Weighted $k$-Path and Other Problems in Almost $O^*(2^k)$ Deterministic Time via Dynamic Representative Sets

TL;DR

This paper introduces a Dynamic Representative Set data structure that enables faster deterministic algorithms for weighted $k$-Path and related problems, achieving near $O^*(2^k)$ time complexity.

Contribution

The paper presents a novel Dynamic Representative Set data structure with efficient update and query operations, leading to improved deterministic algorithms for weighted $k$-Path and similar problems.

Findings

Achieves $2^{k+O(\sqrt{k}\log^2k)}(n+m)\log n$ time for weighted directed $k$-Path.

Provides a data structure with $2^{k+O(\sqrt{k}\log^2k)}\log n$ query time after preprocessing.

Answers a major open problem by approaching near $O^*(2^k)$ deterministic time complexity.

Abstract

We present a data structure that we call a Dynamic Representative Set. In its most basic form, it is given two parameters $0< k < n$ and allows us to maintain a representation of a family $\mathcal{F}$ of subsets of $\{1,\ldots,n\}$. It supports basic update operations (unioning of two families, element convolution) and a query operation that determines for a set $B \subseteq \{1,\ldots,n\}$ whether there is a set $A \in \mathcal{F}$ of size at most $k-|B|$ such that $A$ and $B$ are disjoint. After $2^{k+O(\sqrt{k}\log^2k)}n \log n$ preprocessing time, all operations use $2^{k+O(\sqrt{k}\log^2k)}\log n$ time. Our data structure has many algorithmic consequences that improve over previous works. One application is a deterministic algorithm for the Weighted Directed $k$-Path problem, one of the central problems in parameterized complexity. Our algorithm takes as input an $n$-vertex directed graph $G=(V,E)$ with edge lengths and an integer $k$, and it outputs the minimum edge length of a path on $k$ vertices in $2^{k+O(\sqrt{k}\log^2k)}(n+m)\log n$ time (in the word RAM model where weights fit into a single word). Modulo the lower order term $2^{O(\sqrt{k}\log^2k)}$, this answers a question that has been repeatedly posed as a major open problem in the field.