一种新的三次图非Hamilton性检测方法
A New Heuristic for Detecting Non-Hamiltonicity in Cubic Graphs
摘要：
本文分析了一个多面体，它包含一个给定的无向连通三次图的所有哈密顿圈的凸包。我们构造的多面体是由多项式多连续（松弛）变量的多项式多线性约束定义的。显然，所构造的多面体的空性意味着该图是非哈密尔顿的。然而，只要构造的多面体是非空的，结果是不确定的。因此，下面的自然问题出现了：如果我们假设一个非空的多面体蕴含着哈密顿性，那么这个诊断有多频繁是不正确的？我们证明了，在桥图的情况下，所构造的多面体总是空的。我们还证明了一些非桥非Hamilton三次图也诱导空多面体。我们比较我们的方法著名的Dantzig-Fulkerson-Johnson松弛的TSP，并给出经验证据表明，后者是不可行的，当且仅当我们构造的多面体也是空的。通过考虑特殊的边割集是目前在大多数三次图，我们描述了一个启发式的方法，建立在我们构建的多面体，不正确的诊断非哈密顿图的哈密尔顿是非常罕见的。特别是，对于包含多达18个顶点的三次图，45，982个无向连通三次图中只有4个被误诊。通过分析，我们证明了一个等价的启发式，当建立在丹兹-富克尔森-约翰逊松弛的TSP，大多是不成功的，在确定额外的非哈密顿图。这些经验结果表明，多项式算法的基础上，我们构造的多面体可能能够正确地识别一个三次图的哈密尔顿性，但在极少数情况下。
英文摘要：
We analyse a polyhedron which contains the convex hull of all Hamiltonian cycles of a given undirected connected cubic graph. Our constructed polyhedron is defined by polynomially-many linear constraints in polynomially-many continuous (relaxed) variables. Clearly, the emptiness of the constructed polyhedron implies that the graph is non-Hamiltonian. However, whenever a constructed polyhedron is non-empty, the result is inconclusive. Hence, the following natural question arises: if we assume that a non-empty polyhedron implies Hamiltonicity, how frequently is this diagnosis incorrect? We prove that, in the case of bridge graphs, the constructed polyhedron is always empty. We also demonstrate that some non-bridge non-Hamiltonian cubic graphs induce empty polyhedra as well. We compare our approach to the famous Dantzig-Fulkerson-Johnson relaxation of a TSP, and give empirical evidence which suggests that the latter is infeasible if and only if our constructed polyhedron is also empty. By considering special edge cut sets which are present in most cubic graphs, we describe a heuristic approach, built on our constructed polyhedron, for which incorrect diagnoses of non-Hamiltonian graphs as Hamiltonian appear to be very rare. In particular, for cubic graphs containing up to 18 vertices, only four out of 45,982 undirected connected cubic graphs were so misdiagnosed. By constrast, we demonstrate that an equivalent heuristic, when built on the Dantzig-Fulkerson-Johnson relaxation of a TSP, is mostly unsuccessful in identifying additional non-Hamiltonian graphs. These empirical results suggest that polynomial algorithms based on our constructed polyhedron may be able to correctly identify Hamiltonicity of a cubic graph in all but rare cases.
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