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143
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 70 (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
Bidimensionality and Kernels
, 2010
"... Bidimensionality theory appears to be a powerful framework in the development of metaalgorithmic techniques. It was introduced by Demaine et al. [J. ACM 2005] as a tool to obtain subexponential time parameterized algorithms for bidimensional problems on Hminor free graphs. Demaine and Hajiaghayi ..."
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Cited by 58 (23 self)
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Bidimensionality theory appears to be a powerful framework in the development of metaalgorithmic techniques. It was introduced by Demaine et al. [J. ACM 2005] as a tool to obtain subexponential time parameterized algorithms for bidimensional problems on Hminor free graphs. Demaine and Hajiaghayi [SODA 2005] extended the theory to obtain polynomial time approximation schemes (PTASs) for bidimensional problems. In this paper, we establish a third metaalgorithmic direction for bidimensionality theory by relating it to the existence of linear kernels for parameterized problems. In parameterized complexity, each problem instance comes with a parameter k and the parameterized problem is said to admit a linear kernel if there is a polynomial time algorithm, called
Satisfiability Allows No Nontrivial Sparsification Unless The PolynomialTime Hierarchy Collapses
 ELECTRONIC COLLOQUIUM ON COMPUTATIONAL COMPLEXITY, REPORT NO. 38 (2010)
, 2010
"... Consider the following twoplayer communication process to decide a language L: The first player holds the entire input x but is polynomially bounded; the second player is computationally unbounded but does not know any part of x; their goal is to cooperatively decide whether x belongs to L at small ..."
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Cited by 56 (2 self)
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Consider the following twoplayer communication process to decide a language L: The first player holds the entire input x but is polynomially bounded; the second player is computationally unbounded but does not know any part of x; their goal is to cooperatively decide whether x belongs to L at small cost, where the cost measure is the number of bits of communication from the first player to the second player. For any integer d ≥ 3 and positive real ǫ we show that if satisfiability for nvariable dCNF formulas has a protocol of cost O(n d−ǫ) then coNP is in NP/poly, which implies that the polynomialtime hierarchy collapses to its third level. The result even holds when the first player is conondeterministic, and is tight as there exists a trivial protocol for ǫ = 0. Under the hypothesis that coNP is not in NP/poly, our result implies tight lower bounds for parameters of interest in several areas, namely sparsification, kernelization in parameterized complexity, lossy compression, and probabilistically checkable proofs. By reduction, similar results hold for other NPcomplete problems. For the vertex cover problem on nvertex duniform hypergraphs, the above statement holds for any integer d ≥ 2. The case d = 2 implies that no NPhard vertex deletion problem based on a graph property that is inherited by subgraphs can have kernels consisting of O(k 2−ǫ) edges unless coNP is in NP/poly, where k denotes the size of the deletion set. Kernels consisting of O(k 2) edges are known for several problems in the class, including vertex cover, feedback vertex set, and boundeddegree deletion.
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 46 (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.
CrossComposition: A New Technique for Kernelization Lower Bounds
, 2011
"... We introduce a new technique for proving kernelization lower bounds, called crosscomposition. A classical problem L crosscomposes into a parameterized problem Q if an instance of Q with polynomially bounded parameter value can express the logical OR of a sequence of instances of L. Building on wor ..."
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Cited by 38 (8 self)
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We introduce a new technique for proving kernelization lower bounds, called crosscomposition. A classical problem L crosscomposes into a parameterized problem Q if an instance of Q with polynomially bounded parameter value can express the logical OR of a sequence of instances of L. Building on work by Bodlaender et al. (ICALP 2008) and using a result by Fortnow and Santhanam (STOC 2008) we show that if an NPhard problem crosscomposes into a parameterized problem Q then Q does not admit a polynomial kernel unless the polynomial hierarchy collapses. Our technique generalizes and strengthens the recent techniques of using orcomposition algorithms and of transferring the lower bounds via polynomial parameter transformations. We show its applicability by proving kernelization lower bounds for a number of important graphs problems with structural (nonstandard) parameterizations, e.g., Chromatic Number, Clique, and Weighted Feedback Vertex Set do not admit polynomial kernels with respect to the vertex cover number of the input graphs unless the polynomial hierarchy collapses, contrasting the fact that these problems are trivially fixedparameter tractable for this parameter. We have similar lower bounds for Feedback Vertex Set.
