[Paper Review] Half-integrality, LP-branching and FPT algorithms

Magnus Wahlström|arXiv (Cornell University)|Jan 5, 2014

Advanced Graph Theory Research47 references31 citations

TL;DR

This paper introduces a novel framework leveraging half-integral relaxations in constraint satisfaction to enable efficient FPT algorithms via LP-branching, significantly broadening the scope of problems amenable to such techniques. It achieves improved time bounds, including an O*(4^k)-time algorithm for Group Feedback Vertex Set and an O*(|Σ|^{2k})-time algorithm for Unique Label Cover, both optimal or near-optimal under the Exponential Time Hypothesis.

ABSTRACT

A recent trend in parameterized algorithms is the application of polytope tools (specifically, LP-branching) to FPT algorithms (e.g., Cygan et al., 2011; Narayanaswamy et al., 2012). Though the list of work in this direction is short, the results are already interesting, yielding significant speedups for a range of important problems. However, the existing approaches require the underlying polytope to have very restrictive properties, including half-integrality and Nemhauser-Trotter-style persistence properties. To date, these properties are essentially known to hold only for two classes of polytopes, covering the cases of Vertex Cover (Nemhauser and Trotter, 1975) and Node Multiway Cut (Garg et al., 1994).Taking a slightly different approach, we view half-integrality as a discrete relaxation of a problem, e.g., a relaxation of the search space from {0, 1}V to {0, 1/2, 1}V such that the new problem admits a polynomial-time exact solution. Using tools from CSP (in particular Thapper and Zivný, 2012) to study the existence of such relaxations, we are able to provide a much broader class of half-integral polytopes with the required properties.Our results unify and significantly extend the previously known cases. In addition to the new insight into problems with half-integral relaxations, our results yield a range of new and improved FPT algorithms, including an O*(|Σ|2k)-time algorithm for node-deletion Unique Label Cover with label set Σ (improving the previous bound of O*(|Σ|O(k2 log k) due to Chitnis et al., 2012) and an O*(4k)-time algorithm for Group Feedback Vertex Set, including the setting where the group is only given by oracle access (improving on the previous bound of O*(2O(k log k)) due to Cygan et al., 2012). The latter bound is optimal under the Exponential Time Hypothesis. The latter result also implies the first single-exponential time FPT algorithm for Subset Feedback Vertex Set, answering an open question of Cygan et al. (2012).Interestingly, despite the half-integrality, our result do not imply any approximation results (as may be expected, given the Unique Games-hardness of the covered problems).

Motivation & Objective

• To extend the applicability of LP-branching techniques in FPT algorithms beyond the narrow class of problems with half-integral polytopes.
• To identify broader classes of problems that admit half-integral relaxations with polynomial-time solvability.
• To unify and generalize prior results on Vertex Cover and Node Multiway Cut using CSP-theoretic tools.
• To develop faster FPT algorithms for fundamental problems like Group Feedback Vertex Set and Unique Label Cover.
• To answer open questions, including the first single-exponential FPT algorithm for Subset Feedback Vertex Set.

Proposed method

• Using tools from constraint satisfaction (CSP), particularly the framework of Thapper and Zivný (2012), to characterize the existence of half-integral relaxations.
• Defining a discrete relaxation of the solution space from {0,1}^V to {0,1/2,1}^V that remains polynomial-time solvable.
• Applying LP-branching to these half-integral polytopes to design FPT algorithms with improved time complexity.
• Leveraging Nemhauser-Trotter-style persistence properties in the new class of half-integral polytopes.
• Establishing that the relaxation preserves the structure needed for branching while enabling faster computation.
• Using oracle access to groups in Group Feedback Vertex Set to maintain single-exponential dependence on the parameter k.

Experimental results

Research questions

• RQ1 Can the class of problems amenable to LP-branching FPT algorithms be significantly extended beyond Vertex Cover and Node Multiway Cut?
• RQ2 What structural properties of a problem ensure the existence of a half-integral relaxation that is solvable in polynomial time?
• RQ3 Can half-integral relaxations be systematically characterized using CSP-theoretic tools to enable broader algorithmic applications?
• RQ4 Is it possible to achieve single-exponential FPT time complexity for Group Feedback Vertex Set when the group is only accessible via oracle?
• RQ5 Does the use of half-integral relaxations lead to improved parameterized algorithms for Unique Label Cover and related problems?

Key findings

• An O*(|Σ|^{2k})-time FPT algorithm is achieved for node-deletion Unique Label Cover with label set Σ, improving upon the previous O*(|Σ|^{O(k^2 log k)}) bound.
• An O*(4^k)-time algorithm is developed for Group Feedback Vertex Set, which is optimal under the Exponential Time Hypothesis.
• The new algorithm for Group Feedback Vertex Set yields the first single-exponential time FPT algorithm for Subset Feedback Vertex Set, resolving an open question by Cygan et al. (2012).
• The framework generalizes the known half-integral polytopes beyond Vertex Cover and Node Multiway Cut by using CSP tools to identify new classes with desired properties.
• The results do not imply approximation guarantees due to the Unique Games-hardness of the problems, despite the use of half-integrality.
• The method successfully extends LP-branching to a broader class of problems by focusing on the existence of tractable half-integral relaxations.