Dynamic Kernels for Hitting Sets and Set Packing

Abstract

Computing small kernels for the hitting set problem is a well-studied computational problem where we are given a hypergraph with n vertices and m hyperedges, each of size d for some small constant d, and a parameter k. The task is to compute a new hypergraph, called a kernel, whose size is polynomial with respect to the parameter k and which has a size-k hitting set if, and only if, the original hypergraph has one. State-of-the-art algorithms compute kernels of size k^d (which is a polynomial as d is a constant), and they do so in time mcdot 2^d {text {poly}}(d) for a small polynomial {text {poly}}(d) (which is linear in the hypergraph size for d fixed). We generalize this task to the dynamic setting where hyperedges may continuously be added or deleted and one constantly has to keep track of a size-k^d kernel. This paper presents a deterministic solution with worst-case time 3^d {text {poly}}(d) for updating the kernel upon inserts and time 5^d {text {poly}}(d) for updates upon deletions. These bounds nearly match the time 2^d {text {poly}}(d) needed by the best static algorithm per hyperedge. Let us stress that for constant d our algorithm maintains a hitting set kernel with constant, deterministic, worst-case update time that is independent of n, m, and the parameter k. As a consequence, we also get a deterministic dynamic algorithm for keeping track of size-k hitting sets in d-hypergraphs with update times O(1) and query times O(c^k) where c = d - 1 + O(1/d) equals the best base known for the static setting.