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Backdoor Branching

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

作者： Fischetti, Matteo Monaci, Michele Univ Padua DEI Via Gradenigo 6-A I-35131 Padua Italy

ISBN: (纸本)9783642208072

Which is the minimum number of variables that need branching for a given MIP instance? Can this information be effective in producing compact branching trees, hence improving the performance of a state-of-the-art solver? In this paper we present a restart exact MIP solution scheme where a set covering model is used to find a small set of variables (a "backdoor", in the terminology of [8]) to be used as first-choice variables for branching. In a preliminary "sampling" phase, our method quickly collects a number of relevant low-cost fractional solutions that qualify as obstacles for LP bound improvement. then a set covering model is solved to detect a small subset of variables (the backdoor) that "cover the fractionality" of the collected fractional solutions. these backdoor variables are put in a priority branching list, and a black-box MIP solver is eventually run-in its default mode-by taking this list into account, thus avoiding any other interference with its highly-optimized internal mechanisms. Computational results on a large set of instances from MIPLIB 2010 are presented, showing that some speedup can be achieved even with respect to a state-of-the-art solver such as IBM ILOG Cplex 12.2.

关键词： integer programming

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Continuous Facility Location on Graphs 1

引用

21st international conference on integer programming and combinatorial optimization (ipco)

作者： Hartmann, Tim A. Lendl, Stefan Woeginger, Gerhard J. Rhein Westfal TH Aachen Dept Comp Sci Aachen Germany Graz Univ Technol Dept Math Graz Austria

ISBN: (数字)9783030457716

ISBN: (纸本)9783030457716;9783030457709

We study a continuous facility location problem on undirected graphs where all edges have unit length and where the facilities may be positioned at the vertices as well as at interior points of the edges. the goal is to cover the entire graph with a minimum number of facilities with covering range (delta > 0. In other words, we want to position as few facilities as possible subject to the condition that every point on every edge is at distance at most delta from one of these facilities. We investigate this covering problem in terms of the rational parameter delta. We prove that the problem is polynomially solvable whenever delta is a unit fraction, and that the problem is NP-hard for all non unit fractions delta. We also analyze the parametrized complexity with the solution size as parameter: the resulting problem is fixed parameter tractable for all delta < 3/2, and it is W[2]-hard for all delta >= 3/2.

关键词： Location theory Graph theory Parametrized complexity

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integer programming and Incidence Treedepth 1

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20th international conference on integer programming and combinatorial optimization (ipco)

作者： Eiben, Eduard Ganian, Robert Knop, Dusan Ordyniak, Sebastian Pilipczuk, Michal Wrochna, Marcin Univ Bergen Dept Informat Bergen Norway Tech Univ Wien Algorithms & Complex Grp Vienna Austria TU Berlin Fac 4 Algorithm & Computat Complex Berlin Germany Czech Tech Univ Dept Theoret Comp Sci Fac Informat Technol Prague Czech Republic Univ Sheffield Dept Comp Sci Algorithms Grp Sheffield S Yorkshire England Univ Warsaw Inst Informat Warsaw Poland Univ Oxford Oxford England

ISBN: (数字)9783030179533

ISBN: (纸本)9783030179533;9783030179526

Recently a strong connection has been shown between the tractability of integer programming (IP) with bounded coefficients on the one side and the structure of its constraint matrix on the other side. To that end, integer linear programming is fixed-parameter tractable with respect to the primal (or dual) treedepth of the Gaifman graph of its constraint matrix and the largest coefficient (in absolute value). Motivated by this, Koutecky, Levin, and Onn [ICALP 2018] asked whether it is possible to extend these result to a more broader class of integer linear programs. More formally, is integer linear programming fixed-parameter tractable with respect to the incidence treedepth of its constraint matrix and the largest coefficient (in absolute value)? We answer this question in negative. We prove that deciding the feasibility of a system in the standard form, Ax = b, l <= x <= u, is NP-hard even when the absolute value of any coefficient in A is 1 and the incidence treedepth of A is 5. Consequently, it is not possible to decide feasibility in polynomial time even if both the assumed parameters are constant, unless P = NP.

