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A unified approach for scheduling with convex resource consumption functions using positional penalties

EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 2010年第2期206卷 301-312页

作者： Leyvand, Yaron Shabtay, Dvir Steiner, George McMaster Univ DeGroote Sch Business Operat Management Area Hamilton ON Canada Ben Gurion Univ Negev Dept Ind Engn & Management IL-84105 Beer Sheva Israel

We provide a unified model for solving single machine scheduling problems with controllable processing times in polynomial time using positional penalties. We show how this unified model can be useful in solving three different groups of scheduling problems. The first group includes four different due date assignment problems to minimize an objective function which includes costs for earliness, tardiness, due date assignment, makespan and total resource consumption. The second group includes three different due date assignment problems to minimize an objective function which includes the weighted number of tardy jobs, due date assignment costs, makespan and total resource consumption costs. The third group includes various scheduling problems which do not involve due date assignment decisions. We show that each of the problems from the first and the third groups can be reduced to a special case of our unified model and thus can be solved in O(n(3)) time. Furthermore, we show how the unified model can housed repeatedly as a subroutine to solve all problems from the second group in O(n(4)) time. In addition, we also show that faster algorithms exist for several special cases. Crown Copyright (C) 2010 Published by Elsevier B.V. All rights reserved.

关键词： Single machine scheduling Controllable processing times Resource allocation Due date assignment Positional penalties polynomial-time algorithm

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A quadratic time algorithm for computing the optimal landing times of a fixed sequence of planes

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 2018年第3期270卷 1148-1157页

作者： Faye, Alain ENSILE Lab CEDRIC 1 Sq Resistance F-91025 Evry France

This paper considers the Aircraft Landing Problem. The aim is to schedule arriving aircraft at the airport under the condition of safe landing. Landing times lie within predefined time windows and safety separation constraints between two successive aircraft landing on the same runway, must be satisfied. The objective is to minimize the total cost of landing deviation from predefined target times within the time windows. On each runway, for a fixed landing sequence, the problem can be solved by a linear program. When safety separation times satisfy the Triangle Inequality, we propose an O(n(2)) algorithm for solving the problem where n is the number of planes of the sequence. We compare the performance of this polynomial time algorithm to the linear programming method. The two methods are embedded in an iterative algorithm, based on simulated annealing, for solving the single runway case. Computational tests, performed on publicly available problems, show that the heuristic based on our algorithm is much faster than the one using linear programming. This gain in CPU time allows to obtain better solutions in a fixed amount of time. (C) 2018 Elsevier B.V. All rights reserved.

关键词： OR in airlines Aircraft Landing Problem Landing times Dynamic programming polynomial-time algorithm

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On decomposability of Multilinear sets

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MATHEMATICAL PROGRAMMING 2018年第2期170卷 387-415页

作者： Del Pia, Alberto Khajavirad, Aida Univ Wisconsin Dept Ind & Syst Engn Madison WI 53706 USA Univ Wisconsin Wisconsin Inst Discovery Madison WI 53706 USA Carnegie Mellon Univ Dept Chem Engn Pittsburgh PA 15213 USA

We consider the Multilinear set defined as the set of binary points (x, y) satisfying a collection of multilinear equations of the form , , where denotes a family of subsets of of cardinality at least two. Such sets appear in factorable reformulations of many types of nonconvex optimization problems, including binary polynomial optimization. A great simplification in studying the facial structure of the convex hull of the Multilinear set is possible when is decomposable into simpler Multilinear sets ,;namely, the convex hull of can be obtained by convexifying each , separately. In this paper, we study the decomposability properties of Multilinear sets. Utilizing an equivalent hypergraph representation for Multilinear sets, we derive necessary and sufficient conditions under which is decomposable into , , based on the structure of pair-wise intersection hypergraphs. Our characterizations unify and extend the existing decomposability results for the Boolean quadric polytope. Finally, we propose a polynomial-time algorithm to optimally decompose a Multilinear set into simpler subsets. Our proposed algorithm can be easily incorporated in branch-and-cut based global solvers as a preprocessing step for cut generation.

关键词： Multilinear functions Convex hull Decomposition Zero-one polynomial optimization Factorable relaxations polynomial-time algorithm

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Combinatorics and algorithms for Quasi-Chain Graphs

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algorithmICA 2023年第3期85卷 642-664页

作者： Alecu, Bogdan Atminas, Aistis Lozin, Vadim Malyshev, Dmitriy Univ Leeds Sch Comp Leeds W Yorkshire England Xian Jiaotong Liverpool Univ Dept Math Sci 111 Renai Rd Suzhou 215123 Peoples R China Univ Warwick Math Inst Coventry CV4 7AL W Midlands England Natl Res Univ Lab Algorithms & Technol Networks Anal Higher Sch Econ 136 Rodionova Str Nizhnii Novgorod 603093 Russia

The class of quasi-chain graphs is an extension of the well-studied class of chain graphs. This latter class enjoys many nice and important properties, such as bounded clique-width, implicit representation, well-quasi-ordering by induced subgraphs, etc. The class of quasi-chain graphs is substantially more complex. In particular, this class is not well-quasi-ordered by induced subgraphs, and the clique-width is not bounded in it. In the present paper, we show that the universe of quasi-chain graphs is at least as complex as the universe of permutations by establishing a bijection between the class of all permutations and a subclass of quasi-chain graphs. This implies, in particular, that the induced subgraph isomorphism problem is NP-complete for quasi-chain graphs. On the other hand, we propose a decomposition theorem for quasi-chain graphs that implies an implicit representation for graphs in this class and efficient solutions for some algorithmic problems that are generally intractable.

