Efficient Computation of Representative Families with Applications in Parameterized and Exact Algorithms

Abstract

Let M =( E , I ) be a matroid and let S ={ S 1 , ċ , S t } be a family of subsets of E of size p . A subfamily Ŝ ⊆ S is q - representative for S if for every set Y ⊆ E of size at most q , if there is a set X ∈ S disjoint from Y with X ∪ Y ∈ I , then there is a set Xˆ ∈ Ŝ disjoint from Y with Xˆ ∪ Y ∈ I . By the classic result of Bollobás, in a uniform matroid, every family of sets of size p has a q -representative family with at most ( p + q p ) sets. In his famous “two families theorem” from 1977, Lovász proved that the same bound also holds for any matroid representable over a field F. We give an efficient construction of a q -representative family of size at most ( p + q p ) in time bounded by a polynomial in ( p + q p ), t , and the time required for field operations. We demonstrate how the efficient construction of representative families can be a powerful tool for designing single-exponential parameterized and exact exponential time algorithms. The applications of our approach include the following: —In the L ong D irected C ycle problem, the input is a directed n -vertex graph G and the positive integer k . The task is to find a directed cycle of length at least k in G , if such a cycle exists. As a consequence of our 6.75 k + o ( k ) n O (1) time algorithm, we have that a directed cycle of length at least log n , if such a cycle exists, can be found in polynomial time. —In the M inimum E quivalent G raph (MEG) problem, we are seeking a spanning subdigraph D ′ of a given n -vertex digraph D with as few arcs as possible in which the reachability relation is the same as in the original digraph D . —We provide an alternative proof of the recent results for algorithms on graphs of bounded treewidth showing that many “connectivity” problems such as H amiltonian C ycle or S teiner T ree can be solved in time 2 O ( t ) n on n -vertex graphs of treewidth at most t . For the special case of uniform matroids on n elements, we give a faster algorithm to compute a representative family. We use this algorithm to provide the fastest known deterministic parameterized algorithms for k -P ath , k -T ree , and, more generally, k -S ubgraph I somorphism , where the k -vertex pattern graph is of constant treewidth.