关键词： integer programming Incidence treedepth Gaifman graph Computational complexity

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Improving the Integrality Gap for Multiway Cut 1

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20th international conference on integer programming and combinatorial optimization (ipco)

作者： Berczi, Kristof Chandrasekaran, Karthekeyan Kiraly, Tamas Madan, Vivek Eotvos Lorand Univ Dept Operat Res MTA ELTE Egervary Res Grp Budapest Hungary Univ Illinois Urbana IL 61801 USA

ISBN: (数字)9783030179533

ISBN: (纸本)9783030179533;9783030179526

In the multiway cut problem, we are given an undirected graph with non-negative edge weights and a collection of k terminal nodes, and the goal is to partition the node set of the graph into k non-empty parts each containing exactly one terminal so that the total weight of the edges crossing the partition is minimized. the multiway cut problem for k >= 3 is APX-hard. For arbitrary k, the best-known approximation factor is 1.2965 due to Sharma and Vondrak [12] while the best known inapproximability result due to Angelidakis, Makarychev and Manurangsi [1] rules out efficient algorithms to achieve an approximation factor that is less than 1.2. In this work, we improve on the lower bound to 1.20016 by constructing an integrality gap instance for the CKR relaxation. A technical challenge in improving the gap has been the lack of geometric tools to understand higher-dimensional simplices. Our instance is a non-trivial 3-dimensional instance that overcomes this technical challenge. We analyze the gap of the instance by viewing it as a convex combination of 2-dimensional instances and a uniform 3-dimensional instance. We believe that this technique could be exploited further to construct instances with larger integrality gap. One of the ingredients of our proof technique is a generalization of a result on Sperner admissible labelings due to Mirzakhani and Vondrak [11] that might be of independent combinatorial interest.

关键词： combinatorial optimization Multiway cut Integrality gap Approximation

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A Better-than-1.6-Approximation for Prize-Collecting TSP 25th

A Better-Than-1.6-Approximation for Prize-Collecting TSP

引用

25th international conference on integer programming and combinatorial optimization (ipco)

作者： Blauth, Jannis Klein, Nathan Nagele, Martin Univ Bonn Res Inst Discrete Math Bonn Germany Univ Bonn Hausdorff Ctr Math Bonn Germany Inst Adv Study Olden Lane Princeton NJ 08540 USA

ISBN: (纸本)9783031598340;9783031598357

Prize-Collecting TSP is a variant of the traveling salesperson problem where one may drop vertices from the tour at the cost of vertex-dependent penalties. the quality of a solution is then measured by adding the length of the tour and the sum of all penalties of vertices that are not visited. We present a polynomial-time approximation algorithm with an approximation guarantee slightly below 1.6, where the guarantee is with respect to the natural linear programming relaxation of the problem. this improves upon the previous best-known approximation ratio of 1.774. Our approach is based on a known decomposition for solutions of this linear relaxation into rooted trees. Our algorithm takes a tree from this decomposition and then performs a pruning step before doing parity correction on the remainder. Using a simple analysis, we bound the approximation guarantee of the proposed algorithm by (1 + root 5)/2 approximate to 1.618, the golden ratio. With some additional technical care we further improve it to 1.599.

关键词： combinatorial optimization Approxim. Algorithms TSP

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On Perturbation Spaces of Minimal Valid Functions: Inverse Semigroup theory and Equivariant Decomposition theorem 1

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20th international conference on integer programming and combinatorial optimization (ipco)

作者： Hildebrand, Robert Koppe, Matthias Zhou, Yuan Virginia Tech Grado Dept Ind & Syst Engn Blacksburg VA USA Univ Calif Davis Dept Math Davis CA 95616 USA Univ Kentucky Dept Math Lexington KY 40506 USA

ISBN: (数字)9783030179533

ISBN: (纸本)9783030179533;9783030179526

the non-extreme minimal valid functions for the Gomory-Johnson infinite group problem are those that admit effective perturbations. For a class of piecewise linear functions for the 1-row problem we give a precise description of the space of these perturbations as a direct sum of certain finite- and infinite-dimensional subspaces. the infinite-dimensional subspaces have partial symmetries;to describe them, we develop a theory of inverse semigroups of partial bijections, interacting with the functional equations satisfied by the perturbations. Our paper provides the foundation for grid-free algorithms for testing extremality and for computing liftings of non-extreme functions. the grid-freeness makes the algorithms suitable for piecewise linear functions whose breakpoints are rational numbers with huge denominators.

