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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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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.
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
Weak Compositions and Their Applications to Polynomial Lower Bounds for Kernelization
"... Abstract. We introduce a new form of composition called weak composition that allows us to obtain polynomial kernelization lowerbounds for several natural parameterized problems. Let d ≥ 2 be some constant and let L1, L2 ⊆ {0, 1} ∗ × N be two parameterized problems where the unparameterized versi ..."
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Abstract. We introduce a new form of composition called weak composition that allows us to obtain polynomial kernelization lowerbounds for several natural parameterized problems. Let d ≥ 2 be some constant and let L1, L2 ⊆ {0, 1} ∗ × N be two parameterized problems where the unparameterized version of L1 is NPhard. Assuming coNP ̸ ⊆ NP/poly, our framework essentially states that composing t L1instances each with parameter k, to an L2instance with parameter k ′ ≤ t 1/d k O(1) , implies that L2 does not have a kernel of size O(k d−ε) for any ε> 0. We show two examples of weak composition and derive polynomial kernelization lower bounds for dBipartite Regular Perfect Code and dDimensional Matching, parameterized by the solution size k. By reduction, using linear parameter transformations, we then derive the following lowerbounds for kernel sizes when the parameter is the solution size k (assuming coNP ̸ ⊆ NP/poly): – dSet Packing, dSet Cover, dExact Set Cover, Hitting Set with dBounded Occurrences, and Exact Hitting Set with dBounded Occurrences have no kernels of size O(k d−3−ε) for any ε> 0. – Kd Packing and Induced K1,d Packing have no kernels of size O(k d−4−ε) for any ε> 0. – dRedBlue Dominating Set and dSteiner Tree have no kernels of sizes O(k d−3−ε) and
Conondeterminism in compositions: A kernelization lower bound for a Ramseytype problem
, 2012
"... Until recently, techniques for obtaining lower bounds for kernelization were one of the most sought after tools in the field of parameterized complexity. Now, after a strong influx of techniques, we are in the fortunate situation of having tools available that are even stronger than what has been re ..."
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Until recently, techniques for obtaining lower bounds for kernelization were one of the most sought after tools in the field of parameterized complexity. Now, after a strong influx of techniques, we are in the fortunate situation of having tools available that are even stronger than what has been required in their applications so far. Based on a result of Fortnow and Santhanam (STOC 2008, JCSS 2011), Bodlaender et al. (ICALP 2008, JCSS 2009) showed that, unless NP ⊆ coNP/poly, the existence of a deterministic polynomialtime composition algorithm, i.e., an algorithm which outputs an instance of bounded parameter value which is yes if and only if one of t input instances is yes, rules out the existence of polynomial kernels for a problem. Dell and van Melkebeek (STOC 2010) continued this line
Parameterized complexity and kernelizability of max ones and exact ones problems
 In Proc. of the 35th international symposium on Mathematical Foundations of Computer Science (MFCS
, 2010
"... Abstract. For a finite set Γ of Boolean relations, Max Ones SAT(Γ) and Exact Ones SAT(Γ) are generalized satisfiability problems where every constraint relation is from Γ, and the task is to find a satisfying assignment with at least/exactly k variables set to 1, respectively. We study the parameter ..."
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Cited by 6 (3 self)
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Abstract. For a finite set Γ of Boolean relations, Max Ones SAT(Γ) and Exact Ones SAT(Γ) are generalized satisfiability problems where every constraint relation is from Γ, and the task is to find a satisfying assignment with at least/exactly k variables set to 1, respectively. We study the parameterized complexity of these problems, including the question whether they admit polynomial kernels. For Max Ones SAT(Γ), we give a classification into 5 different complexity levels: polynomialtime solvable, admits a polynomial kernel, fixedparameter tractable, solvable in polynomial time for fixed k, and NPhard already for k = 1. For Exact Ones SAT(Γ), we refine the classification obtained earlier by having a closer look at the fixedparameter tractable cases and classifying the sets Γ for which Exact Ones SAT(Γ) admits a polynomial kernel. 1
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 ..."
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Cited by 3 (0 self)
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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