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

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

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 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

Tight Kernel Bounds for Problems on Graphs with Small Degeneracy

In this paper we consider kernelization for problems on d-degenerate graphs, i.e. graphs such that any subgraph contains a vertex of degree at most $d$. This graph class generalizes many classes of graphs for which effective kernelization is known to exist, e.g. planar graphs, H-minor free graphs, and H-topological-minor free graphs. We show that for several natural problems on d-degenerate graphs the best known kernelization upper bounds are essentially tight.Comment: Full version of ESA 201

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)

Polynomial Kernels for Weighted Problems

Kernelization is a formalization of efficient preprocessing for NP-hard problems using the framework of parameterized complexity. Among open problems in kernelization it has been asked many times whether there are deterministic polynomial kernelizations for Subset Sum and Knapsack when parameterized by the number $n$ of items. We answer both questions affirmatively by using an algorithm for compressing numbers due to Frank and Tardos (Combinatorica 1987). This result had been first used by Marx and V\'egh (ICALP 2013) in the context of kernelization. We further illustrate its applicability by giving polynomial kernels also for weighted versions of several well-studied parameterized problems. Furthermore, when parameterized by the different item sizes we obtain a polynomial kernelization for Subset Sum and an exponential kernelization for Knapsack. Finally, we also obtain kernelization results for polynomial integer programs

Guarantees and Limits of Preprocessing in Constraint Satisfaction and Reasoning

We present a first theoretical analysis of the power of polynomial-time preprocessing for important combinatorial problems from various areas in AI. We consider problems from Constraint Satisfaction, Global Constraints, Satisfiability, Nonmonotonic and Bayesian Reasoning under structural restrictions. All these problems involve two tasks: (i) identifying the structure in the input as required by the restriction, and (ii) using the identified structure to solve the reasoning task efficiently. We show that for most of the considered problems, task (i) admits a polynomial-time preprocessing to a problem kernel whose size is polynomial in a structural problem parameter of the input, in contrast to task (ii) which does not admit such a reduction to a problem kernel of polynomial size, subject to a complexity theoretic assumption. As a notable exception we show that the consistency problem for the AtMost-NValue constraint admits a polynomial kernel consisting of a quadratic number of variables and domain values. Our results provide a firm worst-case guarantees and theoretical boundaries for the performance of polynomial-time preprocessing algorithms for the considered problems.Comment: arXiv admin note: substantial text overlap with arXiv:1104.2541, arXiv:1104.556