关键词： integer programs Cutting planes Group relaxations

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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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Online combinatorial Assignment in Independence Systems 25th

Online Combinatorial Assignment in Independence Systems

引用

25th international conference on integer programming and combinatorial optimization (ipco)

作者： Marinkovic, Javier Soto, Jose A. Verdugo, Victor Univ Chile Dept Math Engn Santiago Chile Univ Chile Ctr Math Modeling IRL CNRS 2807 Santiago Chile Pontificia Univ Catalica Chile Inst Math & Computat Engn Santiago Chile Pontificia Univ Catalica Chile Dept Ind & Syst Engn Santiago Chile

ISBN: (纸本)9783031598340;9783031598357

We consider an online multi-weighted generalization of several classic online optimization problems called the online combinatorial assignment problem. We are given an independence system over a ground set of elements and agents that arrive online one by one. Upon arrival, each agent reveals a weight function over the elements of the ground set. If the independence system is given by the matchings of a hypergraph, we recover the combinatorial auction problem, where every node represents an item to be sold, and every edge represents a bundle of items. For combinatorial auctions, Kesselheim et al. showed upper bounds of O(log log(k)/ log(k)) and O(log log(n)/ log(n)) on the competitiveness of any online algorithm, even in the random order model, where k is the maximum bundle size and n is the number of items. We provide an exponential improvement by giving upper bounds of O(log(k)/k), and O(log(n)/root n) for the prophet IID setting. Furthermore, using linear programming, we provide new and improved guarantees for the k-bounded online combinatorial auction problem (i.e., bundles of size at most k). We show a (1-e(-k))/k-competitive algorithm in the prophet IID model, a 1/(k + 1)-competitive algorithm in the prophet-secretary model using a single sample per agent, and a k(-k/(k-1))-competitive algorithm in the secretary model. Our algorithms run in polynomial time and work in more general independence systems where the offline combinatorial assignment problem admits the existence of a polynomial-time randomized algorithm that we call certificate sampler. these systems include some classes of matroids, matroid intersections, and matchoids.

关键词： Online Algorithms Linear programming combinatorial Auctions

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Lower bounds on the sizes of integer programs without additional variables

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

作者： Kaibel, Volker Weltge, Stefan Otto-von-Guericke-Universität Magdeburg Germany

ISBN: (纸本)9783319075563

For a given set X ⊆ Z d of integer points, we investigate the smallest number of facets of any polyhedron whose set of integer points is conv(X) ∩ Z d . this quantity, which we call the relaxation complexity of X, corresponds to the smallest number of linear inequalities of any integer program having X as the set of feasible solutions that does not use auxiliary variables. We show that the use of auxiliary variables is essential for constructing polynomial size integer programming formulations in many relevant cases. In particular, we provide asymptotically tight exponential lower bounds on the relaxation complexity of the integer points of several well-known combinatorial polytopes, including the traveling salesman polytope and the spanning tree polytope. © 2014 Springer international Publishing Switzerland.

关键词： integer programming

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Fixed-Charge Transportation on a Path: Linear programming Formulations

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

作者： Van Vyve, Mathieu Ctr Operat Res & Econometr Voie Roman Pays 34 B-1348 Louvain La Neuve Belgium Louvain Sch Management Voie Roman Pays 34 B-1348 Louvain La Neuve Belgium

ISBN: (纸本)9783642208072

the fixed-charge transportation problem is a fixed-charge network flow problem on a bipartite graph. this problem appears as a subproblem in many hard transportation problems, and has also strong links with the challenging big-bucket multi-item lot-sizing problem. We provide a polyhedral analysis of the polynomially solvable special case in which the associated bipartite graph is a path. We describe a new class of inequalities that we call "path-modular" inequalities. We give two distinct proofs of their validity. the first one is direct and crucially relies on sub-and super-modularity of an associated set function, thereby providing an interesting link with flow-cover type inequalities. the second proof is by projecting a tight extended formulation, therefore also showing that these inequalities suffice to describe the convex hull of the feasible solutions to this problem. We finally show how to solve the separation problem associated to the path-modular inequalities in O(n(3)) time.

关键词： mixed-integer programming lot-sizing transportation convex hull extended formulation

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