Efficient computation of representative sets with applications in parameterized and exact algorithms

Let M = (E, I) be a matroid and let S = {S1, ...,St} be a family of subsets of E of size p. A subfamily S ⊆ S is q-representative for S if for every set Y ⊆ E of size at most q, if there is a set X ε S disjoint from Y with X ∪ Y ε I, then there is a set X ε S disjoint from Y with X ∪ Y ε I. By the classical result of Bollobás, in a uniform matroid, every family of sets of size p has a q-representative family with at most (p+qp) 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. As observed by Marx, Lovász's proof is constructive. In this paper we show how Lovász's proof can be turned into an algorithm constructing a q-representative family of size at most (p+qp) in time bounded by a polynomial in (p+qp), 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 (2O(k)) and exact exponential time (2O(k)) algorithms. The applications of our approach include the following. • In the Long Directed Cycle 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 8k+onO(1) time algorithm, we have that a directed cycle of length at least log n, if such cycle exists, can be found in polynomial time. As it was shown by Björklund, Husfeldt, and Khanna [ICALP 2004], under an appropriate complexity assumption, it is impossible to improve this guarantee by more than a constant factor. Thus our algorithm not only improves over the best previous log n/log log n bound of Gabow and Nie [SODA 2004] but also closes the gap between known lower and upper bounds for this problem. • In the Minimum Equivalent Graph (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. The existence of a single-exponential cn-time algorithm for some constant c > 1 for MEG was open since the work of Moyles and Thompson [JACM 1969]. • To demonstrate the diversity of applications of the approach, we provide an alternative proof of the results recently obtained by Bodlaender, Cygan, Kratsch and Nederlof for algorithms on graphs of bounded treewidth, who showed that many \"connectivity\" problems such as Hamiltonian Cycle or Steiner Tree can be solved in time 2O(t)n on n-vertex graphs of treewidth at most t. We believe that expressing graph problems in \"matroid language\" shed light on what makes it possible to solve connectivity problems single-exponential time parameterized by treewidth. For the special case of uniform matroids on n elements, we give a faster algorithm computing a representative family in time O((p+q/q)q · 2o(p+q) · t · log n). We use this algorithm to provide the fastest known deterministic parameterized algorithms for k-Path, k-Tree, and more generally, for k-Subgraph Isomorphism, where the k-vertex pattern graph is of constant treewidth. For example, our k-Path algorithm runs in time O(2.851k n log2 n log W) on weighted graphs with maximum edge weight W.