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integer programs with prescribed number of solutions and a weighted version of Doignon-bell-scarf's theorem

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17th international conference on integer programming and combinatorial optimization, ipco 2014

作者： Aliev, Iskander De Loera, Jesús A. Louveaux, Quentin Cardiff University United Kingdom University of California Davis United States Université de Liège Belgium

ISBN: (纸本)9783319075563

In this paper we study a generalization of the classical feasibility problem in integer linear programming, where an ILP needs to have a prescribed number of solutions to be considered solved. We first provide a generalization of the famous Doignon-Bell-Scarf theorem: Given an integer k, we prove that there exists a constant c(k,n), depending only on the dimension n and k, such that if a polyhedron {x: Ax≤b} contains exactly k integer solutions, then there exists a subset of the rows of cardinality no more than c(k,n), defining a polyhedron that contains exactly the same k integer solutions. the second contribution of the article presents a structure theory that characterizes precisely the set sg ≥ k (A) of all vectors b such that the problem Ax = b, x ≥ 0, x Ε Zn, has at least k-solutions. We demonstrate that this set is finitely generated, a union of translated copies of a semigroup which can be computed explicitly via Hilbert bases computation. Similar results can be derived for those right-hand-side vectors that have exactly k solutions or fewer than k solutions. Finally we show that, when n, k are fixed natural numbers, one can compute in polynomial time an encoding of sg ≥ (A) as a generating function, using a short sum of rational functions. As a consequence, one can identify all right-hand-side vectors that have exactly k solutions (similarly for at least k or less than k solutions). Under the same assumptions we prove that the k-Frobenius number can be computed in polynomial time. © 2014 Springer international Publishing Switzerland.

关键词： Bells

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Adaptive embedded systems: New composed technical solutions for feasible low-power and real-time flexible OS tasks

Adaptive embedded systems: New composed technical solutions ...

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international conference on Informatics in Control, Automation and Robotics (ICINCO)

作者： Hamza Chniter Mohamed Khalgui Fethi Jarray INSAT Institute University of Carthage Carthage Tunisia ISI Institute Mednine Tunisia FST faculty University of Elmanar Elmanar Tunisia Laboratoire CEDRIC CNAM France

the paper deals with low-power adaptive scheduling of synchronous and flexible real-time OS tasks. A software reconfiguration scenario is assumed to be any run-time operation allowing the addition-removal-update of OS tasks to adapt the system to its environment under well-defined conditions. the problem is that any reconfiguration can push the system to an unfeasible behavior where temporal properties are violated or the energy consumption is possibly high and unacceptable. A task in the system can change its characteristics at any time when a reconfiguration scenario is applied, it can also be stopped or replaced by another one. the difficulty is how to find the new temporal parameters of the systems tasks after any reconfiguration. We use a DVS processor which is with a variable speed to support run-time solutions that re-obtain the system's feasibility. the challenge is how to compute the best combinations between available processor speeds for a good compromise between execution time and energy consumption. We propose a combinatorial optimization method based on integer programming and heuristics. We propose also a solution when the available speeds do not allow the feasibility of the system. Both approaches include a mechanism to adjust the deadlines of tasks to satisfy the feasibility conditions and overcome the problem of rejected tasks. this mechanism makes the scheduling more flexible and able to react in accordance with its environment.

关键词： Real-time systems Energy consumption Embedded systems Linear programming Processor scheduling Time factors Computational modeling

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Cut-generating functions

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16th conference on integer programming and combinatorial optimization, ipco 2013

作者： Conforti, Michele Cornuéjols, Gérard Daniilidis, Aris Lemaréchal, Claude Malick, Jérôme University of Padova Italy Carnegie Mellon University United States Autonomous University of Barcelona Spain INRIA Grenoble France CNRS Grenoble France

ISBN: (纸本)9783642366932

In optimization problems such as integer programs or their relaxations, one encounters feasible regions of the form {x ∈ +n : Rx ∈ S} where R is a general real matrix and S ⊂ q is a specific closed set with 0 ∉ S. For example, in a relaxation of integer programs introduced in [ALWW2007], S is of the form q - b where b ∉ q. One would like to generate valid inequalities that cut off the infeasible solution x = 0. Formulas for such inequalities can be obtained through cut-generating functions. this paper presents a formal theory of minimal cut-generating functions and maximal S-free sets which is valid independently of the particular S. this theory relies on tools of convex analysis. © 2013 Springer-Verlag.

