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.

Fixed-Parameter Algorithms for Maximum-Profit Facility Location Under Matroid Constraints

“…We now prove Proposition 4.8. Like its implicit ad-hoc proofs in the literature [15,16,36], we will prove it by iterative application of the following known result. .…”

“…Fomin et al[15] require the weight function to be non-negative. Yet their proof does not exploit non-negativity.…”

On r-Simple k-Path and Related Problems Parameterized by k/r

“…Finally, let us remark that k-Path (on both directed and undirected graph) and p-Set q-Packing are both among the most extensively studied problems in Parameterized Complexity. In particular, after a long sequence of works during the past three decades, the current best known parameterized algorithms for k-Path have running times 1.657 k n O(1) (randomized, undirected only) [10,9] (extended in [11]), 2 k n O(1) (randomized) [43] and 2.597 k n O(1) (deterministic) [44,20,40]. In addition, k-Path is known not to admit any polynomial kernel unless NP ⊆ coNP/poly [12].…”

Extensor-coding

“…Without the promise on the number of subgraphs, Fomin et al [28] detect subgraphs in randomized timeÕ(2 k n tw(H)+1 ) and Fomin et al [27] do so in deterministic time 2.619 k n O(tw(H)) . For C ≤ O(1), or C ≤ poly(n, k) when ignoring polynomial factors, we thus match the running time of the fastest randomized algorithm, but do so deterministically, and for C ≤ O(1.144 k ), our algorithm is the fastest deterministic algorithm for this problem.…”

“…Clearly, this problem generalizes the NP-hard Hamiltonian path problem [30]. We consider the decision version, the k-path problem, in which we wish to find a path of length k in a given graph G. It was proved fixed-parameter tractable avant la lettre [50], and a sequence of both iterative improvements and conceptual breakthroughs [11,4,7,40,16,27,63] have lead to the current state-of-the-art for undirected graphs: a randomized algorithm by Björklund et al [9] in time 1.66 k · poly(n). For directed graphs, the fastest known randomized algorithm is by Koutis and Williams [43] in time 2 k · poly(n), whereas the fastest deterministic algorithm is due to Zehavi [66] in time 2.5961 k · poly(n).Subgraph isomorphism.…”