In this paper the symmetric traveling salesman problem (STSP) is modeled as a problem of discrete semidefinite programming. A class of semidefinite relaxations of STSP model is defined and two variants of a branch-and...
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Given a factorable function f, we propose a procedure that constructs a concave underestimator of f that is tight at a given point. these underestimators can be used to generate intersection cuts. A peculiarity of the...
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ISBN:
(纸本)9783030179533;9783030179526
Given a factorable function f, we propose a procedure that constructs a concave underestimator of f that is tight at a given point. these underestimators can be used to generate intersection cuts. A peculiarity of these underestimators is that they do not rely on a bounded domain. We propose a strengthening procedure for the intersection cuts that exploits the bounds of the domain. Finally, we propose an extension of monoidal strengthening to take advantage of the integrality of the non-basic variables.
Consider a matroid where all elements are labeled with an element in Z. We are interested in finding a base where the sum of the labels is congruent to g (mod m). We show that this problem can be solved in (O) over ti...
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ISBN:
(纸本)9783031598340;9783031598357
Consider a matroid where all elements are labeled with an element in Z. We are interested in finding a base where the sum of the labels is congruent to g (mod m). We show that this problem can be solved in (O) over tilde (2(4m)nr(5/6)) time for a matroid with n elements and rank r, when m is either the product of two primes or a prime power. the algorithm can be generalized to all moduli and, in fact, to all abelian groups if a classic additive combinatorics conjecture by Schrijver and Seymour holds true. We also discuss the optimization version of the problem.
We obtain new transference bounds that connect two active areas of research: proximity and sparsity of solutions to integer programs. Specifically, we study the additive integrality gap of the integer linear programs ...
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ISBN:
(纸本)9783031598340;9783031598357
We obtain new transference bounds that connect two active areas of research: proximity and sparsity of solutions to integer programs. Specifically, we study the additive integrality gap of the integer linear programs min{c.x : x is an element of P boolean AND Z(n)}, where P = {x is an element of R-n : Ax = b, x >= 0} is a polyhedron in the standard form determined by an integer m x n matrix A and an integer vector b. the main result of the paper gives an upper bound for the integrality gap that drops exponentially in the size of support of the optimal solutions corresponding to the vertices of the integer hull of P. Additionally, we obtain a new proximity bound that estimates the l(2)-distance from a vertex of P to its nearest integer point in the polyhedron P. the proofs make use of the results from the geometry of numbers and convex geometry.
One of the most famous conjectures in combinatorialoptimization is the four-thirds conjecture, which states that the integrality gap of the subtour LP relaxation of the TSP is equal to 4/3. For 40 years, the best kno...
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ISBN:
(纸本)9783031598340;9783031598357
One of the most famous conjectures in combinatorialoptimization is the four-thirds conjecture, which states that the integrality gap of the subtour LP relaxation of the TSP is equal to 4/3. For 40 years, the best known upper bound was 1.5, due to Wolsey [Wol80]. Recently, Karlin, Klein, and Oveis Gharan [KKO22] showed that the max entropy algorithm for the TSP gives an improved bound of 1.5 - 10(-36). In this paper, we show that the approximation ratio of the max entropy algorithm is at least 1.375, even for graphic TSP. thus the max entropy algorithm does not appear to be the algorithm that will ultimately resolve the four-thirds conjecture in the affirmative, should that be possible.
the exact solution of bilevel optimization problems is a very challenging task that received more and more attention in recent years, as witnessed by the flourishing recent literature on this topic. In this paper we p...
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ISBN:
(数字)9783319334615
ISBN:
(纸本)9783319334615;9783319334608
the exact solution of bilevel optimization problems is a very challenging task that received more and more attention in recent years, as witnessed by the flourishing recent literature on this topic. In this paper we present ideas and algorithms to solve to proven optimality generic Mixed-integer Bilevel Linear Programs (MIBLP's) where all constraints are linear, and some/all variables are required to take integer values. In doing so, we look for a general-purpose approach applicable to any MIBLP (under mild conditions), rather than ad-hoc methods for specific cases. Our approach concentrates on minimal additions required to convert an effective branch-and-cut MILP exact code into a valid MIBLP solver, thus inheriting the wide arsenal of MILP tools (cuts, branching rules, heuristics) available in modern solvers.
Linear programming is a foundational tool for many aspects of integer and combinatorialoptimization. this work studies the complexity of solving linear programs exactly over the rational numbers through use of an ora...
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ISBN:
(纸本)9783030179533;9783030179526
Linear programming is a foundational tool for many aspects of integer and combinatorialoptimization. this work studies the complexity of solving linear programs exactly over the rational numbers through use of an oracle capable of returning limited-precision LP solutions. Under mild assumptions, it is shown that a polynomial number of calls to such an oracle and a polynomial number of bit operations, is sufficient to compute an exact solution to an LP. Previous work has often considered oracles that provide solutions of an arbitrary specified precision. While this leads to polynomial-time algorithms, the level of precision required is often unrealistic for practical computation. In contrast, our work provides a foundation for understanding and analyzing the behavior of the methods that are currently most effective in practice for solving LPs exactly.
Modern parcel logistic networks are designed to ship demand between given origin, destination pairs of nodes in an underlying directed network. Efficiency dictates that volume needs to be consolidated at intermediate ...
ISBN:
(纸本)9783031598340;9783031598357
Modern parcel logistic networks are designed to ship demand between given origin, destination pairs of nodes in an underlying directed network. Efficiency dictates that volume needs to be consolidated at intermediate nodes in a typical hub-and-spoke fashion. In practice, such consolidation requires parcel sortation. In this work, we propose a mathematical model for the physical requirements, and limitations of parcel sortation. We then show that it is NP-hard to determine whether a feasible sortation plan exists. We discuss several settings, where (near-) feasibility of a given sortation instance can be determined efficiently. the algorithms we propose are fast and build on combinatorial witness set type lower bounds that are reminiscent and extend those used in earlier work on degree-bounded spanning trees and arborescences.
the simplex algorithm is one of the most popular algorithms to solve linear programs (LPs). Starting at an extreme point solution of an LP, it performs a sequence of basis exchanges (called pivots) that allows one to ...
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ISBN:
(纸本)9783031598340;9783031598357
the simplex algorithm is one of the most popular algorithms to solve linear programs (LPs). Starting at an extreme point solution of an LP, it performs a sequence of basis exchanges (called pivots) that allows one to move to a better extreme point along an improving edge-direction of the underlying polyhedron. A key issue in the simplex algorithm's performance is degeneracy, which may lead to a (potentially long) sequence of basis exchanges which do not change the current extreme point solution. In this paper, we prove that it is always possible to limit the number of consecutive degenerate pivots that the simplex algorithm performs to n - m - 1, where n is the number of variables and m is the number of equality constraints of a given LP in standard equality form.
We extend the theoretical foundations of the branch-and-cut method using lift-and-project cuts for a broader class of disjunctive constraints, and also present a new, substantially improved disjunctive cut generator. ...
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ISBN:
(纸本)354064590X
We extend the theoretical foundations of the branch-and-cut method using lift-and-project cuts for a broader class of disjunctive constraints, and also present a new, substantially improved disjunctive cut generator. Employed together with an efficient commercial MIP solver, our code is a robust, general purpose method for solving mixed integer programs. We present extensive computational experience withthe most difficult problems in the MIPLIB library.
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