The union of minimal hitting sets: Parameterized combinatorial bounds and counting

Abstract

A k-hitting set in a hypergraph is a set of at most k vertices that intersects all hyperedges. We study the union of all inclusion-minimal k-hitting sets in hypergraphs of rank r (where the rank is the maximum size of hyperedges). We show that this union is relevant for certain combinatorial inference problems and give worst-case bounds on its size, depending on r and k. For r = 2 our result is tight, and for each r 3 we have an asymptotically optimal bound and make progress regarding the constant factor. The exact worst-case size for r3 remains an open problem. We also propose an algorithm for counting all k-hitting sets in hypergraphs of rank r. Its asymptotic runtime matches the best one known for the much more special problem of finding one k-hitting set. The results are used for efficient counting of k-hitting sets that contain any particular vertex.

On the Parameterized Complexity of Reconfiguration Problems

“…To show that Q Reconfiguration is in FPT, we observe that the number of nodes in the reconfiguration graph for Q is bounded by a function of k: each solution of Q is a subset of M , yielding at most 2 |M | nodes, and |M | is bounded by a function of k. For Bounded Hitting Set, the proof of Theorem 4 can be strengthened to develop a polynomial reconfiguration kernel. In fact, we use the ideas in Theorem 4 to adapt a special kernel that retains all minimal k-hitting sets in the reduced instances [16].…”

“…We let (G, S, T, k, ) be an instance of Bounded Hitting Set Reconfiguration: G is a family of sets of vertices of size at most r and each of S and T is a hitting set of size at most k, that is, a set of vertices intersecting each set in G. We form a reconfiguration kernel using the reduction algorithm A of Damaschke and Molokov [16]: G = A(G) contains all minimal hitting set solutions of size at most k, and is of size at most (r − 1)k r + k.…”

“…To handle these difficulties, we introduce a general approach for parameterized reconfiguration problems. We use a reconfiguration kernel, showing how to adapt Bodlaender's cubic kernel [15] for Feedback Vertex Set, and a special kernel by Damaschke and Molokov [16] for Bounded Hitting Set (where the cardinality of each input set is bounded) to obtain polynomial reconfiguration kernels, with respect to k. These results can be considered as interesting applications of kernelization, and a general approach for other similar reconfiguration problems.…”

Some Fixed-Parameter Tractable Classes of Hypergraph Duality and Related Problems

“…Other related FPT results were obtained by Damaschke who studied counting and generating minimal transversals of size up to k and showed both problems to be FPT if hyperedges have constantly bounded size [Dam06,Dam07].…”

“…Both, Dual and Dualization have many applications in such different fields like artificial intelligence and logic [EG95,EG02], database theory [MR92], data mining and machine learning [GKM + 03], computational biology [Dam06,Dam07], mobile communication systems [SS98], distributed systems [GB85], and graph theory [JPY88,LLK80]. The currently best known algorithms for Dual run in quasi-polynomial time or use O(log 2 n) nondeterministic bits [EGM03,FK96,KS03].…”