Given a matroid $M=(E,{\cal I})$, and a total ordering over the elements $E$, a broken circuit is a circuit where the smallest element is removed and an NBC independent set is an independent set in ${\cal I}$ with no broken circuit. The set of NBC independent sets of any matroid $M$ define a simplicial complex called the broken circuit complex which has been the subject of intense study in combinatorics. Recently, Adiprasito, Huh and Katz showed that the face of numbers of any broken circuit complex form a log-concave sequence, proving a long-standing conjecture of Rota. We study counting and optimization problems on NBC bases of a generic matroid. We find several fundamental differences with the independent set complex: for example, we show that it is NP-hard to find the max-weight NBC base of a matroid or that the convex hull of NBC bases of a matroid has edges of arbitrary large length. We also give evidence that the natural down-up walk on the space of NBC bases of a matroid may not mix rapidly by showing that for some family of matroids it is NP-hard to count the number of NBC bases after certain conditionings.

翻译：给定拟阵 $M=(E,{\cal I})$ 及其元素集 $E$ 上的全序关系，断裂回路指移除最小元素后的回路，而非断裂回路独立集（NBC独立集）是 ${\cal I}$ 中不含断裂回路的独立集。任意拟阵 $M$ 的NBC独立集构成称为断裂回路复形的单纯复形，该复形是组合学领域的重要研究对象。近期，Adiprasito、Huh与Katz证明了任意断裂回路复形的面数构成对数凹序列，从而验证了Rota的长期猜想。本文研究一般拟阵NBC基的计数与优化问题，发现其与独立集复形存在若干根本性差异：例如，我们证明寻找拟阵的最大权重NBC基是NP-hard问题，且拟阵NBC基的凸包可存在任意大长度的边。通过展示在某些条件限制下，对某些拟阵族而言计数NBC基的数量是NP-hard问题，我们同时证明在NBC基空间上的自然上下行走可能无法快速混合。