构造4-正则重图Hamilton分解的回溯算法
Backtracking algorithms for constructing the Hamiltonian decomposition of a 4-regular multigraph
摘要：
本文考虑了一个将正则图划分为边不相交的哈密尔顿圈的哈密尔顿分解问题。已知在对称和非对称的旅行商多面体的1-骨架中验证顶点不邻接是NP-完全的。另一方面，两个顶点不相邻的一个充分条件可以表述为一个求4-正则多重图的第二Hamilton分解的组合问题。我们提出了两个回溯算法构造的第二个哈密尔顿分解和验证顶点的非邻接性：一个算法的基础上的一个简单的路径扩展和算法的基础上的链边固定过程。对无向多重图的计算实验结果表明，这两种回溯算法都不敌已知的一般变邻域搜索算法。然而，对于有向多重图，基于链固定的边缘的算法表现出的结果与现有的解决方案和更好的结果的不可行的情况下，不存在的哈密尔顿分解的实例上的实例的并行计算。
英文摘要：
We consider a Hamiltonian decomposition problem of partitioning a regular graph into edge-disjoint Hamiltonian cycles. It is known that verifying vertex non-adjacency in the 1-skeleton of the symmetric and asymmetric traveling salesperson polytopes is NP-complete. On the other hand, a sufficient condition for two vertices to be non-adjacent can be formulated as a combinatorial problem of finding a second Hamiltonian decomposition of a 4-regular multigraph. We present two backtracking algorithms for constructing a second Hamiltonian decomposition and verifying vertex non-adjacency: an algorithm based on a simple path extension and an algorithm based on the chain edge fixing procedure. Based on the results of computational experiments for undirected multigraphs, both backtracking algorithms lost to the known general variable neighborhood search heuristics. However, for directed multigraphs, the algorithm based on chain fixing of edges showed results comparable to heuristics on instances with an existing solution and better results on infeasible instances where the Hamiltonian decomposition does not exist.
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