Results 1  10
of
16
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 ..."
Abstract

Cited by 46 (5 self)
 Add to MetaCart
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 Complexity of Some Colorful Problems Parameterized by Treewidth
, 2007
"... We study the complexity of several coloring problems on graphs, parameterized by the treewidth t of the graph: (1) The list chromatic number χl(G) of a graph G is defined to be the smallest positive integer r, such that for every assignment to the vertices v of G, of a list Lv of colors, where eac ..."
Abstract

Cited by 20 (10 self)
 Add to MetaCart
(Show Context)
We study the complexity of several coloring problems on graphs, parameterized by the treewidth t of the graph: (1) The list chromatic number χl(G) of a graph G is defined to be the smallest positive integer r, such that for every assignment to the vertices v of G, of a list Lv of colors, where each list has length at least r, there is a choice of one color from each vertex list Lv yielding a proper coloring of G. We show that the problem of determining whether χl(G) ≤ r, the LIST CHROMATIC NUMBER problem, is solvable in linear time for every fixed treewidth bound t. The method by which this is shown is new and of general applicability. (2) The LIST COLORING problem takes as input a graph G, together with an assignment to each vertex v of a set of colors Cv. The problem is to determine whether it is possible to choose a color for vertex v from the set of permitted colors Cv, for each vertex, so that the obtained coloring of G is proper. We show that this problem is W [1]hard, parameterized by the treewidth of G. The closely related PRECOLORING EXTENSION problem is also shown to be W [1]hard, parameterized
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 ..."
Abstract

Cited by 20 (6 self)
 Add to MetaCart
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.
Cliquewidth: On the Price of Generality
, 2009
"... Many hard problems can be solved efficiently when the input is restricted to graphs of bounded treewidth. By the celebrated result of Courcelle, every decision problem expressible in monadic second order logic is fixed parameter tractable when parameterized by the treewidth of the input graph. Moreo ..."
Abstract

Cited by 11 (1 self)
 Add to MetaCart
(Show Context)
Many hard problems can be solved efficiently when the input is restricted to graphs of bounded treewidth. By the celebrated result of Courcelle, every decision problem expressible in monadic second order logic is fixed parameter tractable when parameterized by the treewidth of the input graph. Moreover, for every fixed k ≥ 0, such problems can be solved in linear time on graphs of treewidth at most k. In particular, this implies that basic problems like Dominating Set, Graph Coloring, Clique, and Hamiltonian Cycle are solvable in linear time on graphs of bounded treewidth. A significant amount of research in graph algorithms has been devoted to extending this result to larger classes of graphs. It was shown that some of the algorithmic metatheorems for treewidth can be carried over to graphs of bounded cliquewidth. Courcelle, Makowsky, and Rotics proved that the analogue of Courcelle’s result holds for graphs of bounded cliquewidth when the logical formulas do not use edge set quantifications. Despite of its generality, this does not resolve the parameterized complexity of many basic problems concerning edge subsets (like Edge Dominating Set), vertex
INTRACTABILITY OF CLIQUEWIDTH PARAMETERIZATIONS
, 2009
"... We show that Edge Dominating Set, Hamiltonian Cycle, and Graph Coloring are W [1]hard parameterized by cliquewidth. It was an open problem, explicitly mentioned in several papers, whether any of these problems is fixed parameter tractable when parameterized by the cliquewidth, that is, solvable i ..."
Abstract

Cited by 9 (2 self)
 Add to MetaCart
(Show Context)
We show that Edge Dominating Set, Hamiltonian Cycle, and Graph Coloring are W [1]hard parameterized by cliquewidth. It was an open problem, explicitly mentioned in several papers, whether any of these problems is fixed parameter tractable when parameterized by the cliquewidth, that is, solvable in time g(k) · nO(1) on nvertex graphs of cliquewidth k, where g is some function of k only. Our results imply that the running time O(nf(k) ) of many cliquewidth based algorithms is essentially the best we can hope for (up to a widely believed assumption from parameterized complexity, namely F P T ̸ = W [1]).
Planar Capacitated Dominating Set is W[1]hard
, 2009
"... Given a graph G together with a capacity function c: V (G) â N, we call S â V (G) a capacitated dominating set if there exists a mapping f: (V (G) \ S) â S which maps every vertex in (V (G) \ S) to one of its neighbors such that the total number of vertices mapped by f to any vertex v â ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
(Show Context)
Given a graph G together with a capacity function c: V (G) â N, we call S â V (G) a capacitated dominating set if there exists a mapping f: (V (G) \ S) â S which maps every vertex in (V (G) \ S) to one of its neighbors such that the total number of vertices mapped by f to any vertex v â S does not exceed c(v). In the Planar Capacitated Dominating Set problem we are given a planar graph G, a capacity function c and a positive integer k and asked whether G has a capacitated dominating set of size at most k. In this paper we show that Planar Capacitated Dominating Set is W[1]hard, resolving an open problem of Dom et al. [IWPEC, 2008]. This is the first bidimensional problem to be shown W[1]hard. Thus Planar Capacitated Dominating Set can become a useful starting point for reductions showing parameterized intractablility of planar graph problems.
Guard Games on Graphs: Keep the Intruder Out!
"... A team of mobile agents, called guards, tries to keep an intruder out of an assigned area by blocking all possible attacks. In a graph model for this setting, the agents and the intruder are located on the vertices of a graph, and they move from node to node via connecting edges. The area protected ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
A team of mobile agents, called guards, tries to keep an intruder out of an assigned area by blocking all possible attacks. In a graph model for this setting, the agents and the intruder are located on the vertices of a graph, and they move from node to node via connecting edges. The area protected by the guards is a subgraph of the given graph. We investigate the algorithmic aspects of finding the minimum number of guards sufficient to patrol the area. We show that this problem is PSPACEhard in general and proceed to investigate a variant of the game where the intruder must reach the guarded area in a single step in order to win. We show that this case approximates the general problem, and that for graphs without cycles of length 5 the minimum number of required guards in both games coincides. We give a polynomial time algorithm for solving the onestep guarding problem in graphs of bounded treewidth, and complement this result by showing that the problem is W [1]hard parameterized by the treewidth of the input graph. We conclude the study of the onestep guarding problem in bounded treewidth graphs by showing that the problem is fixed parameter tractable (FPT) parameterized by the treewidth and maximum degree of the input graph. Finally, we turn our attention to a large class of sparse graphs, including planar graphs and graphs of bounded genus, namely graphs excluding some fixed apex graph as a minor. We prove that the problem is FPT and give a PTAS on apexminorfree graphs.
Kernelization Lower Bounds through Colors and IDs
, 2009
"... In parameterized complexity each problem instance comes with a parameter k, and a parameterized problem is said to admit a polynomial kernel if there are polynomial time preprocessing rules that reduce the input instance to an instance with size polynomial in k. Many problems have been shown to admi ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
In parameterized complexity each problem instance comes with a parameter k, and a parameterized problem is said to admit a polynomial kernel if there are polynomial time preprocessing rules that reduce the input instance 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 [17]. 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 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 k, and the CONNECTED VERTEX COVER and CAPACITATED VERTEX COVER problems do not admit a polynomial kernel. The two latter results are surpris