[Paper Review] On the Power of Tree-Depth for Fully Polynomial FPT Algorithms

Yoichi Iwata, Tomoaki Ogasawara+1 more|arXiv (Cornell University)|Oct 12, 2017

Advanced Graph Theory Research18 references3 citations

TL;DR

This paper presents fully polynomial FPT algorithms for several graph problems in P, parameterized by tree-depth. By leveraging a divide-and-conquer approach on elimination forests, the authors achieve $O( ext{td} imes m)$ or $O( ext{td} imes (m+n/log n))$ time bounds, solving open problems including fully polynomial FPT algorithms for Weighted Matching and 2-hop Cover under tree-depth parameterization.

ABSTRACT

There are many classical problems in P whose time complexities have not been improved over the past decades. Recent studies of "Hardness in P" have revealed that, for several of such problems, the current fastest algorithm is the best possible under some complexity assumptions. To bypass this difficulty, Fomin et al. (SODA 2017) introduced the concept of fully polynomial FPT algorithms. For a problem with the current best time complexity $O(n^c)$, the goal is to design an algorithm running in $k^{O(1)}n^{c'}$ time for a parameter $k$ and a constant $c'

Motivation & Objective

• Address the challenge of designing faster algorithms for classical P-complete graph problems that have resisted improvement for decades.
• Overcome hardness-in-P barriers by introducing fully polynomial FPT algorithms parameterized by tree-depth.
• Resolve open problems in parameterized complexity, including the first fully polynomial FPT algorithm for Weighted Matching parameterized by tree-width.
• Provide a unifying framework using divide-and-conquer on elimination forests to achieve efficient, parameterized solutions across multiple problems.

Proposed method

• Apply a divide-and-conquer strategy on the elimination forest of a graph, exploiting small balanced separators derived from tree-depth.
• Use dynamic programming with incremental updates to maintain and merge solutions over subsets of vertices in the elimination order.
• Leverage a min-heap to efficiently compute minimum values across disjoint subgraphs during union operations.
• Design two key algorithms: <code>Increment</code> for adding a vertex to a subset and <code>Union</code> for merging solutions from disjoint subsets.
• Utilize Dijkstra’s algorithm in the <code>Increment</code> step to compute shortest paths from and to the new vertex in the extended subgraph.
• Ensure correctness by proving that the 2-hop cover and other functions are preserved under union and incremental extension operations.

Experimental results

Research questions

• RQ1 Can we design fully polynomial FPT algorithms for fundamental graph problems in P using tree-depth as a parameter?
• RQ2 Does tree-depth offer a more effective parameter than tree-width for achieving faster, parameterized algorithms in P?
• RQ3 Can the divide-and-conquer approach on elimination forests be generalized to yield efficient algorithms for multiple graph problems?
• RQ4 Is it possible to achieve a fully polynomial FPT algorithm for Weighted Matching parameterized by tree-width, as posed by Fomin et al.?
• RQ5 What is the optimal time complexity achievable for problems like 2-hop cover and Minimum Weight Cycle under tree-depth parameterization?

Key findings

• The paper presents an $O( ext{td} imes m)$-time algorithm for Negative Cycle Detection and Minimum Weight Cycle, improving upon previous bounds.
• An $O( ext{td} imes (m+n/log n))$-time algorithm is developed for Weighted Matching, resolving an open problem posed by Fomin et al.
• A fully polynomial FPT algorithm for 2-hop Cover is achieved in $O( ext{td} imes (m+n/log n))$ time with maximum label size $2 ext{td}$.
• The algorithm for Replacement Paths runs in $O( ext{td} imes (m+n/log n))$ time, matching or improving prior tree-width-based results.
• The framework achieves $O( ext{tw} imes m imes ext{polylog} ext{ }n)$ time for all problems when parameterized by tree-width, since $ ext{td} riangleq O( ext{tw} imes ext{log }n)$.
• The correctness and efficiency of the algorithms are proven via a general framework using <code>Increment</code> and <code>Union</code> operations on subsets of vertices in elimination order.