in planar graphs and in graphs excluding a fixed graph H as a minor. However, the techniques developed for parameterized version of nonpartial covering problems cannot be applied directly to their partial counterparts. The approach we use, to show that various partial covering problems are fixed parameter tractable on planar graphs, graphs of bounded local treewidth and graph excluding some graph as a minor, is quite different from previously known techniques. The main idea behind our approach is the concept of implicit branching. We find implicit branching technique to be interesting on its own and believe that it can be used for some other problems.
Parameterized Algorithmics for Finding Connected Motifs in Biological Networks
 IEEE TRANSACTIONS ON COMPUTATIONAL BIOLOGY AND BIOINFORMATICS
"... We study the NPhard LISTCOLORED GRAPH MOTIF problem which, given an undirected listcolored graph G = (V, E) and a multiset M of colors, asks for maximumcardinality sets S ⊆ V and M ′ ⊆ M such that G[S] is connected and contains exactly (with respect to multiplicity) the colors in M ′. LISTCOLO ..."
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Cited by 8 (0 self)
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We study the NPhard LISTCOLORED GRAPH MOTIF problem which, given an undirected listcolored graph G = (V, E) and a multiset M of colors, asks for maximumcardinality sets S ⊆ V and M ′ ⊆ M such that G[S] is connected and contains exactly (with respect to multiplicity) the colors in M ′. LISTCOLORED GRAPH MOTIF has applications in the analysis of biological networks. We study LISTCOLORED GRAPH MOTIF with respect to three different parameterizations. For the parameters motif size M  and solution size S  we present fixedparameter algorithms, whereas for the parameter V −M  we show W[1]hardness for general instances and achieve fixedparameter tractability for a special case of LISTCOLORED GRAPH MOTIF. We implemented the fixedparameter algorithms for parameters M  and S, developed further speedup heuristics for these algorithms, and applied them in the context of querying proteininteraction networks, demonstrating their usefulness for realistic instances. Furthermore, we show that extending the request for motif connectedness to stronger demands such as biconnectedness or bridgeconnectedness leads to W[1]hard problems when the parameter is the motif size M.
Parameterized algorithmics for computational social choice: nine research challenges
 Tsinghua Science and Technology
, 2014
"... Computational Social Choice is an interdisciplinary research area involving Economics, Political Science, and Social Science on the one side, and Mathematics and Computer Science (including Artificial Intelligence and Multiagent Systems) on the other side. Typical computational problems studied in ..."
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Cited by 6 (3 self)
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Computational Social Choice is an interdisciplinary research area involving Economics, Political Science, and Social Science on the one side, and Mathematics and Computer Science (including Artificial Intelligence and Multiagent Systems) on the other side. Typical computational problems studied in this field include the vulnerability of voting procedures against attacks, or preference aggregation in multiagent systems. Parameterized Algorithmics is a subfield of Theoretical Computer Science seeking to exploit meaningful problemspecific parameters in order to identify tractable special cases of in general computationally hard problems. In this paper, we propose nine of our favorite research challenges concerning the parameterized complexity of problems appearing in this context.
FPTalgorithms for connected feedback vertex set
, 2009
"... We study the recently introduced CONNECTED FEEDBACK VERTEX SET (CFVS) problem from the viewpoint 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 ..."
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Cited by 5 (3 self)
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We study the recently introduced CONNECTED FEEDBACK VERTEX SET (CFVS) problem from the viewpoint 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 F ⊆ V, F  ≤ 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( √ k log k) n O(1)) on graphs excluding a fixed graph H as a minor. Our result on general undirected graphs uses, as a 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 could be useful for obtaining parameterized algorithms for other connectivity problems.
CONNECTED VERTEX COVERS IN DENSE GRAPHS
"... Abstract. We consider the variant of the minimum vertex cover problem in which we require that the cover induces a connected subgraph. We give new approximation results for this problem in dense graphs, in which either the minimum or the average degree is linear. In particular, we prove tight parame ..."
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Abstract. We consider the variant of the minimum vertex cover problem in which we require that the cover induces a connected subgraph. We give new approximation results for this problem in dense graphs, in which either the minimum or the average degree is linear. In particular, we prove tight parameterized upper bounds on the approximation returned by Savage’s algorithm, and extend a vertex cover algorithm from Karpinski and Zelikovsky to the connected case. The new algorithm approximates the minimum connected vertex cover problem within a factor strictly less than 2 on all dense graphs. All these results are shown to be tight. Finally, we introduce the price of connectivity for the vertex cover problem, defined as the worstcase ratio between the sizes of a minimum connected vertex cover and a minimum vertex cover. We prove that the price of connectivity is bounded by 2/(1 + ε) in graphs with average degree εn, and give a family of neartight examples. Key words: approximation algorithm, vertex cover, connected vertex cover, dense graph. 1.
Combinatorial Voter Control in Elections
"... Voter control problems model situations such as an external agent trying to affect the result of an election by adding voters, for example by convincing some voters to vote who would otherwise not attend the election. Traditionally, voters are added one at a time, with the goal of making a disting ..."
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Cited by 3 (3 self)
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Voter control problems model situations such as an external agent trying to affect the result of an election by adding voters, for example by convincing some voters to vote who would otherwise not attend the election. Traditionally, voters are added one at a time, with the goal of making a distinguished alternative win by adding a minimum number of voters. In this paper, we initiate the study of combinatorial variants of control by adding voters: In our setting, when we choose to add a voter v, we also have to add a whole bundle κ(v) of voters associated with v. We study the computational complexity of this problem for two of the most basic voting rules, namely the Plurality rule and the Condorcet rule.
Parameterized Algorithm for Eternal Vertex Cover
, 2009
"... In this paper we initiate the study of a “dynamic ” variant of the classical Vertex Cover problem, the Eternal Vertex Cover problem, from parameterized algorithmic perspective. Klostermeyer and Mynhardt introduced the Eternal Vertex Cover problem, which consists in placing a minimum number of guards ..."
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In this paper we initiate the study of a “dynamic ” variant of the classical Vertex Cover problem, the Eternal Vertex Cover problem, from parameterized algorithmic perspective. Klostermeyer and Mynhardt introduced the Eternal Vertex Cover problem, which consists in placing a minimum number of guards on the vertices of a graph such that these guards can protect the graph from any sequence of attacks on its edges. In response to an attack, each guard is allowed either to stay in his vertex, or to move to a neighboring vertex. However, at least one guard has to fix the attacked edge by moving along it. The other guards may move to reconfigure and prepare for the next attack. Thus at every step the vertices occupied by guards form a vertex cover. We show that the problem admits a kernel of size 4 k (k +1)+2k, which shows that the problem is fixed parameter tractable when parameterized by the number of available guards k. Finally, we also provide an algorithm with running time O(2 O(k2) + (1.2738) k m + n) for Eternal Vertex Cover, where n is the number of vertices and m the number of edges of the input graph. In passing we also observe that Eternal Vertex Cover is NPhard but that there is a polynomial time 2approximation algorithm. 1