Kernelization Lower Bounds By Cross-Composition

We introduce the cross-composition framework for proving kernelization lower bounds. A classical problem L AND/OR-cross-composes into a parameterized problem Q if it is possible to efficiently construct an instance of Q with polynomially bounded parameter value that expresses the logical AND or 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) with a refinement by Dell and van Melkebeek (STOC 2010), we show that if an NP-hard problem OR-cross-composes into a parameterized problem Q then Q does not admit a polynomial kernel unless NP \subseteq coNP/poly and the polynomial hierarchy collapses. Similarly, an AND-cross-composition for Q rules out polynomial kernels for Q under Bodlaender et al.'s AND-distillation conjecture. Our technique generalizes and strengthens the recent techniques of using composition 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 (non-standard) parameterizations, e.g., Clique, Chromatic Number, Weighted Feedback Vertex Set, and Weighted Odd Cycle Transversal 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 fixed-parameter tractable for this parameter. After learning of our results, several teams of authors have successfully applied the cross-composition framework to different parameterized problems. For completeness, our presentation of the framework includes several extensions based on this follow-up work. For example, we show how a relaxed version of OR-cross-compositions may be used to give lower bounds on the degree of the polynomial in the kernel size.Comment: A preliminary version appeared in the proceedings of the 28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011) under the title "Cross-Composition: A New Technique for Kernelization Lower Bounds". Several results have been strengthened compared to the preliminary version (http://arxiv.org/abs/1011.4224). 29 pages, 2 figure

Cross-Composition: A New Technique for Kernelization Lower Bounds

We introduce a new technique for proving kernelization lower bounds, called cross-composition. A classical problem L cross-composes 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 NP-complete problem cross-composes 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 OR-composition 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 (non-standard) 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 fixed-parameter tractable for this parameter. We have similar lower bounds for Feedback Vertex Set.Comment: Updated information based on final version submitted to STACS 201

Crossing Paths with Hans Bodlaender:A Personal View on Cross-Composition for Sparsification Lower Bounds

On the occasion of Hans Bodlaender’s 60th birthday, I give a personal account of our history and work together on the technique of cross-composition for kernelization lower bounds. I present several simple new proofs for polynomial kernelization lower bounds using cross-composition, interlaced with personal anecdotes about my time as Hans’ PhD student at Utrecht University. Concretely, I will prove that Vertex Cover, Feedback Vertex Set, and the H-Factor problem for every graph H that has a connected component of at least three vertices, do not admit kernels of (formula presented) bits when parameterized by the number of vertices n for any (formula presented), unless (formula presented). These lower bounds are obtained by elementary gadget constructions, in particular avoiding the use of the Packing Lemma by Dell and van Melkebeek.</p

Kernel Bounds for Structural Parameterizations of Pathwidth

Assuming the AND-distillation conjecture, the Pathwidth problem of determining whether a given graph G has pathwidth at most k admits no polynomial kernelization with respect to k. The present work studies the existence of polynomial kernels for Pathwidth with respect to other, structural, parameters. Our main result is that, unless NP is in coNP/poly, Pathwidth admits no polynomial kernelization even when parameterized by the vertex deletion distance to a clique, by giving a cross-composition from Cutwidth. The cross-composition works also for Treewidth, improving over previous lower bounds by the present authors. For Pathwidth, our result rules out polynomial kernels with respect to the distance to various classes of polynomial-time solvable inputs, like interval or cluster graphs. This leads to the question whether there are nontrivial structural parameters for which Pathwidth does admit a polynomial kernelization. To answer this, we give a collection of graph reduction rules that are safe for Pathwidth. We analyze the success of these results and obtain polynomial kernelizations with respect to the following parameters: the size of a vertex cover of the graph, the vertex deletion distance to a graph where each connected component is a star, and the vertex deletion distance to a graph where each connected component has at most c vertices.Comment: This paper contains the proofs omitted from the extended abstract published in the proceedings of Algorithm Theory - SWAT 2012 - 13th Scandinavian Symposium and Workshops, Helsinki, Finland, July 4-6, 201

FPT is Characterized by Useful Obstruction Sets

Many graph problems were first shown to be fixed-parameter tractable using the results of Robertson and Seymour on graph minors. We show that the combination of finite, computable, obstruction sets and efficient order tests is not just one way of obtaining strongly uniform FPT algorithms, but that all of FPT may be captured in this way. Our new characterization of FPT has a strong connection to the theory of kernelization, as we prove that problems with polynomial kernels can be characterized by obstruction sets whose elements have polynomial size. Consequently we investigate the interplay between the sizes of problem kernels and the sizes of the elements of such obstruction sets, obtaining several examples of how results in one area yield new insights in the other. We show how exponential-size minor-minimal obstructions for pathwidth k form the crucial ingredient in a novel OR-cross-composition for k-Pathwidth, complementing the trivial AND-composition that is known for this problem. In the other direction, we show that OR-cross-compositions into a parameterized problem can be used to rule out the existence of efficiently generated quasi-orders on its instances that characterize the NO-instances by polynomial-size obstructions.Comment: Extended abstract with appendix, as accepted to WG 201

Kernel Bounds for Path and Cycle Problems

Connectivity problems like k-Path and k-Disjoint Paths relate to many important milestones in parameterized complexity, namely the Graph Minors Project, color coding, and the recent development of techniques for obtaining kernelization lower bounds. This work explores the existence of polynomial kernels for various path and cycle problems, by considering nonstandard parameterizations. We show polynomial kernels when the parameters are a given vertex cover, a modulator to a cluster graph, or a (promised) max leaf number. We obtain lower bounds via cross-composition, e.g., for Hamiltonian Cycle and related problems when parameterized by a modulator to an outerplanar graph

