顶点覆盖数为二时时间路径覆盖的紧致困难性
Tight Hardness for Temporal Path Covers at Vertex-Cover Number Two
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中文总结 AI 辅助
本文通过从相异数值匹配归约，证明顶点覆盖数为二时时间路径覆盖与时间不相交路径覆盖均为NP完全，并给出时间有向星上基于跨度参数的XP算法。
中文摘要 AI 辅助
时间路径覆盖（TPC）和时间不相交路径覆盖（TDPC）问题要求用时间路径对时间有向图进行最小基数覆盖，其中TDPC还额外要求路径两两时间不相交。Cioni等人证明了这两个问题在底层无向图顶点覆盖数为三的时间有向无环图（DAG）上是NP困难的，并留下了顶点覆盖数为二的情况未解决。我们肯定地回答了这两个问题的这一开放情况。通过从具有目标和的相异数值匹配（Distinct Numerical Matching with Target Sums）进行多项式时间归约，我们证明当底层无向图的顶点覆盖数恰好为二时，TPC和TDPC已经是NP完全的。对于TPC，这给出了一个精确的顶点覆盖阈值：顶点覆盖数至多为一时可多项式时间求解，而为二时是NP完全的。对于TDPC，我们进一步研究了时间有向星（temporal oriented stars）的边界情况。一个重用归一化论证将最优解归约为叶子唯一的双边合并。如果入边侧或出边侧是单标签的，即使另一侧是多标签的，最优解仍可在多项式时间内计算。我们还为可行的合并实现引入了一个跨度参数，并证明任何可行的k个合并族都强制要求跨度至少为2k-2的实现。因此，时间有向星上的TDPC以最大跨度为参数属于XP类，并且对于每个固定的跨度界限都是多项式时间可解的。无限制的多标签星形情况仍然开放。
英文摘要
Temporal Path Cover (TPC) and Temporally Disjoint Path Cover (TDPC) ask for minimum-cardinality covers of a temporal digraph by temporal paths, with TDPC additionally requiring pairwise temporal disjointness. Cioni et al. proved both problems NP-hard on temporal DAGs whose underlying undirected graph has vertex-cover number three and left the vertex-cover-two case open. We resolve this question affirmatively for both problems. Using polynomial-time reductions from Distinct Numerical Matching with Target Sums, we show that TPC and TDPC are NP-complete already when the underlying undirected graph has vertex-cover number exactly two. For TPC, this yields an exact vertex-cover threshold: polynomial-time solvability at vertex-cover number at most one and NP-completeness at two. For TDPC, we further study the boundary case of temporal oriented stars. A reuse-normalization argument reduces optimum solutions to leaf-unique two-edge merges. If either the incoming or outgoing side is single-label, the optimum remains polynomial-time computable even when the opposite side is multi-label. We also introduce a span parameter for feasible merge realizations and prove that any feasible family of k merges forces a realization of span at least 2k-2. Hence TDPC on temporal oriented stars is in XP parameterized by maximum span, and is polynomial-time solvable for every fixed span bound. The unrestricted multi-label star case remains open.
发表机构
- Istanbul Topkapı University（伊斯坦布尔托普卡帕大学）
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