Dynamic Kernels for Hitting Sets and Set Packing

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