The Parameterized Complexity of Finding Minimum Bounded Chains

TL;DR

This paper investigates the computational complexity of the Minimum Bounded Chain problem in simplicial complexes, proving W[1]-hardness for higher dimensions, and introduces fixed parameter tractable algorithms for solving it.

Contribution

It establishes the W[1]-hardness of MBC$_d$ for all $d extgreater 1$, and presents new FPT algorithms for solving MBC$_d$ across all dimensions.

Findings

MBC$_d$ is NP-hard and W[1]-hard for all $d extgreater 1$.

Polynomial-time algorithm for MBC$_1$.

Two FPT algorithms for MBC$_d$ for all $d$, including a generalized Dijkstra's and a treewidth-based dynamic programming.

Abstract

Finding the smallest $d$-chain with a specific $(d-1)$-boundary in a simplicial complex is known as the \textsc{Minimum Bounded Chain} (MBC$_d$) problem. The MBC$_d$ problem is NP-hard for all $d\geq 2$. In this paper, we prove that it is also W[1]-hard for all $d\geq 2$, if we parameterize the problem by solution size. We also give an algorithm solving the MBC$_1$ problem in polynomial time and introduce and implemented two fixed parameter tractable (FPT) algorithms solving the MBC$_d$ problem for all $d$. The first algorithm is a generalized version of Dijkstra's algorithm and is parameterized by solution size and coface degree. The second algorithm is a dynamic programming approach based on treewidth, which has the same runtime as a lower bound we prove under the exponential time hypothesis.