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21
Kernelization: New Upper and Lower Bound Techniques
 In Proc. of the 4th International Workshop on Parameterized and Exact Computation (IWPEC), volume 5917 of LNCS
, 2009
"... Abstract. In this survey, we look at kernelization: algorithms that transform in polynomial time an input to a problem to an equivalent input, whose size is bounded by a function of a parameter. Several results of recent research on kernelization are mentioned. This survey looks at some recent resu ..."
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Abstract. In this survey, we look at kernelization: algorithms that transform in polynomial time an input to a problem to an equivalent input, whose size is bounded by a function of a parameter. Several results of recent research on kernelization are mentioned. This survey looks at some recent results where a general technique shows the existence of kernelization algorithms for large classes of problems, in particular for planar graphs and generalizations of planar graphs, and recent lower bound techniques that give evidence that certain types of kernelization algorithms do not exist.
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.
On the computation of fully proportional representation
 JOURNAL OF AI RESEARCH
, 2013
"... We investigate two systems of fully proportional representation suggested by Chamberlin & Courant and Monroe. Both systems assign a representative to each voter so that the “sum of misrepresentations” is minimized. The winner determination problem for both systems is known to be NPhard, hence t ..."
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Cited by 20 (6 self)
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We investigate two systems of fully proportional representation suggested by Chamberlin & Courant and Monroe. Both systems assign a representative to each voter so that the “sum of misrepresentations” is minimized. The winner determination problem for both systems is known to be NPhard, hence this work aims at investigating whether there are variants of the proposed rules and/or specific electorates for which these problems can be solved efficiently. As a variation of these rules, instead of minimizing the sum of misrepresentations, we considered minimizing the maximalmisrepresentationintroducingeffectively two new rules. In the general case these “minimax ” versions of classical rules appeared to be still NPhard. We investigated the parameterized complexity of winner determination of the two classical and two new rules with respect to several parameters. Here we have a mixture of positive and negative results: e.g., we proved fixedparameter tractability for the parameter the number of candidates but fixedparameter intractability for the number of winners. For singlepeaked electorates our results are overwhelmingly positive: we provide polynomialtime algorithms for most of the considered problems. The only rule that remains NPhard for singlepeaked electorates is the classical Monroe rule. 1.
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
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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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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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.
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
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.
FPT algorithms for connected feedback vertex set
 In Proc. WALCOM 2010
, 2010
"... Abstract. 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 th ..."
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Abstract. 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 ) 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.
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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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.