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Designing optimization Problems with Diverse Solutions 1

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24th international conference on integer programming and combinatorial optimization (ipco)

作者： Hanguir, Oussama Ma, Will Ryan, Christopher thomas Columbia Univ Ind Engn & Operat Res New York NY 10027 USA Columbia Univ Grad Sch Business New York NY 10027 USA Univ British Columbia UBC Sauder Sch Business Vancouver BC V6T 1Z2 Canada

ISBN: (数字)9783031327261

ISBN: (纸本)9783031327254;9783031327261

We consider the problem of designing a linear program that has diverse solutions as the right-hand side varies. this problem arises in video game settings where designers aim to have players use different "weapons" or "tactics" as they progress. We model this design question as a choice over the constraint matrix A and cost vector c to maximize the number of possible supports of unique optimal solutions (what we call "loadouts") of Linear Programs max{c(inverted perpendicular) x vertical bar Ax <= b, x >= 0} with nonnegative data considered over all resource vectors b. We provide an upper bound on the optimal number of loadouts and provide a family of constructions that have an asymptotically optimal number of loadouts. the upper bound is based on a connection between our problem and the study of triangulations of point sets arising from polyhedral combinatorics, and specifically the combinatorics of the cyclic polytope. Our asymptotically optimal construction also draws inspiration from the properties of the cyclic polytope.

关键词： linear programming triangulations diversity

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From Approximate to Exact integer programming 1

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24th international conference on integer programming and combinatorial optimization (ipco)

作者： Dadush, Daniel Eisenbrand, Friedrich Rothvoss, thomas CWI Amsterdam Netherlands Ecole Polytech Fed Lausanne Lausanne Switzerland Univ Washington Seattle WA 98195 USA

ISBN: (数字)9783031327261

ISBN: (纸本)9783031327254;9783031327261

Approximate integer programming is the following: For a given convex body K subset of R-n, either determine whether K boolean AND Z(n) is empty, or find an integer point in the convex body 2 center dot (K - c) + c which is K, scaled by 2 from its center of gravity c. Approximate integer programming can be solved in time 2(O(n)) while the fastest known methods for exact integer programming run in time 2(O(n)) center dot n(n). So far, there are no efficientmethods for integer programming known that are based on approximate integer programming. Our main contribution are two such methods, each yielding novel complexity results. First, we showthat an integer point x * is an element of (K boolean AND Z(n)) can be found in time 2(O(n)), provided that the remainders of each component x(i)* mod l for some arbitrarily fixed l >= 5(n + 1) of x * are given. the algorithm is based on a cutting-plane technique, iteratively halving the volume of the feasible set. the cutting planes are determined via approximate integer programming. Enumeration of the possible remainders gives a 2(O(n))n(n) algorithm for general integer programming. this matches the current best bound of an algorithm by Dadush (2012) that is considerably more involved. Our algorithm also relies on a new asymmetric approximate Caratheodory theorem that might be of interest on its own. Our second method concerns integer programming problems in standard equation form Ax = b, 0 <= x <= u, x is an element of Z(n). Such a problem can be reduced to the solution of Pi(i) O(logu(i) + 1) approximate integer programming problems. this implies, for example that knapsack or subset-sum problems with polynomial variable range 0 <= x(i) <= p(n) can be solved in time (logn)(O(n)). For these problems, the best running time so far was n(n) center dot 2(O(n)).

关键词： integer programming

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Optimizing Low Dimensional Functions over the integers 1

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24th international conference on integer programming and combinatorial optimization (ipco)

作者： Dadush, Daniel Leonard, Arthur Rohwedder, Lars Verschae, Jose CWI Amsterdam Netherlands ENS Paris France Maastricht Univ Maastricht Netherlands Pontificia Univ Catolica Chile Santiago Chile

