4 research outputs found
Linear Time Subgraph Counting, Graph Degeneracy, and the Chasm at Size Six
We consider the problem of counting all k-vertex subgraphs in an input graph, for any constant k. This problem (denoted SUB-CNT_k) has been studied extensively in both theory and practice. In a classic result, Chiba and Nishizeki (SICOMP 85) gave linear time algorithms for clique and 4-cycle counting for bounded degeneracy graphs. This is a rich class of sparse graphs that contains, for example, all minor-free families and preferential attachment graphs. The techniques from this result have inspired a number of recent practical algorithms for SUB-CNT_k. Towards a better understanding of the limits of these techniques, we ask: for what values of k can SUB_CNT_k be solved in linear time?
We discover a chasm at k=6. Specifically, we prove that for k < 6, SUB_CNT_k can be solved in linear time. Assuming a standard conjecture in fine-grained complexity, we prove that for all k ? 6, SUB-CNT_k cannot be solved even in near-linear time
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The Role of Degeneracy in Real-World Subgraph Counting
Many real-world phenomena are modeled by large graphs. Subgraph counting, the problem of counting occurrences of small target pattern graphs in large input graphs is a fundamental algorithmic task in network analysis. Subraph counting has been extensively studied in both theory and practice and has found applications in areas such as network analysis, social sciences, and bioinformatics. Graph orientation techniques for subgraph counting based on vertex orderings such as degeneracy ordering is a classical idea. These techniques have inspired many recent practical subraph counting algorithms. In this thesis we analyze the role of graph orientation and degeneracy in subgraph counting, both in theory and practice. Based on these techniques, we present efficient algorithms for getting local subgraph counts (orbits counts) of all 5-vertex patterns, and counting triangles in temporal networks. In modern applications, input graphs are large and one desires (near) linear time algorithms. We focus on the case where the input graph is in the class of bounded degeneracy graphs. This is a rich class of sparse graphs that is practically relevant as real-world graphs such as social networks have been shown to have low degeneracy. We consider the problem of counting all connected subgraphs with vertices, and determine for what values of this problem is solvable in linear time, assuming a standard conjecture in fine-grained complexity. We also give a clean characterizations of all subgraph patterns whose homomorphisms could be counted in near linear time in bounded degeneracy graphs