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A Polyhedral Approach to the Single Row Facility Layout Problem
, 2011
"... The Single Row Facility Layout Problem (SRFLP) is the ..."
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Cited by 5 (1 self)
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The Single Row Facility Layout Problem (SRFLP) is the
Semidefinite Relaxations of Ordering Problems
, 2011
"... Ordering problems assign weights to each ordering and ask to find an ordering of maximum weight. We consider problems where the cost function is either linear or quadratic. In the first case, there is a given profit if the element u is before v in the ordering. In the second case, the profit depends ..."
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Cited by 4 (2 self)
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Ordering problems assign weights to each ordering and ask to find an ordering of maximum weight. We consider problems where the cost function is either linear or quadratic. In the first case, there is a given profit if the element u is before v in the ordering. In the second case, the profit depends on whether u is before v and r is before s. The linear ordering problem is well studied, with exact solution methods based on polyhedral relaxations. The quadratic ordering problem does not seem to have attracted similar attention. We present a systematic investigation of semidefinite optimization based relaxations for the quadratic ordering problem, extending and improving existing approaches. We show the efficiency of our relaxations by providing computational experience on a variety of problem classes.
Exact Approaches to Multilevel Vertical Orderings
, 2011
"... We present a semidefinite programming (SDP) approach for the problem of ordering vertices of a layered graph such that the edges of the graph are drawn as vertical as possible. This multilevel vertical ordering (MLVO) problem falls into the class of quadratic ordering problems. It is conceptually r ..."
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Cited by 1 (1 self)
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We present a semidefinite programming (SDP) approach for the problem of ordering vertices of a layered graph such that the edges of the graph are drawn as vertical as possible. This multilevel vertical ordering (MLVO) problem falls into the class of quadratic ordering problems. It is conceptually related to the wellstudied problem of multilevel crossing minimization (MLCM), but offers certain interesting novel properties: we not only have to consider the pure relative ordering of the nodes, but their final absolute ranks (i.e., positions) within the ordered levels. Furthermore, MLVO is a natural quadratic problem that does not only consist of multiple sequentially linked bilevel quadratic ordering problems, but is a genuine multilevel quadratic ordering problem. This allows us to describe the graphs’ structures more compactly and therefore obtain (near)optimal, (well)readable drawings of graphs too large for MLCM. We show (theoretically and experimentally) that these properties lead to the situation that approaches based on ILPs and QPs are inapplicable, even for small sparse graphs, while the SDP works surprisingly well in practice. This is in stark contrast to other ordering problems as, e.g., MLCM, where such graphs are typically solved more efficiently with ILPs. In this paper we present a motivation, mathematical models, strengthening constraints for ILPs and QPs, and an SDP relaxation for MLVO. We compare the relevant models from the polyhedral point of view, and conduct a series of experiments (including a comparison to MLCM) to showcase our SDP’s applicability. We conclude with sketching further applications in scheduling and ranking problems, where MLVO occurs apart from graph drawing.
WillemJan van Hoeve
, 2014
"... We propose a general branchandbound algorithm for discrete optimization in which binary decision diagrams (BDDs) play the role of the traditional linear programming relaxation. In particular, relaxed BDD representations of the problem provide bounds and guidance for branching, while restricted BD ..."
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We propose a general branchandbound algorithm for discrete optimization in which binary decision diagrams (BDDs) play the role of the traditional linear programming relaxation. In particular, relaxed BDD representations of the problem provide bounds and guidance for branching, while restricted BDDs supply a primal heuristic. Each problem is given a dynamic programming model that allows one to exploit recursive structure, even though the problem is not solved by dynamic programming. A novel search scheme branches within relaxed BDDs rather than on values of variables. Preliminary testing shows that a rudimentary BDDbased solver is competitive with or superior to a leading commercial integer programming solver for the maximum stable set problem, the maximum cut problem on a graph, and the maximum 2satisfiability problem. Specific to the maximum cut problem, we tested the BDDbased solver on a classical benchmark set and identified better solutions and tighter relation bounds than have ever been identified by any technique, nearly closing the entire optimality gap on several largescale instances.
ON A CLASS OF METRICS RELATED TO GRAPH LAYOUT PROBLEMS
, 2010
"... We examine the metrics that arise when a finite set of points is embedded in the real line, in such a way that the distance between each pair of points is at least 1. These metrics are closely related to some other known metrics in the literature, and also to a class of combinatorial optimization ..."
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We examine the metrics that arise when a finite set of points is embedded in the real line, in such a way that the distance between each pair of points is at least 1. These metrics are closely related to some other known metrics in the literature, and also to a class of combinatorial optimization problems known as graph layout problems. We prove several results about the structure of these metrics. In particular, it is shown that their convex hull is not closed in general. We then show that certain linear inequalities define facets of the closure of the convex hull. Finally, we characterise the unbounded edges of the convex hull and of its closure.