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

Jesper Nederlof

TL;DR

This work introduces Dynamic Representative Sets, a deterministic framework to speed up dynamic programming by manipulating rep-sets over idempotent semirings and exploiting a low-rank factorization of the disjointness matrix D_{n,k}. The authors prove a main result that D_{n,k} can be factored as L R with rank roughly 2^k (up to poly(log n) factors) and provide a convolve operation that updates representations efficiently, enabling a deterministic $2^{k+O(\u221a{k}1- c^2 k)}(n+m) ext{log}n$-time algorithm for Weighted Directed k-Path. They extend the framework to Skewed Multilinear Monomial Summation over idempotent semirings, yielding fast deterministic solutions for problems such as Subgraph Isomorphism and related counting/detection tasks. Overall, the paper presents a general, algebraically grounded tool that derandomizes a broad class of DP-based algorithms by maintaining compact, representative DP states and offers concrete algorithmic gains for key parameterized problems.

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.

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

TL;DR

This work introduces Dynamic Representative Sets, a deterministic framework to speed up dynamic programming by manipulating rep-sets over idempotent semirings and exploiting a low-rank factorization of the disjointness matrix D_{n,k}. The authors prove a main result that D_{n,k} can be factored as L R with rank roughly 2^k (up to poly(log n) factors) and provide a convolve operation that updates representations efficiently, enabling a deterministic -time algorithm for Weighted Directed k-Path. They extend the framework to Skewed Multilinear Monomial Summation over idempotent semirings, yielding fast deterministic solutions for problems such as Subgraph Isomorphism and related counting/detection tasks. Overall, the paper presents a general, algebraically grounded tool that derandomizes a broad class of DP-based algorithms by maintaining compact, representative DP states and offers concrete algorithmic gains for key parameterized problems.

Abstract

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

Paper Structure

This paper contains 16 sections, 10 theorems, 34 equations, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. The Weighted $k$-Path Problem.
  3. Representative Sets.
  4. Representative sets for $k$-Path.
  5. Intuition behind our proof for Theorem \ref{['thm:kpath']}.
  6. The algorithmic tool of "Dynamic Representative Sets" and its applications beyond $k$-Path
  7. Related work and comparison with our approach.
  8. Preliminaries and notation
  9. Idempotent Semirings.
  10. Linear Algebra.
  11. Splitters and Universal Families.
  12. Dynamic Representative Sets: Statement and Applications
  13. Application to Weighted Directed $k$-Path.
  14. Application to Skewed Multilinear Monomial Summation over $\mathbb{S}$.
  15. Dynamic Representative Sets: Implementation
  16. ...and 1 more sections

Key Result

Theorem 1.1

There is a deterministic algorithm for the Weighted Directed $k$-path problem that runs in $2^{k+O(\sqrt{k}\log^2k)}(n+m)\log n$ time on the word RAM model, where each weight fits into a word.

Theorems & Definitions (24)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Lemma 2.7: DBLP:journals/jacm/AlonYZ95
  • Lemma 2.8: DBLP:conf/focs/NaorSS95, Theorem 3(i)
  • ...and 14 more