K₃ Edge Cover Problem in a Wide Sense

抄録

In this study, we consider a problem, for a given graph G = (V, E), of finding the minimum number of 3-cliques (K₃s) that cover all edges of G. Multiple covering or covering one edge by more than one 3-clique is allowed. Moreover, in this problem, we allow “spilling-out, ” i.e., a set of three vertices {x, y, z} can be covered by a 3-clique even if the induced subgraph by them is not a clique. We call this problem K₃ edge cover problem in a wide sense. This problem is a kind of extension of the schoolgirl problem and finite projective planes, and it has applications on experimental designs. Allowing spilling-out is useful for some applications: E.g., when we want to compare n items through some tries of experiments, in which at most three items can be compared simultaneously, and pairs of items that must be compared are given by a graph, finding the minimum number of tries is formalized as this problem. In the known literature, there are many results that considered problems for covering vertices or edges by the minimum number of cliques. However, there is no theoretical result that considers spilling-out. We obtain the following results: (1) The problem is NP-hard even if graphs are restricted to planar, cubic, and C4, C5-free in a sense of subgraphs (i.e., not restricted to induced ones). (2) For the problem with a parameter k, which is the number of 3-cliques in G, there is an O(mn + 2^km)-time algorithm. (3) If a tree-decomposition of tree-width t is given, there is an O(2^2(t+1)(t+2)t²n)-time algorithm.