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

2025-12-09

AI Summary

This paper studies the parameterized $k$-path problem on weighted directed graphs: given an $n$-vertex directed graph with edge weights and an integer $k$, find a shortest path on exactly $k$ vertices. For this classical problem, the authors introduce the **Dynamic Representative Set**, a novel data structure supporting union, element-wise convolution, and disjointness queries over families of sets, integrated with preprocessing and logarithmic-time updates. They present the first **deterministic** algorithm solving the problem in $2^{k+O(sqrt{k}log^2 k)}(n+m)log n$ time—significantly improving upon prior deterministic bounds and approaching the theoretical lower bound of $2^k$. This is the first deterministic algorithm for the weighted directed $k$-path problem running in nearly $O^*(2^k)$ time. The result provides both a powerful new tool and a conceptual paradigm for parameterized path problems.

Technology Category

• Planning, Routing, and Scheduling: Deterministic Planning
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

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.

Problem

• Develops a Dynamic Representative Set data structure for efficient subset queries.
• Provides a deterministic algorithm for the Weighted Directed k-Path problem.
• Improves time complexity to almost O*(2^k) for parameterized graph problems.

Innovation

• Dynamic Representative Set data structure for subset families
• Deterministic algorithm for Weighted Directed k-Path problem
• Almost O*(2^k) time complexity via improved preprocessing