Clique cover and graph separation: New incompressibility results

The field of kernelization studies polynomial-time preprocessing routines for hard problems in the framework of parameterized complexity. Although a framework for proving kernelization lower bounds has been discovered in 2008 and successfully applied multiple times over the last three years, establishing kernelization complexity of many important problems remains open. In this paper we show that, unless NP is a subset of coNP/poly and the polynomial hierarchy collapses up to its third level, the following parameterized problems do not admit a polynomial-time preprocessing algorithm that reduces the size of an instance to polynomial in the parameter: - EDGE CLIQUE COVER, parameterized by the number of cliques, - DIRECTED EDGE/VERTEX MULTIWAY CUT, parameterized by the size of the cutset, even in the case of two terminals, - EDGE/VERTEX MULTICUT, parameterized by the size of the cutset, and - k-WAY CUT, parameterized by the size of the cutset. The existence of a polynomial kernelization for EDGE CLIQUE COVER was a seasoned veteran in open problem sessions. Furthermore, our results complement very recent developments in designing parameterized algorithms for cut problems by Marx and Razgon [STOC'11], Bousquet et al. [STOC'11], Kawarabayashi and Thorup [FOCS'11] and Chitnis et al. [SODA'12].

💡 Research Summary

The paper investigates the limits of kernelization – a polynomial‑time preprocessing technique that reduces an instance of a parameterized problem to an equivalent instance whose size is bounded by a function of the parameter – for four fundamental graph problems: Edge Clique Cover, Directed Multiway Cut (with two terminals), Multicut, and k‑Way Cut. Using the modern cross‑composition framework introduced by Bodlaender et al., the authors construct, for each problem, a polynomial‑time reduction that merges many instances of an NP‑complete source language into a single instance of the target parameterized problem. The reduction guarantees that the parameter of the combined instance grows only polynomially with the size of the largest source instance (plus a logarithmic term in the number of instances).

The key theoretical tool is the following: if an NP‑hard language L cross‑composes into a parameterized problem Q that admits a polynomial kernel, then NP ⊆ coNP/poly, which would collapse the polynomial hierarchy to its third level (by Fortnow and Santhanam). Therefore, demonstrating a cross‑composition for a problem rules out the existence of a polynomial kernel unless a widely believed complexity‑theoretic collapse occurs.

For Edge Clique Cover, the authors start from a suitably restricted version of 3‑SAT and encode each formula as a graph whose edges can be covered by cliques only if the formula is satisfiable. They design two distinct compositions: an OR‑composition (standard for kernel lower bounds) and an AND‑composition (where the combined instance is a YES‑instance only if all source instances are YES). This is the first natural parameterized problem known to admit both types of composition, strengthening the evidence that a polynomial kernel would cause both a PH collapse and violate the “AND‑distillation conjecture”.

For Directed Multiway Cut with two terminals, the reduction uses directed s‑t cut instances. Each source instance is placed in a dedicated “terminal block” that forces any feasible cut to respect the structure of the original instance. By carefully linking the blocks with high‑capacity edges, the authors ensure that a cut of size at most p exists in the combined graph if and only if at least one source instance admits a cut of size p. This yields an OR‑composition and thus excludes a polynomial kernel under the standard assumption.

For Multicut, the construction leverages the concept of important separators (Marx) and starts from the Edge‑Disjoint‑Paths problem. The authors present two independent OR‑compositions: one employing an “instance selector” gadget that encodes a binary choice among source instances, and another using a “color‑labeling” technique that forces the cut to separate specific terminal pairs only when the corresponding source instance is a YES‑instance. Both constructions keep the cut‑size parameter polynomially bounded, leading again to the same lower‑bound conclusion.

Finally, for k‑Way Cut, the authors transform each source instance into a “separable cluster” that can be disconnected from the rest of the graph by removing a bounded number of edges. The target instance asks whether at most s edges can be deleted to obtain at least k components. The parameter s is bounded by a polynomial in the size of the largest source instance, and the combined instance is a YES‑instance precisely when some source instance is a YES‑instance. This OR‑composition rules out a polynomial kernel for k‑Way Cut parameterized by the cut size.

Overall, the paper makes four major contributions: (1) it establishes, via cross‑composition, that Edge Clique Cover, Directed Multiway Cut (two terminals), Multicut, and k‑Way Cut do not admit polynomial kernels unless NP ⊆ coNP/poly; (2) it provides the first natural problem (Edge Clique Cover) known to be both AND‑ and OR‑compositional; (3) it showcases how techniques from parameterized algorithm design—iterative compression for Edge Clique Cover and important separators for Multicut—can be repurposed to build powerful lower‑bound constructions; and (4) it complements recent algorithmic advances for cut problems, clarifying that while fixed‑parameter tractability is achievable for many of them, efficient preprocessing to polynomial size is unlikely. These results guide researchers toward focusing on alternative strategies (e.g., Turing kernels, approximation, or heuristic preprocessing) for the studied problems, as classical kernelization appears fundamentally limited.