Abstract
We introduce the referenced vertex ordering problem (REVORDER) as a combinatorial decision problem generalizing several vertex ordering problems that already appeared in the scientific literature under different guises. In other words, REVORDER is a generic problem with several possible extensions corresponding to various real-life applications. Previous works show that REVORDER has a polynomial complexity, whereas its optimization version, denoted in our work as MIN REVORDER, is NP-hard. We give a survey of methods and algorithms that can be applied to the solution of MIN REVORDER, and we develop a new enumeration scheme for its solution. Our theoretical analysis of this scheme yields several pruning techniques aimed at the reduction of the number of enumeration nodes. We then discuss how upper and lower bounds can be computed during the enumeration to design a branch-and-bound algorithm. Finally, we validate the branch-and-bound by conducting a large set of computational experiments on instances coming from various real-life applications. Our results highlight that the newly introduced pruning techniques allow for major reductions of computational cost, with a constant solution quality. In result, our branch-and-bound outperforms other existing solution methods: among 180 instances with 60 vertices, it solves 179 instances to optimality whereas the best existing method is only able to solve 109 of them. Moreover, our tests show that our algorithm can solve medium-scale instances including up to 500 vertices, which opens the perspective of handling new real-life problems. Our implementation of the branch-and-bound algorithm, together with all instances we have used, is publicly available on GitLab.
Highlights
Consider the problem of an undergraduate or graduate student organizing her University program
We will focus on the cycle-constrained extended formulation of [21] which appeared to be the best performing integer programming (IP) approach in our tests. We first generalize this formulation to min revorder, and we develop two sets of valid inequalities based on cliques and cycles in the subgraph of G induced by low degree vertices
We have introduced a vertex ordering decision problem, revorder, and its optimization counterpart, min revorder
Summary
Consider the problem of an undergraduate or graduate student organizing her University program. From an initial selection of the classes of the very first semester, the student’s problem consists in constructing an order for (a part of) the other graph vertices so that they admit a predetermined number of predecessors (the courses given in the previous semester). Consider the same student achieving a successful academic career Some years later she is faced with the problem of organizing a conference scientific program. The two simple examples given above illustrate this particularity of the problem we consider This problem is not new as it has appeared already in the scientific literature, and we will mainly focus our attention on the following two scientific problems (see Section 2).
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