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.
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.
Robust combinatorialoptimization with budget uncertainty is one of the most popular approaches for integrating uncertainty in optimization problems. the existence of a compact reformulation for (mixed-integer) linear...
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ISBN:
(数字)9783031327261
ISBN:
(纸本)9783031327254;9783031327261
Robust combinatorialoptimization with budget uncertainty is one of the most popular approaches for integrating uncertainty in optimization problems. the existence of a compact reformulation for (mixed-integer) linear programs and positive complexity results give the impression that these problems are relatively easy to solve. However, the practical performance of the reformulation is actually quite poor when solving robust integer problems due to its weak linear relaxation. To overcome the problems arising from the weak formulation, we propose a procedure to derive new classes of valid inequalities for robust binary optimization problems. For this, we recycle valid inequalities of the underlying deterministic problem such that the additional variables from the robust formulation are incorporated. the valid inequalities to be recycled may either be readily available model constraints or actual cutting planes, where we can benefit from decades of research on valid inequalities for classical optimization problems. We first demonstrate the strength of the inequalities theoretically, by proving that recycling yields a facet-defining inequality in surprisingly many cases, even if the original valid inequality was not facet-defining. Afterwards, we show in a computational study that using recycled inequalities leads to a significant improvement of the computation time when solving robust optimization problems.
We introduce and study a two-stage stochastic stable matching problem between students and schools. A decision maker chooses a stable matching in a marriage instance;then, after some agents enter or leave the market f...
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ISBN:
(纸本)9783031598340;9783031598357
We introduce and study a two-stage stochastic stable matching problem between students and schools. A decision maker chooses a stable matching in a marriage instance;then, after some agents enter or leave the market following a probability distribution D, chooses a stable matching in the new instance. the goal is, roughly speaking, to maximize the expected quality of the matchings across the two stages and minimize the expected students' discontent for being downgraded to a less preferred school in the second-stage. We consider boththe case when D is given explicitly and when it is accessed via a sampling oracle. In the former case, we give a polynomial time algorithm. In the latter case, we show that, unless P = NP, no algorithm can find the optimal value or the optimal solution of the problem in polynomial-time. On the positive side, we give a pseudopolynomial algorithm that computes a solution of arbitrarily small additive error. Our techniques include the use of a newly defined poset of stable pairs, which may be of independent interest.
Standard mixed-integerprogramming formulations for the stable set problem on n-node graphs require n integer variables. We prove that this is almost optimal: We give a family of n-node graphs for which every polynomi...
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ISBN:
(纸本)9783031598340;9783031598357
Standard mixed-integerprogramming formulations for the stable set problem on n-node graphs require n integer variables. We prove that this is almost optimal: We give a family of n-node graphs for which every polynomial-sizeMIP formulation requires Omega(n/ log(2) n) integer variables. By a polyhedral reduction we obtain an analogous result for n-item knapsack problems. In both cases, this improves the previously known bounds of Omega(root n/ log n) by Cevallos, Weltge & Zenklusen (SODA 2018). To this end, we show that there exists a family of n-node graphs whose stable set polytopes satisfy the following: any (1 + epsilon/n)-approximate extended formulation for these polytopes, for some constant epsilon > 0, has size 2(Omega(n / log n)). Our proof extends and simplifies the information-theoretic methods due to Goos, Jain & Watson (FOCS 2016, SIAM J. Comput. 2018) who showed the same result for the case of exact extended formulations (i.e. epsilon = 0).
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.
Submodular optimization is a classical problem of combinatorialoptimization. the objective functions of many combinatorialoptimization problems are submodular functions and they also have significant applications in...
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Motivated by applications in e-retail and online advertising, we study the problem of assortment optimization under visibility constraints, referred to as APV. We are given a universe of substitutable products and a s...
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ISBN:
(纸本)9783031598340;9783031598357
Motivated by applications in e-retail and online advertising, we study the problem of assortment optimization under visibility constraints, referred to as APV. We are given a universe of substitutable products and a stream of T customers. the objective is to determine the optimal assortment of products to offer to each customer in order to maximize the total expected revenue, subject to the constraint that each product is required to be shown to a minimum number of customers. the minimum display requirement for each product is given exogenously and we refer to these constraints as visibility constraints. We assume that customer choices follow a Multinomial Logit model. We provide a characterization of the structure of the optimal assortments and present an efficient polynomial time algorithm for solving APV. To accomplish this, we introduce a novel function called the "expanded revenue" of an assortment and establish its supermodularity. Our algorithm takes advantage of this structural property. Additionally, we demonstrate that APV can be formulated as a compact linear program. Finally, we propose a novel, fair strategy for pricing the revenue loss due to the enforcement of visibility constraints.
We consider the bilevel knapsack problem with interdiction constraints, a fundamental bilevel integerprogramming problem which generalizes the 0-1 knapsack problem. In this problem, there are two knapsacks and n item...
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ISBN:
(纸本)9783031327254;9783031327261
We consider the bilevel knapsack problem with interdiction constraints, a fundamental bilevel integerprogramming problem which generalizes the 0-1 knapsack problem. In this problem, there are two knapsacks and n items. the objective is to select some items to pack into the first knapsack such that the maximum profit attainable from packing some of the remaining items into the second knapsack is minimized. We present a combinatorial branch-and-bound algorithm which outperforms the current state-of-the-art solution method in computational experiments by 4.5 times on average for all instances reported in the literature. On many of the harder instances, our algorithm is hundreds of times faster, and we solved 53 of the 72 previously unsolved instances. Our result relies fundamentally on a new dynamic programming algorithm which computes very strong lower bounds. this dynamic program solves a relaxation of the problem from bilevel to 2n-level where the items are processed in an online fashion. the relaxation is easier to solve but approximates the original problem surprisingly well in practice. We believe that this same technique may be useful for other interdiction problems.
We consider sensitivity analysis for Mixed Binary Quadratic Programs (MBQPs) with respect to changing right-hand-sides (rhs). We show that even if the optimal solution of a given MBQP is known, it is NP-hard to approx...
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ISBN:
(纸本)9783031598340;9783031598357
We consider sensitivity analysis for Mixed Binary Quadratic Programs (MBQPs) with respect to changing right-hand-sides (rhs). We show that even if the optimal solution of a given MBQP is known, it is NP-hard to approximate the change in objective function value with respect to changes in rhs. Next, we study algorithmic approaches to obtaining dual bounds for MBQP with changing rhs. We leverage Burer's completely-positive (CPP) reformulation of MBQPs. Its dual is an instance of co-positive programming (COP), and can be used to obtain sensitivity bounds. We prove that strong duality between the CPP and COP problems holds if the feasible region is bounded or if the objective function is convex, while the duality gap can be strictly positive if neither condition is met. We also show that the COP dual has multiple optimal solutions, and the choice of the dual solution affects the quality of the bounds with rhs changes. We finally provide a method for finding good nearly optimal dual solutions, and we present preliminary computational results on sensitivity analysis for MBQPs.
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