Given a k-node pattern graph H and an n-node host graph G, the subgraph counting problem asks to compute the number of copies of H in G. In this work we address the following question: can we count the copies of H faster if G is sparse? We answer in the affirmative by introducing a novel tree-like decomposition for directed acyclic graphs, inspired by the classic tree decomposition for undirected graphs. This decomposition gives a dynamic program for counting the homomorphisms of H in G by exploiting the degeneracy of G, which allows us to beat the state-of-the-art subgraph counting algorithms when G is sparse enough. For example, we can count the induced copies of any k-node pattern H in time \(2^{O(k^2)} O(n^{0.25k + 2} \log n)\) if G has bounded degeneracy, and in time \(2^{O(k^2)} O(n^{0.625k + 2} \log n)\) if G has bounded average degree. These bounds are instantiations of a more general result, parameterized by the degeneracy of G and the structure of H, which generalizes classic bounds on counting cliques and complete bipartite graphs. We also give lower bounds based on the Exponential Time Hypothesis, showing that our results are actually a characterization of the complexity of subgraph counting in bounded-degeneracy graphs.

给定一个ķ -node模式图ħ和Ñ -node主机图形ģ，子图计数的问题询问计算的拷贝数ħ在ģ。在这项工作中，我们解决了以下问题：如果G是稀疏的，我们能否更快地计算H的副本？我们通过为有向无环图引入一种新的树状分解来做出肯定的回答，其灵感来自于无向图的经典树分解。这种分解给出了一个动态程序用于计数的同态ħ在ģ通过利用简并ģ，当G足够稀疏时，这使我们能够击败最先进的子图计数算法。例如，如果G有界，我们可以在时间\(2^{O(k^2)} O(n^{0.25k + 2} \log n)\)计算任何k节点模式H的诱导副本简并性，如果G 的平均度有界，则在时间上\(2^{O(k^2)} O(n^{0.625k + 2} \log n)\)。这些边界是更一般结果的实例化，由G的简并性和H的结构参数化，它概括了计数派和完整二部图的经典界限。我们还给出了基于指数时间假设的下界，表明我们的结果实际上是对有界简并图中子图计数复杂性的表征。