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265
Infeasibility of instance compression and succinct PCPs for NP
 Electronic Colloquium on Computational Complexity (ECCC
"... The ORSAT problem asks, given Boolean formulae φ1,..., φm each of size at most n, whether at least one of the φi’s is satisfiable. We show that there is no reduction from ORSAT to any set A where the length of the output is bounded by a polynomial in n, unless NP ⊆ coNP/poly, and the PolynomialTi ..."
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Cited by 34 (1 self)
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The ORSAT problem asks, given Boolean formulae φ1,..., φm each of size at most n, whether at least one of the φi’s is satisfiable. We show that there is no reduction from ORSAT to any set A where the length of the output is bounded by a polynomial in n, unless NP ⊆ coNP/poly, and the PolynomialTime Hierarchy collapses. This result settles an open problem proposed by Bodlaender et. al. [4] and Harnik and Naor [15] and has a number of implications. • A number of parametric NP problems, including Satisfiability, Clique, Dominating Set and Integer Programming, are not instance compressible or polynomially kernelizable unless NP ⊆ coNP/poly. • Satisfiability does not have PCPs of size polynomial in the number of variables unless NP ⊆ coNP/poly. • An approach of Harnik and Naor to constructing collisionresistant hash functions from oneway functions is unlikely to be viable in its present form. • (BuhrmanHitchcock) There are no subexponentialsize hard sets for NP unless NP is in coNP/poly. We also study probabilistic variants of compression, and show various results about and connections between these variants. To this end, we introduce a new strong derandomization hypothesis, the Oracle Derandomization Hypothesis, and discuss how it relates to traditional derandomization assumptions. Categories and Subject Descriptors
Llull and Copeland voting computationally resist bribery and control
, 2009
"... Control and bribery are settings in which an external agent seeks to influence the outcome of an election. Constructive control of elections refers to attempts by an agent to, via such actions as addition/deletion/partition of candidates or voters, ensure that a given candidate wins. Destructive con ..."
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Cited by 34 (17 self)
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Control and bribery are settings in which an external agent seeks to influence the outcome of an election. Constructive control of elections refers to attempts by an agent to, via such actions as addition/deletion/partition of candidates or voters, ensure that a given candidate wins. Destructive control refers to attempts by an agent to, via the same actions, preclude a given candidate’s victory. An election system in which an agent can sometimes affect the result and it can be determined in polynomial time on which inputs the agent can succeed is said to be vulnerable to the given type of control. An election system in which an agent can sometimes affect the result, yet in which it is NPhard to recognize the inputs on which the agent can succeed, is said to be resistant to the given type of control. Aside from election systems with an NPhard winner problem, the only systems previously known to be resistant to all the standard control types were highly artificial election systems created by hybridization. This paper studies a parameterized version of Copeland voting, denoted by Copeland α, where the parameter α is a rational number between 0 and 1 that specifies how ties are valued in the pairwise comparisons of candidates. In every previously studied constructive or destructive
A more effective linear kernelization for Cluster Editing
 Theor. Comput. Sci
, 2009
"... Abstract. In the NPhard Cluster Editing problem, we have as input an undirected graph G andanintegerk ≥ 0. The question is whether we can transform G, by inserting and deleting at most k edges, into a cluster graph, that is, a union of disjoint cliques. We first confirm a conjecture by Michael Fell ..."
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Cited by 30 (7 self)
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Abstract. In the NPhard Cluster Editing problem, we have as input an undirected graph G andanintegerk ≥ 0. The question is whether we can transform G, by inserting and deleting at most k edges, into a cluster graph, that is, a union of disjoint cliques. We first confirm a conjecture by Michael Fellows [IWPEC 2006] that there is a polynomialtime kernelization for Cluster Editing that leads to a problem kernel with at most 6k vertices. More precisely, we present a cubictime algorithm that, given a graph G andanintegerk ≥ 0, finds a graph G ′ and an integer k ′ ≤ k such that G can be transformed into a cluster graph by at most k edge modifications iff G ′ can be transformed into a cluster graph by at most k ′ edge modifications, and the problem kernel G ′ has at most 6k vertices. So far, only a problem kernel of 24k vertices was known. Second, we show that this bound for the number of vertices of G ′ can be further improved to 4k. Finally, we consider the variant of Cluster Editing where the number of cliques that the cluster graph can contain is stipulated to be a constant d>0. We present a simple kernelization for this variant leaving a problem kernel of at most (d +2)k + d vertices. 1
A Multivariate Complexity Analysis of Determining Possible Winners Given Incomplete Votes
"... The POSSIBLE WINNER problem asks whether some distinguished candidate may become the winner of an election when the given incomplete votes are extended into complete ones in a favorable way. POSSIBLE WINNER is NPcomplete for common voting rules such as Borda, many other positional scoring rules, Bu ..."
