A measure & conquer approach for the analysis of exact algorithms

Abstract

For more than 40 years, Branch & Reduce exponential-time backtracking algorithms have been among the most common tools used for finding exact solutions of NP-hard problems. Despite that, the way to analyze such recursive algorithms is still far from producing tight worst-case running time bounds. Motivated by this, we use an approach, that we call "Measure & Conquer", as an attempt to step beyond such limitations. The approach is based on the careful design of a nonstandard measure of the subproblem size; this measure is then used to lower bound the progress made by the algorithm at each branching step. The idea is that a smarter measure may capture behaviors of the algorithm that a standard measure might not be able to exploit, and hence lead to a significantly better worst-case time analysis.In order to show the potentialities of Measure & Conquer, we consider two well-studied NP-hard problems: minimum dominating set and maximum independent set. For the first problem, we consider the current best algorithm, and prove (thanks to a better measure) a much tighter running time bound for it. For the second problem, we describe a new, simple algorithm, and show that its running time is Preliminary parts of this article appeared in competitive with the current best time bounds, achieved with far more complicated algorithms (and standard analysis).Our examples show that a good choice of the measure, made in the very first stages of exact algorithms design, can have a tremendous impact on the running time bounds achievable.

Branch-and-reduce exponential/FPT algorithms in practice: A case study of vertex cover

“…In our main implementation, we completely use the same strategy as the one used in the theoretical exact exponential algorithm by Fomin et al [7] for selecting a vertex to branch on. Basically, a vertex of the maximum degree is selected.…”

“…In our experiments (Section 7.3.3), we compare this strategy to the random selection strategy and the minimum degree selection strategy. For the mirror branching rule by Fomin et al [7], we branch into two cases: 1) including M[v] to the vertex cover or 2) discarding v while including N (v) to the vertex cover. In our implementation, we use this branching rule when the selected vertex v has mirrors.…”

“…First, we introduce four reduction rules from the exact exponential algorithm by Fomin et al [7]. Three of them are quite simple.…”

“…Lemma 1 (Degree-2 Folding [7]). Let v be a vertex of degree two whose two neighbors are not adjacent, and let G ′ be a graph obtained from G by removing N [v], introducing a new vertex w which is connected to N 2 (v).…”

Computing Optimal Steiner Trees in Polynomial Space

“…For this reason we use the Measure & Conquer analytical technique described in [21] (see also [17,18]), which is based on the quasiconvex analysis of multivariate recurrences by Eppstein [13]. The basic idea is designing a convenient (non-trivial) measure of the size of the problem.…”