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arXiv · 2108.04563

The Parameterized Complexity of Finding Minimum Bounded Chains

Nello Blaser·Morten Brun·Lars M. Salbu·Erlend Raa Vågset
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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.

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Nello Blaser, Morten Brun, Lars M. Salbu, Erlend Raa Vågset. 2021-08-10. The Parameterized Complexity of Finding Minimum Bounded Chains. https://doi.org/10.1016/j.comgeo.2024.102102

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