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Approximation bounds for a general class of precedence constrained parallel machine scheduling problems

SIAM JOURNAL ON COMPUTING 2006年第5期35卷 1241-1253页

作者： Queyranne, M Schulz, AS Univ British Columbia Sauder Sch Business Vancouver BC V6T 1Z2 Canada MIT Alfred P Sloan Sch Management Cambridge MA 02139 USA

An important class of scheduling problems concerns parallel machines and precedence constraints. We consider precedence delays, which associate with each precedence constraint a certain amount of time that must elapse between the completion and start times of the corresponding jobs. Together with ordinary precedence constraints, release dates and delivery times can be modeled in this manner. We present a 4-approximation algorithm for the total weighted completion time objective for this general class of problems. The algorithm is a rather simple form of list scheduling. The list is in order of job midpoints derived from a linear programming relaxation. Our analysis unifies and simplifies that of a number of special cases heretofore separately studied, while actually improving many of the former approximation results.

关键词： approximation algorithm linear programming relaxation performance guarantee precedence constraints scheduling

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Solving pooling problems with time discretization by LP and SOCP relaxations and rescheduling methods

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JOURNAL OF GLOBAL OPTIMIZATION 2019年第3期75卷 631-654页

作者： Kimizuka, Masaki Kim, Sunyoung Yamashita, Makoto Tokyo Inst Technol Dept Math & Comp Sci Meguro Ku 2-12-1 Oh Okayama Tokyo 1528552 Japan Ewha W Univ Dept Math 52 Ewhayeodae Gil Seoul 03760 South Korea

The pooling problem is an important industrial problem in the class of network flow problems for allocating gas flow in pipeline transportation networks. For the pooling problem with time discretization, we propose second order cone programming (SOCP) and linear programming (LP) relaxations and prove that they obtain the same optimal value as the semidefinite programming relaxation. Moreover, a rescheduling method is proposed to efficiently refine the solution obtained by the SOCP or LP relaxation. The efficiency of the SOCP and the LP relaxation and the proposed rescheduling method is illustrated with numerical results on the test instances from the work of Nishi in 2010, some large instances and Foulds 3, 4 and 5 test problems.

关键词： Pooling problem Semidefinite relaxation Second order cone relaxation linear programming relaxation Rescheduling method Computational efficiency

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Optimality Bounds for a Variational relaxation of the Image Partitioning Problem

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JOURNAL OF MATHEMATICAL IMAGING AND VISION 2013年第3期47卷 239-257页

作者： Lellmann, Jan Lenzen, Frank Schnoerr, Christoph Univ Cambridge Ctr Math Sci Dept Appl Math & Theoret Phys Cambridge CB3 0WA England Heidelberg Univ Image & Pattern Anal Grp Heidelberg Germany Heidelberg Univ HCI Dept Math & Comp Sci Heidelberg Germany

We consider a variational convex relaxation of a class of optimal partitioning and multiclass labeling problems, which has recently proven quite successful and can be seen as a continuous analogue of linear programming (LP) relaxation methods for finite-dimensional problems. While for the latter several optimality bounds are known, to our knowledge no such bounds exist in the infinite-dimensional setting. We provide such a bound by analyzing a probabilistic rounding method, showing that it is possible to obtain an integral solution of the original partitioning problem from a solution of the relaxed problem with an a priori upper bound on the objective. The approach has a natural interpretation as an approximate, multiclass variant of the celebrated coarea formula.

关键词： Convex relaxation Multiclass labeling Approximation bound Combinatorial optimization Total variation linear programming relaxation

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The Gap Function: Evaluating Integer programming Models over Multiple Right-Hand Sides

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OPERATIONS RESEARCH 2022年第2期70卷 1259-1270页

作者： Ajayi, Temitayo Thomas, Christopher Schaefer, Andrew J. Nat Source Improved Plants Ithaca NY 14850 USA Bazean Corp Houston TX 77002 USA Rice Univ Dept Computat & Appl Math Houston TX 77005 USA

For an integer programming model with fixed data, the linear programming relaxation gap is considered one of the most important measures of model quality. There is no consensus, however, on appropriate measures of model quality that account for data variation. In particular, when the right-hand side is not known exactly, one must assess a model based on its behavior over many right-hand sides. Gap functions are the linear programming relaxation gaps parametrized by the right-hand side. Despite drawing research interest in the early days of integer programming, the properties and applications of these functions have been little studied. In this paper, we construct measures of integer programming model quality over sets of right-hand sides based on the absolute and relative gap functions. In particular, we formulate optimization problems to compute the expectation and extrema of gap functions over finite discrete sets and bounded hyperrectangles. These optimization problems are linear programs (albeit of an exponentially large size) that contain at most one special ordered-set constraint. These measures for integer programming models, along with their associated formulations, provide a framework for determining a model's quality over a range of right-hand sides.

