We propose and study the graph-theoretical problem EXISTS-PMVC: the existence of perfect matching under vertex-color constraints on graphs with bi-colored edges. EXISTS-PMVC is of special interest because of its motivation from quantum-state identification and quantum-experiment design, as well as its rich expressiveness, i.e., EXISTS-PMVC naturally subsumes many constrained matching problems, such as exact perfect matching. We give complexity and algorithmic results for EXISTS-PMVC under two types of vertex color constraints: 1) symmetric constraints (EXISTS-PMVC-Sym) and 2) decision-diagram constraints (EXISTS-PMVC-DD). For EXISTS-PMVC-DD, we reveal its NP-hardness by a graph-gadget-based technique. We prove that EXISTS-PMVC-Sym with a bounded number of colors (EXISTS-PMVC-Sym-Bounded) is as hard as Exact Perfect Matching (XPM), which indicates EXISTS-PMVC-Sym-Bounded is in RNC on general graphs and PTIME on planar graphs. Directly applying algorithms for XPM to solve EXISTS-PMVC-Sym-Bounded is, however, impractical due to the overhead brought by the reduction. Therefore, we propose algorithms that natively handle EXISTS-PMVC-Sym-Bounded with significantly better efficiency. Our novel results for EXISTS-PMVC provide insights into both constrained matching and scalable quantum experiment design.

翻译：摘要：我们提出并研究了图论问题EXISTS-PMVC：在双色边图上，顶点颜色约束下完美匹配的存在性。该问题因源于量子态识别与量子实验设计，且具有丰富的表达能力（即EXISTS-PMVC自然涵盖精确完美匹配等许多受约束匹配问题）而备受关注。针对两类顶点颜色约束，我们给出了EXISTS-PMVC的复杂度与算法结果：1) 对称约束（EXISTS-PMVC-Sym）和2) 决策图约束（EXISTS-PMVC-DD）。对于EXISTS-PMVC-DD，我们通过基于图灵机技术的图构造方法揭示了其NP困难性。我们证明，有界颜色数的EXISTS-PMVC-Sym（EXISTS-PMVC-Sym-Bounded）与精确完美匹配（XPM）问题难度相当，这表明EXISTS-PMVC-Sym-Bounded在一般图上属于RNC复杂度类，在平面图上属于PTIME复杂度类。然而，直接应用XPM算法求解EXISTS-PMVC-Sym-Bounded会因归约过程产生的额外开销而变得不可行。因此，我们提出了原生处理EXISTS-PMVC-Sym-Bounded的高效算法，其效率显著优于间接方法。本文关于EXISTS-PMVC的新结果不仅为约束匹配问题提供了新见解，也为可扩展量子实验设计奠定了基础。