We consider the Vehicle Routing Problem, in which a fixed fleet of delivery vehicles of uniform capacity must service known customer demands for a single commodity from a common depot at minimum transit cost. This dif...
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We consider the Vehicle Routing Problem, in which a fixed fleet of delivery vehicles of uniform capacity must service known customer demands for a single commodity from a common depot at minimum transit cost. This difficult combinatorial problem contains both the Bin Packing Problem and the Traveling Salesman Problem (TSP) as special cases and conceptually lies at the intersection of these two well-studied problems. The capacity constraints of the integer programming formulation of this routing model provide the link between the underlying routing and packing structures. We describe a decomposition-based separation methodology for the capacity constraints that takes advantage of our ability to solve small instances of the TSP efficiently. Specifically, when standard procedures fail to separate a candidate point, we attempt to decompose it into a convex combination of TSP tours;if successful, the tours present in this decomposition are examined for violated capacity constraints;if not, the Farkas Theorem provides a hyperplane separating the point from the TSP polytope. We present some extensions of this basic concept and a general framework within which it can be applied to other combinatorial models. Computational results are given for an implementation within the parallel branch, cut, and price framework SYMPHONY.
We consider the multilinear polytope which arises naturally in binary polynomial optimization. Del Pia and Di Gregorio introduced the class of odd beta-cycle inequalities valid for this polytope, showed that these gen...
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ISBN:
(数字)9783031069017
ISBN:
(纸本)9783031069017;9783031069000
We consider the multilinear polytope which arises naturally in binary polynomial optimization. Del Pia and Di Gregorio introduced the class of odd beta-cycle inequalities valid for this polytope, showed that these generally have Chvatal rank 2 with respect to the standard relaxation and that, together with flower inequalities, they yield a perfect formulation for cycle hypergraph instances. Moreover, they describe a separation algorithm in case the instance is a cycle hypergraph. We introduce a weaker version, called simple odd beta-cycle inequalities, for which we establish a strongly polynomial-time separation algorithm for arbitrary instances. These inequalities still have Chvatal rank 2 in general and still suffice to describe the multilinear polytope for cycle hypergraphs.
We consider generalizations of parity polytopes whose variables, in addition to a parity constraint, satisfy certain ordering constraints. More precisely, the variable domain is partitioned into k contiguous groups, a...
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We consider generalizations of parity polytopes whose variables, in addition to a parity constraint, satisfy certain ordering constraints. More precisely, the variable domain is partitioned into k contiguous groups, and within each group, we require x(i) >= x(i+1) for all relevant i. Such constraints are used to break symmetry after replacing an integer variable by a sum of binary variables, so-called binarization. We provide extended formulations for such polytopes, derive a complete outer description, and present a separation algorithm for the new constraints. It turns out that applying binarization and only enforcing parity constraints on the new variables is often a bad idea. For our application, an integer programming model for the graphic traveling salesman problem, we observe that parity constraints do not improve the dual bounds, and we provide a theoretical explanation of this effect. (C) 2018 Elsevier B.V. All rights reserved.
The heart rate increases during inspiration and decreases during expiration;the study of this variation and the change of the second heart sound split (a change related to inspiration and expiration) can determine at ...
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The heart rate increases during inspiration and decreases during expiration;the study of this variation and the change of the second heart sound split (a change related to inspiration and expiration) can determine at what time in a cardiac cycle is the inspiration and the expiration. It would also be interesting to study the variation in systolic pulmonary artery pressure (SPAP) estimated over several cardiac cycles and to understand its evolution as its variation is related to the pulmonary valve, on the one hand, and inspiration and expiration, on the other hand. The algorithm developed based on the Hilbert transform and the energy of Shannon gives the second heart sound split. The SPAP will be estimated from spectral parameters of the second heart S2. The results show an excellent performance of the algorithm proposed to extract different information on the variation of heart rate. The results of the change in pressure and split are encouraging and promising for the use of the proposed method in a clinical context of hypertension in non-invasive pulmonary pathways, for example.
Given a graph G = (V, E) with nonnegative weights on the edges, the maximum induced bipartite subgraph problem (MIBSP) is to find a maximum weight bipartite subgraph (W, E[W]) of G. Here E[W] is the edge set induced b...
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Given a graph G = (V, E) with nonnegative weights on the edges, the maximum induced bipartite subgraph problem (MIBSP) is to find a maximum weight bipartite subgraph (W, E[W]) of G. Here E[W] is the edge set induced by W. An edge subset F subset of E is called independent if there is an induced bipartite subgraph of G whose edge set contains F. Otherwise, it is called dependent. In this paper we characterize the minimal dependent sets, that is, the dependent sets that are not contained in any other dependent set. Using this, we give an integer linear programming formulation for MIBSP in the natural variable space, based on an associated class of valid inequalities called dependent set inequalities. Moreover, we show that the minimum dependent set problem with nonnegative weights can be reduced to the minimum circuit problem in a directed graph, and can then be solved in polynomial time. This yields a polynomial-time separation algorithm for the dependent set inequalities as well as a polynomial-time cutting plane algorithm for solving the linear relaxation of the problem. We also discuss some polyhedral consequences.
The min knapsack problem appears as a major component in the structure of capacitated covering problems. Its polyhedral relaxations have been extensively studied, leading to strong relaxations for networking, scheduli...
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The min knapsack problem appears as a major component in the structure of capacitated covering problems. Its polyhedral relaxations have been extensively studied, leading to strong relaxations for networking, scheduling and facility location problems. A valid inequality alpha(T) x >= alpha 0 with therefore alpha >= 0 for a min knapsack instance is said to have pitch <= pi (pi a positive integer) if the pi smallest strictly positive alpha(j) sum to at least alpha(0). An inequality with coefficients and right-hand side in {0, 1,...., pi} has pitch <= pi. The notion of pitch has been used for measuring the complexity of valid inequalities for the min knapsack polytope. Separating inequalities of pitch-1 is already NP-Hard. In this paper, we show an algorithm for efficiently separating inequalities with coefficients in {0, 1,..., pi} for any fixed pi up to an arbitrarily small additive error. As a special case, this allows for approximate separation of inequalities with pitch at most 2. We moreover investigate the integrality gap of minimum knapsack instances when bounded pitch inequalities (possibly in conjunction with other inequalities) are added. Among other results, we show that the CG closure of minimum knapsack has unbounded integrality gap even after a constant number of rounds. (C) 2020 Elsevier B.V. All rights reserved.
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