Representative sets and irrelevant vertices: New tools for kernelization (1111.2195v2)

Abstract: The existence of a polynomial kernel for Odd Cycle Transversal was a notorious open problem in parameterized complexity. Recently, this was settled by the present authors (Kratsch and Wahlstr\"om, SODA 2012), with a randomized polynomial kernel for the problem, using matroid theory to encode flow questions over a set of terminals in size polynomial in the number of terminals. In the current work we further establish the usefulness of matroid theory to kernelization by showing applications of a result on representative sets due to Lov\'asz (Combinatorial Surveys 1977) and Marx (TCS 2009). We show how representative sets can be used to give a polynomial kernel for the elusive Almost 2-SAT problem. We further apply the representative sets tool to the problem of finding irrelevant vertices in graph cut problems, i.e., vertices which can be made undeletable without affecting the status of the problem. This gives the first significant progress towards a polynomial kernel for the Multiway Cut problem; in particular, we get a kernel of O(k^{s+1}) vertices for Multiway Cut instances with at most s terminals. Both these kernelization results have significant spin-off effects, producing the first polynomial kernels for a range of related problems. More generally, the irrelevant vertex results have implications for covering min-cuts in graphs. For a directed graph G=(V,E) and sets S, T \subseteq V, let r be the size of a minimum (S,T)-vertex cut (which may intersect S and T). We can find a set Z \subseteq V of size O(|S|*|T|*r) which contains a minimum (A,B)-vertex cut for every A \subseteq S, B \subseteq T. Similarly, for an undirected graph G=(V,E), a set of terminals X \subseteq V, and a constant s, we can find a set Z\subseteq V of size O(|X|^{s+1}) which contains a minimum multiway cut for any partition of X into at most s pairwise disjoint subsets.

• The paper introduces novel kernelization techniques using representative sets and identifying irrelevant vertices, leveraging matroid theory to achieve polynomial kernels for ALMOST 2-SAT and MULTIWAY CUT problems.
• Key results include a randomized polynomial-time compression of size O(k^6) for ALMOST 2-SAT and identifying O(|X|s+1) vertices for minimum multiway cuts in specific graph partition cases.
• These techniques offer practical benefits for processing large instances in SAT solvers and integer programming, while theoretically paving the way for deeper integration of matroid theory into parameterized algorithms.

An Examination of New Kernelization Techniques via Representative Sets and Irrelevant Vertices

The paper "Representative Sets and Irrelevant Vertices: New Tools for Kernelization" by Stefan Kratsch and Magnus Wahlström introduces novel techniques in the field of kernelization within parameterized complexity, focusing particularly on applications of matroid theory. The authors present significant advancements in the provision of polynomial kernels for notable problems such as ALMOST 2-SAT and MULTIWAY CUT, utilizing representative sets and the identification of irrelevant vertices in graph cut problems.

Key Contributions and Methodologies

The authors bring attention to the ALMOST 2-SAT problem, where the task involves removing at most $k$ clauses to achieve satisfiability of a 2-CNF formula. By employing representative sets, derived from matroid theory due to Lovász and Marx, a polynomial kernel is successfully established for ALMOST 2-SAT, resolving a longstanding open challenge in parameterized complexity.

Further, the paper addresses the intricacies associated with identifying irrelevant vertices within graph cut problems, paving the way for progress in kernelization for MULTIWAY CUT problems. This is accomplished through a kernel of $O(k^{s+1})$ vertices for MULTIWAY CUT instances bounded by the number of terminals $s$.

Numerical Results and Assertions

Two crucial results are highlighted:

1. Polynomial compression: ALMOST 2-SAT is shown to have a randomized polynomial-time compression of size $O(k^6)$, with one-sided error probability.
2. Irrelevant vertex results: For any partition of $X$ into up to $s$ subsets, it is demonstrated that a set $Z \subseteq V$ of size $O(|X|s+1)$ can be selected, containing a minimum multiway cut for a specified partition. The implications of this result extend potentially to practical reductions in computational time for min-cut problems.

Implications and Future Directions

Practically, the findings facilitate more efficient handling of SAT solvers and integer programming preprocessing steps, proving useful in real-world applications where large instances need reduction for manageable computation. Theoretically, this work opens avenues for integrating matroid theory further into parameterized algorithms, potentially influencing future developments in fixed-parameter tractability (FPT) techniques.

The paper leaves several open questions, such as the prospects for a general kernelization of MULTIWAY CUT beyond the constant-bound terminals and whether polynomial kernelization is feasible for DIRECTED FEEDBACK VERTEX SET. Additionally, exploring non-uniform kernelizations offers intriguing challenges that remain ripe for investigation.

In summary, Kratsch and Wahlström's work marks a substantial advance in the utilization of representative sets for kernelization, setting a foundation for both practical and theoretical progress in addressing complex problems in parameterized complexity. This technical exposition not only deepens our understanding of matroid applications but also propels forward the search for efficient algorithmic solutions in computational graph theory and beyond.