Lossy Kernelization

In this paper we propose a new framework for analyzing the performance of preprocessing algorithms. Our framework builds on the notion of kernelization from parameterized complexity. However, as opposed to the original notion of kernelization, our definitions combine well with approximation algorithms and heuristics. The key new definition is that of a polynomial size $\alpha$-approximate kernel. Loosely speaking, a polynomial size $\alpha$-approximate kernel is a polynomial time pre-processing algorithm that takes as input an instance $(I,k)$ to a parameterized problem, and outputs another instance $(I',k')$ to the same problem, such that $|I'|+k' \leq k^{O(1)}$. Additionally, for every $c \geq 1$, a $c$-approximate solution $s'$ to the pre-processed instance $(I',k')$ can be turned in polynomial time into a $(c \cdot \alpha)$-approximate solution $s$ to the original instance $(I,k)$. Our main technical contribution are $\alpha$-approximate kernels of polynomial size for three problems, namely Connected Vertex Cover, Disjoint Cycle Packing and Disjoint Factors. These problems are known not to admit any polynomial size kernels unless $NP \subseteq coNP/poly$. Our approximate kernels simultaneously beat both the lower bounds on the (normal) kernel size, and the hardness of approximation lower bounds for all three problems. On the negative side we prove that Longest Path parameterized by the length of the path and Set Cover parameterized by the universe size do not admit even an $\alpha$-approximate kernel of polynomial size, for any $\alpha \geq 1$, unless $NP \subseteq coNP/poly$. In order to prove this lower bound we need to combine in a non-trivial way the techniques used for showing kernelization lower bounds with the methods for showing hardness of approximationComment: 58 pages. Version 2 contain new results: PSAKS for Cycle Packing and approximate kernel lower bounds for Set Cover and Hitting Set parameterized by universe siz

Optimal Sparsification for Some Binary CSPs Using Low-degree Polynomials

This paper analyzes to what extent it is possible to efficiently reduce the number of clauses in NP-hard satisfiability problems, without changing the answer. Upper and lower bounds are established using the concept of kernelization. Existing results show that if NP is not contained in coNP/poly, no efficient preprocessing algorithm can reduce n-variable instances of CNF-SAT with d literals per clause, to equivalent instances with $O(n^{d-e})$ bits for any e > 0. For the Not-All-Equal SAT problem, a compression to size $\~O(n^{d-1})$ exists. We put these results in a common framework by analyzing the compressibility of binary CSPs. We characterize constraint types based on the minimum degree of multivariate polynomials whose roots correspond to the satisfying assignments, obtaining (nearly) matching upper and lower bounds in several settings. Our lower bounds show that not just the number of constraints, but also the encoding size of individual constraints plays an important role. For example, for Exact Satisfiability with unbounded clause length it is possible to efficiently reduce the number of constraints to n+1, yet no polynomial-time algorithm can reduce to an equivalent instance with $O(n^{2-e})$ bits for any e > 0, unless NP is a subset of coNP/poly.Comment: Updated the cross-composition in lemma 18 (minor update), since the previous version did NOT satisfy requirement 4 of lemma 18 (the proof of Claim 20 was incorrect

On Polynomial Kernels for Integer Linear Programs: Covering, Packing and Feasibility

We study the existence of polynomial kernels for the problem of deciding feasibility of integer linear programs (ILPs), and for finding good solutions for covering and packing ILPs. Our main results are as follows: First, we show that the ILP Feasibility problem admits no polynomial kernelization when parameterized by both the number of variables and the number of constraints, unless NP \subseteq coNP/poly. This extends to the restricted cases of bounded variable degree and bounded number of variables per constraint, and to covering and packing ILPs. Second, we give a polynomial kernelization for the Cover ILP problem, asking for a solution to Ax >= b with c^Tx <= k, parameterized by k, when A is row-sparse; this generalizes a known polynomial kernelization for the special case with 0/1-variables and coefficients (d-Hitting Set)

On Sparsification for Computing Treewidth

We investigate whether an n-vertex instance (G,k) of Treewidth, asking whether the graph G has treewidth at most k, can efficiently be made sparse without changing its answer. By giving a special form of OR-cross-composition, we prove that this is unlikely: if there is an e > 0 and a polynomial-time algorithm that reduces n-vertex Treewidth instances to equivalent instances, of an arbitrary problem, with O(n^{2-e}) bits, then NP is in coNP/poly and the polynomial hierarchy collapses to its third level. Our sparsification lower bound has implications for structural parameterizations of Treewidth: parameterizations by measures that do not exceed the vertex count, cannot have kernels with O(k^{2-e}) bits for any e > 0, unless NP is in coNP/poly. Motivated by the question of determining the optimal kernel size for Treewidth parameterized by vertex cover, we improve the O(k^3)-vertex kernel from Bodlaender et al. (STACS 2011) to a kernel with O(k^2) vertices. Our improved kernel is based on a novel form of treewidth-invariant set. We use the q-expansion lemma of Fomin et al. (STACS 2011) to find such sets efficiently in graphs whose vertex count is superquadratic in their vertex cover number.Comment: 21 pages. Full version of the extended abstract presented at IPEC 201