On the Power of Tree-Depth for Fully Polynomial FPT Algorithms

Abstract

There are many classical problems in P whose time complexities have not been improved over the past decades.
\nRecent 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.
\nTo bypass this difficulty, the concept of "FPT inside P" has been introduced.
\nFor 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'<c.
\n
\nIn this paper, we investigate the complexity of graph problems in P parameterized by tree-depth, a graph parameter related to tree-width.
\nWe show that a simple divide-and-conquer method can solve many graph problems, including
\nWeighted Matching, Negative Cycle Detection, Minimum Weight Cycle, Replacement Paths, and 2-hop Cover,
\nin O(td m) time or O(td (m+nlog n)) time, where td is the tree-depth of the input graph.
\nBecause any graph of tree-width tw has tree-depth at most (tw+1)log_2 n, our algorithms also run in O(tw mlog n) time or O(tw (m+nlog n)log n) time.
\nThese results match or improve the previous best algorithms parameterized by tree-width.
\nEspecially, we solve an open problem of fully polynomial FPT algorithm for Weighted Matching parameterized by tree-width posed by Fomin et al. (SODA 2017).