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How to recycle your facets
- DISCRETE OPTIMIZATION
"... We show how to transform any inequality defining a facet of some 0/1-polytope into an inequality defining a facet of the acyclic subgraph polytope. While this facet-recycling procedure can potentially be used to construct ‘nasty’ facets, it can also be used to better understand and extend the polyh ..."
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We show how to transform any inequality defining a facet of some 0/1-polytope into an inequality defining a facet of the acyclic subgraph polytope. While this facet-recycling procedure can potentially be used to construct ‘nasty’ facets, it can also be used to better understand and extend the polyhedral theory of the acyclic subgraph and linear ordering problems.
LAGRANGIAN RELAXATION AND PEGGING TEST FOR LINEAR ORDERING PROBLEMS
, 2011
"... We develop an algorithm for the linear ordering problem, which has a large number of applications such as triangulation of input-output matrices, minimizing total weighted completion time in one-machine scheduling, and aggregation of individual preferences. The algorithm is based on the Lagrangian ..."
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We develop an algorithm for the linear ordering problem, which has a large number of applications such as triangulation of input-output matrices, minimizing total weighted completion time in one-machine scheduling, and aggregation of individual preferences. The algorithm is based on the Lagrangian relaxation of a binary integer linear programming formulation of the problem. Since the number of the constraints is proportional to the third power of the number of items and grows rapidly, we propose a modified subgradient method that temporarily ignores a large part of the constraints and gradually adds constraints whose Lagrangian multipliers are likely to be positive at an optimal multiplier vector. We also propose an improvement on the ordinary pegging test by using the problem structure.
Weighted graphs defining facets: a connection between stable set and linear ordering polytopes
, 2008
"... A graph is α-critical if its stability number increases whenever an edge is removed from its edge set. The class of α-critical graphs has several nice structural properties, most of them related to their defect which is the number of vertices minus two times the stability number. In particular, a re ..."
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A graph is α-critical if its stability number increases whenever an edge is removed from its edge set. The class of α-critical graphs has several nice structural properties, most of them related to their defect which is the number of vertices minus two times the stability number. In particular, a remarkable result of Lovász (1978) is the finite basis theorem for α-critical graphs of a fixed defect. The class of α-critical graphs is also of interest for at least two topics of polyhedral studies. First, Chvátal (1975) shows that each α-critical graph induces a rank inequality which is facet-defining for its stable set polytope. Investigating a weighted generalization, Lipták and Lovász (2000, 2001) introduce critical facet-graphs (which again produce facet-defining inequalities for their stable set polytopes) and they establish a finite basis theorem. Second, Koppen (1995) describes a construction that delivers from any α-critical graph a facet-defining inequality for the linear ordering polytope. Doignon, Fiorini and Joret (2006) handle the weighted case and thus define facet-defining graphs. Here we investigate relationships between the two weighted generalizations of α-critical graphs. We show that facet-defining graphs (for the linear ordering polytope) are obtainable from 1-critical facet-graphs (linked with stable set polytopes). We then use this connection to derive various results on facet-defining graphs, the most prominent one being derived from Lipták and Lovász’s finite basis theorem for critical facet-graphs. At the end of the paper we offer an alternative proof of Lovász’s finite basis theorem for α-critical graphs.
Ranking Hypotheses to Minimize the Search Cost in Probabilistic Inference Models
"... Suppose that we are given n mutually exclusive hypotheses, m mutually exclusive possible observations, the conditional probabilities for each of these observations under each hypothesis, and a method to probe each hypothesis whether it is the true one. We consider the problem of efficient searching ..."
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Suppose that we are given n mutually exclusive hypotheses, m mutually exclusive possible observations, the conditional probabilities for each of these observations under each hypothesis, and a method to probe each hypothesis whether it is the true one. We consider the problem of efficient searching for the true (target) hypothesis given a particular observation. Our objective is to minimize the expected search cost for a large number of instances, and for the worst-case distribution of targets. More precisely, we wish to rank the hypotheses so that probing them in the chosen order is optimal in this sense. Costs grow monotonic with the number of probes. While it is straightforward to formulate this problem as a linear program, we can solve it in polynomial time only after a certain reformulation: We introduce mn 2 so-called rank variables and arrive at another linear program whose solution can be translated afterwards into an optimal mixed strategy of low description complexity: For each observation, at most n rankings, i.e., permutations of hypotheses, appear with positive probabilities. Dimensionality arguments yield further combinatorial bounds. Possible applications of the optimization goal are discussed.