ISBN: (数字)9783031327261

ISBN: (纸本)9783031327254;9783031327261

We consider box-constrained integer programs with objective g(Wx)+ c(inverted perpendicular) x, where g is a "complicated" function with an m dimensional domain. Here we assume we have n >> m variables and that W is an element of Z(mxn) is an integer matrix with coefficients of absolute value at most Delta We design an algorithm for this problem using only the mild assumption that the objective can be optimized efficiently when all but m variables are fixed, yielding a running time of n(m) (m Delta)(O)(m(2)). Moreover, we can avoid the term n(m) in several special cases, in particular when c = 0. Our approach can be applied in a variety of settings, generalizing several recent results. An important application are convex objectives of low domain dimension, where we imply a recent result by Hunkenschroder et al. [SIOPT'22] for the 0-1-hypercube and sharp or separable convex g, assuming W is given explicitly. By avoiding the direct use of proximity results, which only holds when g is separable or sharp, we match their running time and generalize it for arbitrary convex functions. In the case where the objective is only accessible by an oracle and W is unknown, we further show that their proximity framework can be implemented in n(m(Delta))(O)(m(2))- time instead of n(m Delta)(O)(m(3)). Lastly, we extend the result by Eisenbrand andWeismantel [SODA'17, TALG'20] for integer programs with few constraints to a mixed-integer linear program setting where integer variables appear in only a small number of different constraints.

关键词： integer programming

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LEARNING TO GENERATE COLUMNS WIth APPLICATION TO VERTEX COLORING 11

LEARNING TO GENERATE COLUMNS WITH APPLICATION TO VERTEX COLO...

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11th international conference on Learning Representations, ICLR 2023

作者： Sun, Yuan Ernst, Andreas T. Li, Xiaodong Weiner, Jake La Trobe University Australia Monash University Australia RMIT University Australia

We present a new column generation approach based on Machine Learning (ML) for solving combinatorial optimization *** aim of our method is to generate high-quality columns that belong to an optimal integer solution, in contrast to the traditional approach that aims at solving linear programming *** achieve this aim, we design novel features to characterize a column, and develop an effective ML model to predict whether a column belongs to an optimal integer *** then use the ML model as a filter to select high-quality columns generated from a sampling method and use the selected columns to construct an integer *** method is computationally fast compared to the traditional methods that generate columns by repeatedly solving a pricing *** demonstrate the efficacy of our method on the vertex coloring problem, by empirically showing that the columns selected by our ML model are significantly better, in terms of the integer solution that can be constructed from them, than those selected randomly or based only on their reduced ***, we show that the columns generated by our method can be used as a warm start to boost the performance of a column generation-based heuristic. © 2023 11th international conference on Learning Representations, ICLR 2023. All rights reserved.

关键词： integer programming

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Inapproximability of Shortest Paths on Perfect Matching Polytopes 1

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24th international conference on integer programming and combinatorial optimization (ipco)

作者： Cardinal, Jean Steiner, Raphael Univ Libre Bruxelles ULB Brussels Belgium Swiss Fed Inst Technol Zurich Switzerland

ISBN: (数字)9783031327261

ISBN: (纸本)9783031327254;9783031327261

We consider the computational problem of finding short paths in the skeleton of the perfect matching polytope of a bipartite graph. We prove that unless P = NP, there is no polynomial-time algorithm that computes a path of constant length between two vertices at distance two of the perfect matching polytope of a bipartite graph. Conditioned on P not equal NP, this disproves a conjecture by Ito, Kakimura, Kamiyama, Kobayashi and Okamoto [SIAM Journal on Discrete Mathematics, 36(2), pp. 1102-1123 (2022)]. Assuming the Exponential Time Hypothesis we prove the stronger result that there exists no polynomialtime algorithm computing a path of length at most (1/4 - o(1)) log N/log logN between two vertices at distance two of the perfect matching polytope of an N-vertex bipartite graph. these results remain true if the bipartite graph is restricted to be of maximum degree three. the above has the following interesting implication for the performance of pivot rules for the simplex algorithm on simply-structured combinatorial polytopes: If P not equal NP, then for every simplex pivot rule executable in polynomial time and every constant k is an element of N there exists a linear program on a perfect matching polytope and a starting vertex of the polytope such that the optimal solution can be reached using only two monotone non-degenerate steps from the starting vertex, yet the pivot rule will require at least k non-degenerate steps to reach the optimal solution. this result remains true in the more general setting of pivot rules for so-called circuit-augmentation algorithms.

