Kernelization Lower Bounds By Cross-Composition (1206.5941v1)

Abstract: 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.

• The paper introduces cross-composition, a novel framework that proves certain parameterized problems lack polynomial kernels unless key complexity conjectures fail.
• It applies the technique to graph problems like Clique and Chromatic Number, demonstrating that even fixed-parameter tractable problems may not admit efficient kernelizations.
• The study links kernelization lower bounds to broader complexity theory, providing practical insights into data reduction and preprocessing challenges in computational problems.

An Examination of Kernelization Lower Bounds via Cross-Composition

The paper "Kernelization Lower Bounds By Cross-Composition" by Hans L. Bodlaender, Bart M. P. Jansen, and Stefan Kratsch provides a significant contribution to understanding kernelization lower bounds within the field of parameterized complexity. This work introduces a novel framework called cross-composition, an innovative extension of composition techniques used to establish kernelization lower bounds for parameterized problems. Its applications provide insights into various fundamental problems, especially graph problems with structural parameterizations.

The core contribution of the paper is the formulation of cross-composition, which enables proving that certain parameterized problems do not admit polynomial kernelizations unless certain complexity-theoretic conjectures (such as NP ⊆ coNP/poly) fail. This framework has proven to be more versatile than previous techniques by allowing compositions between different problem families, transcending the traditional boundaries of or/and-composition methods.

Key Concepts and Methodology

1. Cross-Composition Framework: This framework generalizes previous methods by enabling compositions where instances of a classical NP-hard problem (source problem) can be composed into instances of a parameterized problem (target problem), even if they do not share the same nature or structure. An important facet of this technique is its allowance for the source and target problems to differ, facilitated by polynomial equivalence relations.
2. Applications to Graph Problems: The authors apply cross-composition to derive kernelization lower bounds for several well-known graph problems, including Clique, Chromatic Number, Weighted Feedback Vertex Set, and Weighted Odd Cycle Transversal. They prove that these problems do not admit polynomial kernels when parameterized by the vertex cover number, despite being fixed-parameter tractable for this parameterization.
3. Implications for Complexity Theory: A notable theoretical implication of this work is that proving certain problems do not have polynomial kernels under specified parameterizations can be linked to broader complexity-theoretic consequences, such as the collapse of the polynomial hierarchy. Thus, cross-composition not only sheds light on kernelization but also intersects with pivotal conjectures in complexity theory.
4. Cross-Composition Techniques: The paper describes specific constructions (e.g., inflations and the K-in-a-box graph) that enable the cross-composition of Vertex Cover into problems like Weighted Feedback Vertex Set and Weighted Odd Cycle Transversal, highlighting the versatility of this method.

Results and Conclusions

The cross-composition technique significantly broadens the scope of kernelization lower bound techniques by providing a common perspective and integrating previous composition methodologies. Notably, the application of this technique has already led to numerous extensions and has been employed by various researchers to address different parametrized problems, reinforcing its utility and impact.

The capability to demonstrate kernelization lower bounds presents practical implications, particularly in optimizing preprocessing and data reduction for complex computational problems. The absence of polynomial kernels in various settings, as shown in this paper, prompts reevaluation of expected preprocessing efficiencies.

Future Directions

Given the robustness of cross-composition, future research may focus on identifying further problems amenable to this technique and establishing tighter bounds on kernel sizes for those problems. The potential for discovering new applications or expanding this framework to accommodate other computational paradigms remains promising. Moreover, the interactions of kernelization lower bounds with central conjectures such as NP ≠ coNP/poly could yield deeper insights into the nature of computational difficulty and tractability.

In summary, the paper by Bodlaender, Jansen, and Kratsch presents a comprehensive and versatile approach within parameterized complexity, enriching the toolkit for analyzing and establishing kernelization lower bounds. The cross-composition framework is a pivotal addition, promising to guide future investigations in this rich and complex field.