最小前向与整数循环基的刻画及复杂性
Characterizations and Complexity of Minimum Forward and Integer Cycle Bases
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中文总结 AI 辅助
本文刻画了允许前向循环基的有向图结构，证明最小权弱基前向循环基问题是APX难的，引入opt-in图并给出其识别算法，确定完全图$K_n$为opt-in当且仅当$n \leq 7$。
中文摘要 AI 辅助
有向图的循环空间由循环基生成，通常循环可包含前向与后向弧；关于最小循环基问题的复杂性，尤其是最小权整数循环基、最小权弱与严格基前向循环基问题，仍存在多个开放问题。本文针对这些开放问题展开研究：首先，研究最小权前向循环基的存在性、结构与计算复杂性，给出允许弱基（故也为整数）前向循环基的有向图的完整结构刻画；进一步刻画强连通有向图允许基前向循环基的情况，证明当且仅当有向循环集的基数等于循环秩时存在此类基，且该基唯一；最后证明，只要存在最小权基前向循环基即可在多项式时间内找到，但最小权弱基前向循环基问题是APX难的，具体通过对带度量权重的有向图上的最小权弱基循环基问题进行L归约得到。其次，引入opt-in图，即对任意权重函数其最小循环基均为整数的图族，证明该图族是子图闭的，故根据Robertson-Seymour定理，其可由有限组禁子图刻画，因此opt-in识别问题可在多项式时间内求解；最后，提出一种算法，用于检查图是否为opt-in，若不是则识别其哪个子图属于禁子图集；应用该算法可证明，完全图$K_n$为opt-in当且仅当$n \leq 7$。
英文摘要
The cycle space of a directed graph is generated by a cycle basis, where, in general, cycles are allowed to have both forward and backward arcs. In a forward cycle, all arcs must follow the given direction. Several open questions remain regarding the complexity of the minimum cycle basis problem, in particular the minimum-weight integral cycle basis problem, and the minimum-weight weakly and strictly fundamental forward cycle basis problems. In this paper, we address these open questions. First, we study the existence, structure, and computational complexity of minimum-weight forward cycle bases. We give a complete structural characterization of digraphs that admit weakly fundamental (and hence integral) forward cycle bases. We further provide a characterization when a strongly connected digraph admits a forward fundamental cycle basis, proving that such a basis exists if and only if the set of directed cycles has cardinality equal to the cycle rank; in this case, the basis is unique. Lastly, we show that while minimum-weight forward fundamental cycle bases can be found in polynomial time whenever they exist, the minimum-weight forward weakly fundamental cycle basis problem is APX-hard via an L-reduction from the minimum-weight weakly fundamental cycle basis problem on digraphs with metric weights. Second, we introduce opt-in graphs, i.e., the family of graphs for which minimum cycle bases are integral for any weight function. We show that this family is minor-closed and hence, by the Robertson-Seymour theorem, is characterized by a finite set of forbidden minors, so that the opt-in recognition problem is solvable in polynomial time. Lastly, we present an algorithm to check whether a graph is opt-in, and if not, to identify which of its minors belong to the set of forbidden minors. Applying this algorithm, we show that the complete graph $K_n$ is opt-in if and only if $n \leq 7$.
发表机构
- University of Pavia（帕维亚大学）
- Freie Universität Berlin（柏林自由大学）
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