关键词： gap function linear programming relaxation modeling superadditive duality

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A new methodology for the open-pit mine production scheduling problem

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OMEGA-INTERNATIONAL JOURNAL OF MANAGEMENT SCIENCE 2018年 81卷 169-182页

作者： Samavati, Mehran Essam, Daryl Nehring, Micah Sarker, Ruhul Univ Sydney Australian Ctr Field Robot Sydney NSW Australia Univ New South Wales Sch Engn & Informat Technol Canberra ACT Australia Univ Queensland Sch Mech & Min Engn Brisbane Qld Australia

The open pit mine production scheduling problem (OPMPSP) consists of scheduling the extraction of a mineral deposit that is broken into a number of smaller segments, or blocks, such that the net present value (NPV) of the operation is maximised. This problem has been formulated as an integer programming (IP) model, involving both knapsack and precedence constraints. However, due to the large number of blocks and precedence constraints, this model has remained impractical in real planning applications. In this paper, we propose a new method to quickly generate near optimum feasible (integer) solutions by using the fractional solutions from the linear programming (LP) relaxation of the IP model. To be applicable to real sized problems, a new heuristic that quickly computes a feasible LP solution is also proposed. Our methodology is tested on a set of both academically designed and real-world mine deposits, and shows better performance than the heuristic used to tackle the same deposits in the literature. Interestingly, the proposed methodology improves the best known solutions for the majority of the instances. (C) 2017 Elsevier Ltd. All rights reserved.

关键词： Open pit mining Precedence constrained knapsack problem Large size scheduling problems linear programming relaxation

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NP-Completeness and an Approximation Algorithm for Rectangle Escape Problem With Application to PCB Routing

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IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN OF INTEGRATED CIRCUITS AND SYSTEMS 2012年第9期31卷 1356-1365页

作者： Ma, Qiang Wong, Martin D. F. Univ Illinois Dept Elect & Comp Engn Urbana IL 61801 USA Synopsys Inc Mountain View CA 94043 USA

In this paper, we introduce and study the rectangle escape problem (REP), which is motivated by printed circuit board (PCB) bus escape routing. Given a rectangular region R and a set S of rectangles within R, the REP is to choose a direction for each rectangle to escape to the boundary of R, such that the resultant maximum density over R is minimized. We prove that the REP is NP-complete, and show that it can be formulated as an integer linear programming (ILP). A provably good approximation algorithm for the REP is developed by applying linear programming (LP) relaxation and a special rounding technique to the ILP. In addition, an iterative refinement procedure is proposed as a postprocessing step to further improve the results. Our approximation algorithm is also shown to work for more general versions of REP: weighted REP and simultaneous REP. Our approach is tested on a set of industrial PCB bus escape routing problems. Experimental results show that the optimal solution can be obtained within several seconds for each of the test cases.

关键词： Approximation algorithm linear programming relaxation NP-completeness printed circuit board (PCB) routing

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Shannon Meets von Neumann: A Minimax Theorem for Channel Coding in the Presence of a Jammer

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IEEE TRANSACTIONS ON INFORMATION THEORY 2020年第5期66卷 2842-2859页

作者： Jose, Sharu Theresa Kulkarni, Ankur A. Indian Inst Technol Syst & Control Engn Grp Mumbai 400076 Maharashtra India Kings Coll London Dept Engn London WC2R 2LS England

We study the setting of channel coding over a family of channels whose state is controlled by an adversarial jammer by viewing it as a zero-sum game between a finite blocklength encoder-decoder team, and the jammer. The encoder-decoder team choose stochastic encoding and decoding strategies to minimize the average probability of error in transmission, while the jammer chooses a distribution on the state-space to maximize this probability. The min-max value of the game is equivalent to channel coding for a compound channel - we call this the Shannon solution of the problem. The max-min value corresponds to finding a mixed channel with the largest value of the minimum achievable probability of error. When the min-max and max-min values are equal, the problem is said to admit a saddle-point or von Neumann solution. While a Shannon solution always exists, the communicating team's problem is nonconvex for finite blocklengths, whereby a von Neumann solution may not exist. Despite this, we show that the min-max and max-min values become equal asymptotically in the large blocklength limit, for all but finitely many rates. We explicitly characterize this limiting value as a function of the rate and obtain tight finite blocklength bounds on the min-max and max-min value. As a corollary we get an explicit expression for the $\epsilon $ -capacity of a compound channel under stochastic codes - the first such result, to the best of our knowledge. Our results demonstrate a deeper relation between the compound channel and mixed channel than was previously known. They also show that the conventional information-theoretic viewpoint, articulated via the Shannon solution, coincides asymptotically with the game-theoretic one articulated via the von Neumann solution. Key to our results is the derivation of new finite blocklength upper bounds on the min-max value of the game via a novel achievability scheme, and lower bounds on the max-min value obtained via the linear programming relaxat