关键词： Perfect matching polytopes Simplex method Pivot rules Circuit augmentations combinatorial reconfiguration

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Constant-Competitiveness for Random Assignment Matroid Secretary Without Knowing the Matroid 24th

Constant-Competitiveness for Random Assignment Matroid Secre...

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24th international conference on integer programming and combinatorial optimization (ipco)

作者： Santiago, Richard Sergeev, Ivan Zenklusen, Rico Swiss Fed Inst Technol Dept Mathemat Zurich Switzerland

ISBN: (纸本)9783031327254;9783031327261

the Matroid Secretary Conjecture is a notorious open problem in online optimization. It claims the existence of an O(1)-competitive algorithm for the Matroid Secretary Problem (MSP). Here, the elements of a weighted matroid appear one-by-one, revealing their weight at appearance, and the task is to select elements online with the goal to get an independent set of largest possible weight. O(1)-competitive MSP algorithms have so far only been obtained for restricted matroid classes and for MSP variations, including Random-Assignment MSP (RA-MSP), where an adversary fixes a number of weights equal to the ground set size of the matroid, which then get assigned randomly to the elements of the ground set. Unfortunately, these approaches heavily rely on knowing the full matroid upfront. this is an arguably undesirable requirement, and there are good reasons to believe that an approach towards resolving the MSP Conjecture should not rely on it. thus, both Soto [SIAM Journal on Computing 2013] and Oveis Gharan & Vondrak [Algorithmica 2013] raised as an open question whether RA-MSP admits an O(1)-competitive algorithm even without knowing the matroid upfront. In this work, we answer this question affirmatively. Our result makes RA-MSP the first well-known MSP variant with an O(1)-competitive algorithm that does not need to know the underlying matroid upfront and without any restriction on the underlying matroid. Our approach is based on first approximately learning the rank-density curve of the matroid, which we then exploit algorithmically.

关键词： combinatorial mathematics

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19th international conference on integer programming and combinatorial optimization, ipco 2017

19th International Conference on Integer Programming and Com...

引用

19th international conference on integer programming and combinatorial optimization, ipco 2017

ISBN: (纸本)9783319592497

the proceedings contain 36 papers. the special focus in this conference is on integer programming and combinatorial optimization. the topics include: On scheduling coflows;integrality gaps of integer knapsack problems;approximation of corner polyhedra with families of intersection cuts;the structure of the infinite models in integer programming;mixed-integer linear representability, disjunctions, and variable elimination;cutting planes from wide split disjunctions;the heterogeneous capacitated k-center problem;local guarantees in graph cuts and clustering;verifying integer programming results;long term behavior of dynamic equilibria in fluid queuing networks;approximation algorithm for the maximum traveling salesman problem;minimizing multimodular functions and allocating capacity in bike-sharing systems;stochastic online scheduling on unrelated machines;beating half for random arrival;an improved deterministic rescaling for linear programming algorithms;a quasi-polynomial approximation for the restricted assignment problem;adaptive submodular ranking;minimum birkhoff-von Neumann decomposition;maximum matching in the online batch-arrival model;budget feasible mechanisms on matroids;deterministic discrepancy minimization via the multiplicative weight update method;mixed-integer convex representability;high degree sum of squares proofs, bienstock-zuckerberg hierarchy and cg cuts;enumeration of integer points in projections of unbounded polyhedra and equilibrium computation in atomic splittable singleton congestion games.

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A GNN-GUIDED PREDICT-AND-SEARCH FRAMEWORK FOR MIXED-integer LINEAR programming 11

A GNN-GUIDED PREDICT-AND-SEARCH FRAMEWORK FOR MIXED-INTEGER ...