New Limits to Classical and Quantum Instance Compression
 ELECTRONIC COLLOQUIUM ON COMPUTATIONAL COMPLEXITY, REPORT NO. 112
, 2012
"... Given an instance of a hard decision problem, a limited goal is to compress that instance into a smaller, equivalent instance of a second problem. As one example, consider the problem where, given Boolean formulas ψ 1,...,ψ t, we must determine if at least one ψ j is satisfiable. An ORcompression s ..."
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Cited by 20 (1 self)
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Given an instance of a hard decision problem, a limited goal is to compress that instance into a smaller, equivalent instance of a second problem. As one example, consider the problem where, given Boolean formulas ψ 1,...,ψ t, we must determine if at least one ψ j is satisfiable. An ORcompression scheme for SAT is a polynomialtime reduction R that maps (ψ 1,...,ψ t) to a string z, such that z lies in some “target ” language L ′ if and only if ∨ j [ψj ∈ SAT] holds. (Here, L ′ can be arbitrarily complex.) ANDcompression schemes are defined similarly. A compression scheme is strong if z  is polynomially bounded in n = maxj ψ j , independent of t. Strong compression for SAT seems unlikely. Work of Harnik and Naor (FOCS ’06/SICOMP ’10) and Bodlaender, Downey, Fellows, and Hermelin (ICALP ’08/JCSS ’09) showed that the infeasibility of strong ORcompression for SAT would show limits to instance compression for a large number of natural problems. Bodlaender et al. also showed that the infeasibility of strong ANDcompression for SAT would have consequences for a different list of problems. Motivated by this, Fortnow and Santhanam (STOC ’08/JCSS ’11) showed that if SAT is strongly ORcompressible,
Kernelization of Packing Problems
, 2011
"... Kernelization algorithms are polynomialtime reductions from a problem to itself that guarantee their output to have a size not exceeding some bound. For example, dSet Matching for integers d ≥ 3 is the problem of nding a matching of size at least k in a given duniform hypergraph and has kernels w ..."
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Cited by 20 (2 self)
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Kernelization algorithms are polynomialtime reductions from a problem to itself that guarantee their output to have a size not exceeding some bound. For example, dSet Matching for integers d ≥ 3 is the problem of nding a matching of size at least k in a given duniform hypergraph and has kernels with O(k d) edges. Recently, Bodlaender et al. [ICALP 2008], Fortnow and Santhanam [STOC 2008], Dell and Van Melkebeek [STOC 2010] developed a framework for proving lower bounds on the kernel size for certain problems, under the complexitytheoretic hypothesis that coNP is not contained in NP/poly. Under the same hypothesis, we show lower bounds for the kernelization of dSet Matching and other packing problems. Our bounds are tight for dSet Matching: It does not have kernels with O(k d−ɛ) edges for any ɛ> 0 unless the hypothesis fails. By reduction, this transfers to a bound of O(k d−1−ɛ) for the problem of nding k vertexdisjoint cliques of size d in standard graphs. It is natural to ask for tight bounds on the kernel sizes of such graph packing problems. We make rst progress in that direction by showing nontrivial kernels with O(k 2.5) edges for the problem of nding k vertexdisjoint paths of three edges each. This does not quite match the best lower bound of O(k 2−ɛ) that we can prove. Most of our lower bound proofs follow a general scheme that we discover: To exclude kernels of size O(k d−ɛ) for a problem in duniform hypergraphs, one should reduce from a carefully chosen dpartite problem that is still NPhard. As an illustration, we apply this scheme to the vertex cover problem, which allows us to replace the numbertheoretical construction by Dell and Van Melkebeek [STOC 2010] with shorter elementary arguments. 1
KERNEL(S) FOR PROBLEMS WITH NO KERNEL: ON OUTTREES WITH MANY LEAVES (EXTENDED ABSTRACT)
 STACS 2009
, 2009
"... The kLeaf OutBranching problem is to find an outbranching, that is a rooted oriented spanning tree, with at least k leaves in a given digraph. The problem has recently received much attention from the viewpoint of parameterized algorithms. Here, we take a kernelization based approach to the kLea ..."
