Dynamic Kernels for Hitting Sets and Set Packing - Bannach, Max; Heinrich, Zacharias; Reischuk, Rüdiger; Tantau, Till (Algorithmica, 2022)
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 m · 2 d poly ( d ) for a small polynomial 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 poly ( d ) for updating the kernel upon inserts and time 5 d poly ( d ) for updates upon deletions. These bounds nearly match the time 2 d 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.