A Constant Amortized Time Enumeration Algorithm For Independent Sets In Graphs With Bounded Clique Number

In this study, we address the independent set enumeration problem. Although several efficient enumeration algorithms and careful analyses have been proposed for maximal independent sets, no fine-grained analysis has been given for the non-maximal variant. As the main result, we propose an enumeration algorithm for the non-maximal variant that runs in O (q) amortized time and linear space, where q is the clique number, i.e., the maximum size of a clique in an input graph. Note that the proposed algorithm works correctly even if the exact value of q is unknown. It is optimal for graphs with a bounded clique number, such as, triangle-free graphs, bipartite graphs, planar graphs, bounded degenerate graphs, nowhere dense graphs, and F-free graphs for any fixed graph F, where a F-free graph is a graph that has no copy of F as a subgraph. Furthermore, with a slight modification of our proposed algorithm, we can enumerate independent sets with the size at most k in the same time and space complexity. This problem is a generalization of the original problem since this is equal to the original problem if k = n. (C) 2021 Elsevier B.V. All rights reserved.