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Cited by 19 (7 self)
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The kLeaf OutBranching problem is to find an outbranching, that is a rooted oriented spanning tree, with at least k leaves in a given digraph. The problem has recently received much attention from the viewpoint of parameterized algorithms. Here, we take a kernelization based approach to the kLeafOutBranching problem. We give the first polynomial kernel for Rooted kLeafOutBranching, a variant of kLeafOutBranching where the root of the tree searched for is also a part of the input. Our kernel has cubic size and is obtained using extremal combinatorics. For the kLeafOutBranching problem, we show that no polynomial kernel is possible unless the polynomial hierarchy collapses to third level by applying a recent breakthrough result by Bodlaender et al. (ICALP 2008) in a nontrivial fashion. However, our positive results for Rooted kLeafOutBranching immediately imply that the seemingly intractable kLeafOutBranching problem admits a data reduction to n independent O(k³) kernels. These two results, tractability and intractability side by side, are the first ones separating manytoone kernelization from Turing kernelization. This answers affirmatively an open problem regarding “cheat kernelization” raised by Mike Fellows and Jiong Guo independently.
Compression via Matroids: A Randomized Polynomial Kernel for Odd Cycle Transversal
"... The Odd Cycle Transversal problem (OCT) asks whether a given graph can be made bipartite by deleting at most k of its vertices. In a breakthrough result Reed, Smith, and Vetta (Operations Research Letters, 2004) gave a O(4 k kmn) time algorithm for it, the first algorithm with polynomial runtime of ..."
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Cited by 19 (4 self)
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The Odd Cycle Transversal problem (OCT) asks whether a given graph can be made bipartite by deleting at most k of its vertices. In a breakthrough result Reed, Smith, and Vetta (Operations Research Letters, 2004) gave a O(4 k kmn) time algorithm for it, the first algorithm with polynomial runtime of uniform degree for every fixed k. It is known that this implies a polynomialtime compression algorithm that turns OCT instances into equivalent instances of size at most O(4 k), a socalled kernelization. Since then the existence of a polynomial kernel for OCT, i.e., a kernelization with size bounded polynomially in k, has turned into one of the main open questions in the study of kernelization. Despite the impressive progress in the area, including the recent development of lower bound techniques (Bodlaender
PlanarF Deletion: Approximation, Kernelization and Optimal FPT Algorithms
"... Let F be a finite set of graphs. In the FDeletion problem, we are given an nvertex graph G and an integer k as input, and asked whether at most k vertices can be deleted from G such that the resulting graph does not contain a graph from F as a minor. FDeletion is a generic problem and by selectin ..."
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Cited by 18 (8 self)
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Let F be a finite set of graphs. In the FDeletion problem, we are given an nvertex graph G and an integer k as input, and asked whether at most k vertices can be deleted from G such that the resulting graph does not contain a graph from F as a minor. FDeletion is a generic problem and by selecting different sets of forbidden minors F, one can obtain various fundamental problems such as Vertex Cover, Feedback Vertex Set or Treewidth ηDeletion. In this paper we obtain a number of generic algorithmic results about FDeletion, when F contains at least one planar graph. The highlights of our work are • A constant factor approximation algorithm for the optimization version of FDeletion; • A linear time and single exponential parameterized algorithm, that is, an algorithm running in time O(2 O(k) n), for the parameterized version of FDeletion where all graphs in F are connected; • A polynomial kernel for parameterized FDeletion. These algorithms unify, generalize, and improve a multitude of results in the literature. Our main results have several direct applications, but also the methods we develop on the way have applicability beyond the scope of this paper. Our results – constant factor approximation, polynomial kernelization and FPT algorithms – are stringed together by a common theme of polynomial time preprocessing.