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Cited by 29 (10 self)
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The POSSIBLE WINNER problem asks whether some distinguished candidate may become the winner of an election when the given incomplete votes are extended into complete ones in a favorable way. POSSIBLE WINNER is NPcomplete for common voting rules such as Borda, many other positional scoring rules, Bucklin, Copeland etc. We investigate how three different parameterizations influence the computational complexity of POSSIBLE WINNER for a number of voting rules. We show fixedparameter tractability results with respect to the parameter “number of candidates ” but intractability results with respect to the parameter “number of votes”. Finally, we derive fixedparameter tractability results with respect to the parameter “total number of undetermined candidate pairs ” and identify an interesting polynomialtime solvable special case for Borda. 1
Pattern Matching for ArcAnnotated Sequences
 In Proc. of 22nd FSTTCS, number 2556 in LNCS
, 2002
"... A study of pattern matching for arcannotated sequences is started. An O(nm) time algorithm is given to determine whether a length m sequence with nested arc annotations is an arcpreserving subsequence of a length n sequence with nested arc annotations, called APS(nested,nested). Arcannotated ..."
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Cited by 27 (2 self)
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A study of pattern matching for arcannotated sequences is started. An O(nm) time algorithm is given to determine whether a length m sequence with nested arc annotations is an arcpreserving subsequence of a length n sequence with nested arc annotations, called APS(nested,nested). Arcannotated sequences and, in particular, those with nested arc structure are motivated by applications in RNA structure comparison. Our algorithm can be used to accelerate a recent fixedparameter algorithm for LAPCS(nested,nested) and generalizes results for ordered tree inclusion problems. In particular, the presented dynamic programming methodology implies a quadratic time algorithm for an open problem posed by Vialette.
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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Cited by 24 (5 self)
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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.
Reflections on multivariate algorithmics and problem parameterization
 In Proceedings of the 27th International Symposium on Theoretical Aspects of Computer Science (STACS ’10), volume 5 of LIPIcs
"... Abstract. Research on parameterized algorithmics for NPhard problems has steadily grown over the last years. We survey and discuss how parameterized complexity analysis naturally develops into the field of multivariate algorithmics. Correspondingly, we describe how to perform a systematic investiga ..."
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Cited by 24 (19 self)
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Abstract. Research on parameterized algorithmics for NPhard problems has steadily grown over the last years. We survey and discuss how parameterized complexity analysis naturally develops into the field of multivariate algorithmics. Correspondingly, we describe how to perform a systematic investigation and exploitation of the “parameter space ” of computationally hard problems.
A quadratic kernel for feedback vertex set
 in Proc. 20th SODA, ACM/SIAM, 2009
"... We prove that given an undirected graph G on n vertices and an integer k, one can compute in polynomial time in n a graph G ′ with at most 5k 2 +k vertices and an integer k ′ such that G has a feedback vertex set of size at most k iff G ′ has a feedback vertex set of size at most k ′. This result im ..."
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Cited by 23 (2 self)
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We prove that given an undirected graph G on n vertices and an integer k, one can compute in polynomial time in n a graph G ′ with at most 5k 2 +k vertices and an integer k ′ such that G has a feedback vertex set of size at most k iff G ′ has a feedback vertex set of size at most k ′. This result improves a previous O(k 11) kernel of Burrage et al. [6], and a more recent cubic kernel of Bodlaender [3]. This problem was communicated by Fellows in [5]. 1
Techniques for Practical FixedParameter Algorithms
, 2007
"... The fixedparameter approach is an algorithm design technique for solving combinatorially hard (mostly NPhard) problems. For some of these problems, it can lead to algorithms that are both efficient and yet at the same time guaranteed to find optimal solutions. Focusing on their application to solv ..."
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Cited by 22 (9 self)
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The fixedparameter approach is an algorithm design technique for solving combinatorially hard (mostly NPhard) problems. For some of these problems, it can lead to algorithms that are both efficient and yet at the same time guaranteed to find optimal solutions. Focusing on their application to solving NPhard problems in practice, we survey three main techniques to develop fixedparameter algorithms, namely: kernelization (data reduction with provable performance guarantee), depthbounded search trees and a new technique called iterative compression. Our discussion is circumstantiated by several concrete case studies and provides pointers to various current challenges in the field.
Combinatorial Optimization on Graphs of Bounded Treewidth
, 2007
"... There are many graph problems that can be solved in linear or polynomial time with a dynamic programming algorithm when the input graph has bounded treewidth. For combinatorial optimization problems, this is a useful approach for obtaining fixedparameter tractable algorithms. Starting from trees an ..."
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Cited by 21 (1 self)
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There are many graph problems that can be solved in linear or polynomial time with a dynamic programming algorithm when the input graph has bounded treewidth. For combinatorial optimization problems, this is a useful approach for obtaining fixedparameter tractable algorithms. Starting from trees and seriesparallel graphs, we introduce the concepts of treewidth and tree decompositions, and illustrate the technique with the Weighted Independent Set problem as an example. The paper surveys some of the latest developments, putting an emphasis on applicability, on algorithms that exploit tree decompositions, and on algorithms that determine or approximate treewidth and find tree decompositions with optimal or close to optimal treewidth. Directions for further research and suggestions for further reading are also given.