寻找最小有界链的参数化复杂度
The Parameterized Complexity of Finding Minimum Bounded Chains
AI总结：
本研究聚焦最小有界链（MBC$_d$）问题的参数化复杂度，证明$d\boldsymbol{\u2265}2$时以解规模为参数的W[1]-难性，提出多项式时间的MBC$_1$算法及两种FPT算法，其中基于树宽的动态规划算法运行时间匹配指数时间假设下的下界。
摘要：
在单纯复形中寻找具有特定$(d-1)$-边界的最小$d$-链的问题被称为\textsc{Minimum Bounded Chain}（最小有界链，MBC$_d$）问题。MBC$_d$问题对所有$d\boldsymbol{\u2265}2$都是NP难的。本文证明，若以解的规模为参数，该问题对所有$d\boldsymbol{\u2265}2$同样是W[1]-难的。我们还提出了一种可在多项式时间内求解MBC$_1$问题的算法，并介绍且实现了两种可求解所有$d$的MBC$_d$问题的fixed parameter tractable（固定参数可解，FPT）算法。第一种算法是Dijkstra算法的推广版本，以解的规模和余面度为参数；第二种是基于树宽的动态规划方法，其运行时间与我们在指数时间假设下证明的下界一致。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
