This paper proposes a new general technique for maximal subgraph enumeration which we call proximity search, whose aim is to design efficient enumeration algorithms for problems that could not be solved by existing frameworks. To support this claim and illustrate the technique we include output-polynomial algorithms for several problems for which output-polynomial algorithms were not known, including the enumeration of Maximal Bipartite Subgraphs, Maximal k-Degenerate Subgraphs (for bounded k), Maximal Induced Chordal Subgraphs, and Maximal Induced Trees. Using known techniques, such as reverse search, the space of all maximal solutions induces an implicit directed graph called "solution graph" or "supergraph", and solutions are enumerated by traversing it; however, nodes in this graph can have exponential out-degree, thus requiring exponential time to be spent on each solution. The novelty of proximity search is a formalization that allows us to define a better solution graph, and a technique, which we call canonical reconstruction, by which we can exploit the properties of given problems to build such graphs. This results in solution graphs whose nodes have significantly smaller (i.e., polynomial) out-degree with respect to existing approaches, but that remain strongly connected, so that all solutions can be enumerated in polynomial delay by a traversal. A drawback of this approach is the space required to keep track of visited solutions, which can be exponential: we further propose a technique to induce a parent-child relationship among solutions and achieve polynomial space when suitable conditions are met.
翻译:本文提出了一种关于最大子图的新的通用技术,我们称之为近距离搜索,目的是为无法通过现有框架解决的问题设计高效的计数算法。 为了支持这项主张并展示我们包括输出-极式算法在内的技术, 这些问题的输出- 极式算法尚不为人知, 包括Maximal Bipartite 分法、 Maximal k- Degenerate 分法( 绑定 k) 、 Maximal Inducal Chordal 分法和 最大导图。 使用已知的技术, 如逆向搜索, 所有最大解决方案的空间空间空间空间空间空间空间空间空间空间空间空间空间空间空间空间空间空间空间空间空间空间空间空间空间空间空间空间空间空间空间空间空间 。 近距离搜索的新形式让我们定义一个更好的解决方案, 并且我们称之为一种技术, 通过这种技术, 我们可以进一步利用给给定的问题特性来构建这样的图形, 或“ 超级图像” 。 这个图中的节点可以具有指数性结果, 与当前路径的精确度图可以保持。