关键词： Channel coding compound channels and mixed channels saddle-point solution finite blocklength bounds linear programming relaxation

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Single-machine scheduling with precedence constraints

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MATHEMATICS OF OPERATIONS RESEARCH 2005年第4期30卷 1005-1021页

作者： Correa, JR Schulz, AS Univ Adolfo Ibanez Sch Business Santiago Chile MIT Alfred P Sloan Sch Management Cambridge MA 02139 USA

We discuss the problem of sequencing precedence-constrained jobs on a single machine to minimize the average weighted completion time. This problem has attracted much attention in the mathematical programming community since Sidney's pioneering work in 1975 (Sidney, J. B. 1975. Decomposition algorithms for single machine scheduling with precedence relations and deferral costs. Operations Research 23 283-298). We look at the problem from a polyhedral perspective and uncover a relation between Sidney's decomposition theorem and different linear programming relaxations. More specifically, we present a generalization of Sidney's result, which particularly allows us to reason that virtually all known 2-approximation algorithms are consistent with his decomposition. Moreover, we establish a connection between the single-machine scheduling problem and the vertex cover problem. Indeed, in the special case of series-parallel precedence constraints, we prove that the sequencing problem can be seen 'as a special case of the vertex cover problem. We also. argue that this result is true for general precedence constraints if one can show that a certain integer program represents a valid formulation of the sequencing problem. Finally, we give a 3/2-approximation algorithm for two-dimensional partial orders, and we also provide a characterization of the active inequalities of a linear programming relaxation in completion time variables.

关键词： linear ordering scheduling vertex cover linear programming relaxation integer programming formulation approximation algorithm partial order

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Improved Moves for Truncated Convex Models

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JOURNAL OF MACHINE LEARNING RESEARCH 2011年第1期12卷 31-67页

作者： Kumar, M. Pawan Veksler, Olga Torr, Philip H. S. Stanford Univ Dept Comp Sci Stanford CA 94305 USA Univ Western Ontario Dept Comp Sci London ON N6A 5B7 Canada Oxford Brookes Univ Dept Comp Oxford OX33 1HX England

We consider the problem of obtaining an approximate maximum a posteriori estimate of a discrete random field characterized by pairwise potentials that form a truncated convex model. For this problem, we propose two st-MINCUT based move making algorithms that we call Range Swap and Range Expansion. Our algorithms can be thought of as extensions of alpha beta-Swap and alpha-Expansion respectively that fully exploit the form of the pairwise potentials. Specifically, instead of dealing with one or two labels at each iteration, our methods explore a large search space by considering a range of labels (that is, an interval of consecutive labels). Furthermore, we show that Range Expansion provides the same multiplicative bounds as the standard linear programming (LP) relaxation in polynomial time. Compared to previous approaches based on the LP relaxation, for example interior-point algorithms or tree-reweighted message passing (TRW), our methods are faster as they use only the efficient st-MINCUT algorithm in their design. We demonstrate the usefulness of the proposed approaches on both synthetic and standard real data problems.

关键词： truncated convex models move making algorithms range moves multiplicative bounds linear programming relaxation

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Bicriteria Multiresource Generalized Assignment Problem

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NAVAL RESEARCH LOGISTICS 2014年第8期61卷 621-636页

作者： Karsu, Ozlem Azizoglu, Meral Bilkent Univ Dept Ind Engn TR-06800 Ankara Turkey Middle E Tech Univ Dept Ind Engn TR-06531 Ankara Turkey

In this study, we consider a bicriteria multiresource generalized assignment problem. Our criteria are the total assignment load and maximum assignment load over all agents. We aim to generate all nondominated objective vectors and the corresponding efficient solutions. We propose several lower and upper bounds and use them in our optimization and heuristic algorithms. The computational results have shown the satisfactory behaviors of our approaches. (c) 2014 Wiley Periodicals, Inc. Naval Research Logistics, 61: 621-636, 2014

关键词： generalized assignment problem linear programming relaxation branch and bound heuristics

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