关键词： integer programming

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A tight MILP formulation based on multi-product valid inequalities for a lot-sizing problem

A tight MILP formulation based on multi-product valid inequa...

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5th international conference on Industrial Engineering and Systems Management (IEEE IESM)

作者： Gicquel, Celine Minoux, Michel Univ Paris 11 Lab Rech Informat F-91405 Orsay France Univ Paris 06 Lab Informat Paris 6 F-75005 Paris France

ISBN: (纸本)9782960053241

We consider a problem arising in the context of industrial production planning, namely the multi-product discrete lot-sizing and scheduling problem with sequence-dependent changeover costs. We aim at developping an exact solution approach based on a standard Branch & Bound procedure for this combinatorial optimization problem. To achieve this, we propose a new family of multi-product valid inequalities which enables us to better take into account in the mixed integer linear programming formulation the conflicts between different products simultaneously requiring production on the resource. We then present both an exact and a heuristic separation algorithm in order to identify the most violated valid inequalities to be added in the initial MILP formulation within a cutting-plane generation algorithm. We finally discuss preliminary computational results which confirm the practical usefulness of the proposed valid inequalities at strengthening the MILP formulation and at reducing the overall computation time.

关键词： inequalities Valid computational time Exact industrial production New families Mixed integer linear programming lot sizing

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integer programming and combinatorial optimization - 14th international conference, ipco 2010, Proceedings

Integer Programming and Combinatorial Optimization - 14th In...

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14th international conference on integer programming and combinatorial optimization, ipco 2010

ISBN: (纸本)3642130356

the proceedings contain 34 papers. the topics discussed include: solving LP relaxations of large-scale precedence constrained problems;computing minimum multiway cuts in hypergraphs from hypertree packings;eigenvalue techniques for convex objective, nonconvex optimization problems;restricted b-matchings in degree-bounded graphs;zero-coefficient cuts;prize-collecting steiner network problems;on lifting integer variables in minimal inequalities;efficient edge splitting-off algorithms maintaining all-pairs edge-connectivities;on generalizations of network design problems with degree bounds;a polyhedral study of the mixed integer cut;symmetry matters for the sizes of extended formulations;a 3-approximation for facility location with uniform capacities;and approximability of 3- and 4-hop bounded disjoint paths problems.

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Computing the Maximum Degree of Minors in Mixed Polynomial Matrices via combinatorial Relaxation

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15th international conference on integer programming and combinatorial optimization (ipco)

作者： Iwata, Satoru Takamatsu, Mizuyo Kyoto Univ Math Sci Res Inst Kyoto 6068502 Japan Chuo Univ Dept Informat & Syst Engn Tokyo 1128551 Japan

ISBN: (纸本)9783642208072

Mixed polynomial matrices are polynomial matrices with two kinds of nonzero coefficients: fixed constants that account for conservation laws and independent parameters that represent physical characteristics. the computation of their maximum degrees of minors is known to be reducible to valuated independent assignment problems, which can be solved by polynomial numbers of additions, subtractions, and multiplications of rational functions. However, these arithmetic operations on rational functions are much more expensive than those on constants. In this paper, we present a new algorithm of combinatorial relaxation type. the algorithm finds a combinatorial estimate of the maximum degree by solving a weighted bipartite matching problem, and checks if the estimate is equal to the true value by solving independent matching problems. the algorithm mainly relies on fast combinatorial algorithms and performs numerical computation only when necessary. In addition, it requires no arithmetic operations on rational functions. As a byproduct, this method yields a new algorithm for solving a linear valuated independent assignment problem.