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11th international conference on Learning Representations, ICLR 2023

作者： Han, Qingyu Yang, Linxin Chen, Qian Zhou, Xiang Zhang, Dong Wang, Akang Sun, Ruoyu Luo, Xiaodong Shenzhen Research Institute of Big Data China Shandong University China School of Data Science The Chinese University of Hong Kong Shenzhen China School of Science and Engineering The Chinese University of Hong Kong Shenzhen China Huawei China Shenzhen International Center for Industrial and Applied Mathematics Shenzhen Research Institute of Big Data China

Mixed-integer linear programming (MILP) is widely employed for modeling combinatorial optimization problems. In practice, similar MILP instances with only coefficient variations are routinely solved, and machine learning (ML) algorithms are capable of capturing common patterns across these MILP instances. In this work, we combine ML with optimization and propose a novel predict-and-search framework for efficiently identifying high-quality feasible solutions. Specifically, we first utilize graph neural networks to predict the marginal probability of each variable, and then search for the best feasible solution within a properly defined ball around the predicted solution. We conduct extensive experiments on public datasets, and computational results demonstrate that our proposed framework achieves 51.1% and 9.9% performance improvements to MILP solvers SCIP and Gurobi on primal gaps, respectively. © 2023 11th international conference on Learning Representations, ICLR 2023. All rights reserved.

关键词： Machine learning

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Monoidal Strengthening of Simple V -Polyhedral Disjunctive Cuts 24th

Monoidal Strengthening of Simple V -Polyhedral Disjunctive ...

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24th international conference on integer programming and combinatorial optimization, ipco 2023

作者： Kazachkov, Aleksandr M. Balas, Egon University of Florida GainesvilleFL United States Carnegie Mellon University PittsburghPA United States

ISBN: (纸本)9783031327254

Disjunctive cutting planes can tighten a relaxation of a mixed-integer linear program. Traditionally, such cuts are obtained by solving a higher-dimensional linear program, whose additional variables cause the procedure to be computationally prohibitive. Adopting a V -polyhedral perspective is a practical alternative that enables the separation of disjunctive cuts via a linear program with only as many variables as the original problem. the drawback is that the classical approach of monoidal strengthening cannot be directly employed without the values of the extra variables appearing in the extended formulation. We derive how to compute these values from a solution to the linear program generating V -polyhedral disjunctive cuts. We then present computational experiments with monoidal strengthening of cuts from disjunctions with as many as 64 terms. Some instances are dramatically impacted, with strengthening increasing the gap closed by the cuts from 0 to 100%. However, for larger disjunctions, monoidal strengthening appears to be less effective, for which we identify a potential cause. © 2023, the Author(s), under exclusive license to Springer Nature Switzerland AG.

关键词： integer programming

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A PTAS for the Horizontal Rectangle Stabbing Problem 23rd

A PTAS for the Horizontal Rectangle Stabbing Problem

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23rd international conference on integer programming and combinatorial optimization (ipco)

作者： Khan, Arindam Subramanian, Aditya Wiese, Andreas Indian Inst Sci Bengaluru India Vrije Univ Amsterdam Amsterdam Netherlands

ISBN: (纸本)9783031069017;9783031069000

We study rectangle stabbing problems in which we are given n axis-aligned rectangles in the plane that we want to stab, i.e., we want to select line segments such that for each given rectangle there is a line segment that intersects two opposite edges of it. In the horizontal rectangle stabbing problem (Stabbing), the goal is to find a set of horizontal line segments of minimum total length such that all rectangles are stabbed. In general rectangle stabbing problem, also known as horizontal-vertical stabbing problem (HV-Stabbing), the goal is to find a set of rectilinear (i.e., either vertical or horizontal) line segments of minimum total length such that all rectangles are stabbed. Both variants are NP-hard. Chan, van Dijk, Fleszar, Spoerhase, and Wolff [5] initiated the study of these problems by providing O(1)-approximation algorithms. Recently, Eisenbrand, Gallato, Svensson, and Venzin [11] have presented a QPTAS and a polynomial-time 8-approximation algorithm for Stabbing but it is open whether the problem admits a PTAS. In this paper, we obtain a PTAS for Stabbing, settling this question. For HV-Stabbing, we obtain a (2 + epsilon)-approximation. We also obtain PTASes for special cases of HV-Stabbing: (i) when all rectangles are squares, (ii) when each rectangle's width is at most its height, and (iii) when all rectangles are delta-large, i.e., have at least one edge whose length is at least delta, while all edge lengths are at most 1. Our result also implies improved approximations for other problems such as generalized minimum Manhattan network.

关键词： Geometric optimization Approximation algorithms Line stabbing Rectangles

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