关键词： Rational functions

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Jump Number of Two-Directional Orthogonal Ray Graphs

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15th international conference on integer programming and combinatorial optimization (ipco)

作者： Soto, Jose A. Telha, Claudio MIT Dept Math Cambridge MA 02139 USA MIT Ctr Operat Res Cambridge MA USA

ISBN: (纸本)9783642208072

We model maximum cross-free matchings and minimum biclique covers of two-directional orthogonal ray graphs (2-dorgs) as maximum independent sets and minimum hitting sets of an associated family of rectangles in the plane, respectively. We then compute the corresponding maximum independent set using linear programming and uncrossing techniques. this procedure motivates an efficient combinatorial algorithm to find a cross-free matching and a biclique cover of the same cardinality, proving the corresponding min-max relation. We connect this min-max relation with thework ofGyori [19], Lubiw[23], and Frank and Jordan [16] on seemingly unrelated problems. Our result can be seen as a non-trivial application of Frank and Jord ' an's theorem. As a direct consequence, we obtain the first polynomial algorithm for the jump number problem on 2-dorgs. For the subclass of convex graphs, our approach is a vast improvement over previous algorithms. Additionally, we prove that the weighted maximum cross-free matching problem is NP-complete for 2-dorgs and give polynomial algorithms for some subclasses.

关键词： Linear programming

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An Exact Rational Mixed-integer programming Solver

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15th international conference on integer programming and combinatorial optimization (ipco)

作者： Cook, William Koch, thorsten Steffy, Daniel E. Wolter, Kati Georgia Inst Technol Sch Ind & Syst Engn Atlanta GA 30332 USA Zuse Inst Berlin Berlin Germany

ISBN: (纸本)9783642208072

We present an exact rational solver for mixed-integer linear programming that avoids the numerical inaccuracies inherent in the floating-point computations used by existing software. this allows the solver to be used for establishing theoretical results and in applications where correct solutions are critical due to legal and financial consequences. Our solver is a hybrid symbolic/numeric implementation of LP-based branch-and-bound, using numerically-safe methods for all binding computations in the search tree. Computing provably accurate solutions by dynamically choosing the fastest of several safe dual bounding methods depending on the structure of the instance, our exact solver is only moderately slower than an inexact floating-point branch-and-bound solver. the software is incorporated into the SCIP optimization framework, using the exact LP solver QSOPT-EX and the GMP arithmetic library. Computational results are presented for a suite of test instances taken from the MIPLIB and Mittelmann collections.

关键词： integer programming

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integer programming and Combinatoral optimization - 15th international conference, ipco 2011, Proceedings

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15th international conference on integer programming and combinatorial optimization, ipco 2011

ISBN: (纸本)9783642208065

the proceedings contain 33 papers. the topics discussed include: an excluded minor characterization of seymour graphs;complexity analyses of bienstock-zuckerberg and lasserre relaxations on the matching and stable set polytopes;a probabilistic analysis of the strength of the split and triangle closures;partial convexification of general MIPs by Dantzig-Wolfe reformulation;lift-and-project cuts for mixed integer convex programs;TSP on cubic and subcubic graphs;approximability of capacitated network design;facility location with client latencies: linear programming based techniques for minimum latency problems;an exact rational mixed-integer programming solver;valid inequalities for the pooling problem with binary variables;design and verify: a new scheme for generating cutting-planes;and contact center scheduling with strict resource requirements.

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A New Approach to the Stable Set Problem Based on Ellipsoids

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15th international conference on integer programming and combinatorial optimization (ipco)

作者： Giandomenico, Monia Letchford, Adam N. Rossi, Fabrizio Smriglio, Stefano Univ Aquila Dept Comp Sci Laquila Italy Univ Lancaster Dept Management Sci Lancaster England

ISBN: (纸本)9783642208072

A new exact approach to the stable set problem is presented, which attempts to avoids the pitfalls of existing approaches based on linear and semidefinite programming. the method begins by constructing an ellipsoid that contains the stable set polytope and has the property that the upper bound obtained by optimising over it is equal to the Lovasz theta number. this ellipsoid is then used to derive cutting planes, which can be used within a linear programming-based branch-and-cut algorithm. Preliminary computational results indicate that the cutting planes are strong and easy to generate.

关键词： stable set problem semidefinite programming convex quadratic programming cutting planes

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