关键词： Bipartite graphs Implicit representation polynomial-time algorithm

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Independent sets in extensions of 2K₂-free graphs

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DISCRETE APPLIED MATHEMATICS 2005年第1期146卷 74-80页

作者： Lozin, VV Mosca, R Rutgers State Univ RUTCOR Piscataway NJ 08854 USA Univ G dAnnunzio Dipartimento Sci I-65127 Pescara Italy

The class of 2K(2)-free graphs includes several interesting subclasses such as split, pseudo-split, threshold graphs, complements to chordal, interval or trivially perfect graphs. The fundamental property of 2K(2)-free graphs is that they contain polynomially many maximal independent sets. As a consequence, several important problems that are NP-hard in general graphs, such as 3-colorability, maximum weight independent set (WIS), minimum weight independent dominating set (WID), become polynomial-time solvable when restricted to the class of 2K(2)-free graphs. In the present paper, we extend 2K(2)-free graphs to larger classes with polynomial-time solvable WIS or WID. In particular, we show that WIS can be solved in polynomial time for (K-2 + K-1,K-3)-free graphs and WID for (K-2 + K-1.2)-free graphs. The latter result is in contrast with the fact that independent domination is NP-hard in the class of 2K(1.2)-free graphs, which has been recently proven by Zverovich. (C) 2004 Elsevier B.V. All rights reserved.

关键词： maximal stable set polynomial-time algorithm

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Computing lower bounds on functional units before scheduling

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IEEE TRANSACTIONS ON VERY LARGE SCALE INTEGRATION (VLSI) SYSTEMS 1996年第2期4卷 273-279页

作者： Chaudhuri, S Walker, RA RENSSELAER POLYTECH INST DEPT ELECT COMP & SYST ENGNTROYNY 12180 RENSSELAER POLYTECH INST DEPT COMP SCITROYNY 12180

This brief presents a new polynomial-time algorithm for computing lower bounds on the number of functional units (FU's) of each type required to schedule a data Bow graph in a specified number of control steps. A formal approach is presented that is guaranteed to find the tightest possible bounds that can be found by relaxing either the precedence constraints or integrality constraints on the scheduling problem, This tight, yet fairly efficient, bounding method can be used to estimate FU area, to generate resource constraints for reducing the search space, or in conjunction with exact techniques for efficient optimal design space exploration.

关键词： VLSI circuit CAD data flow graphs formal specification high level synthesis integrated circuit design scheduling VLSI synthesis data flow graph scheduling functional units lower bounds optimal design space exploration polynomial-time algorithm resource c Lower bound FUNCTIONAL UNITS circuit CAD integrated circuit design Very large scale integration Formal specifications data flow graphs high level synthesis

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Flow shop scheduling problems under machine-dependent precedence constraints

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JOURNAL OF COMBINATORIAL OPTIMIZATION 2004年第1期8卷 13-28页

作者： Gladky, AA Shafransky, YM Strusevich, VA Natl Acad Sci Belarus Inst Engn Cybernet Minsk BELARUS Univ Greenwich Sch Comp & Math Sci London SE18 6PF England

The paper considers the flow shop scheduling problems to minimize the makespan, provided that an individual precedence relation is specified on each machine. A fairly complete complexity classification of problems wit... 详细信息

关键词： flow shop precedence constraints complexity polynomial-time algorithm

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Efficient domination for classes of P₆-free graphs

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DISCRETE APPLIED MATHEMATICS 2017年 223卷 15-27页

作者： Brandstaedt, Andreas Eschen, Elaine M. Friese, Erik Karthick, T. Univ Rostock Fachbereich Informat A Einstein Str 22 D-18051 Rostock Germany West Virginia Univ Lane Dept CSEE Morgantown WV 26506 USA Univ Rostock Fachbereich Math Ulmenstr 69 D-18057 Rostock Germany Indian Stat Inst Comp Sci Unit Chennai Ctr Madras 600113 Tamil Nadu India

In a finite undirected graph G, a vertex dominates itself and all its neighbors in G. A vertex set D is an efficient dominating set (e.d. for short) of G if every vertex of G is dominated by exactly one vertex of D. The Efficient Domination (ED) problem, which asks for the existence of an e.d. in G, is known to be NP-complete even for very restricted graph classes such as P-7-free chordal graphs;its complexity was an open question for P-6-free graphs and was open even for the subclass of P-6-free chordal graphs. Recently, Lokshtanov et al., and independently, Brandstadt and Mosca showed that ED is solvable in polynomial time for P-6-free graphs. The ED problem on a graph G can be reduced to the Maximum Weight Independent Set problem on the square of G. In this paper, we show that squares of P-6-free chordal graphs that have an e.d. are chordal;this, even holds for the larger class of (P-6, house, hole, domino)-free graphs. Thus, ED/WeightedED is solvable in polynomial time for (P-6, house, hole, domino)-free graphs, and in particular, for P-6-free chordal graphs. Also, the time bound achieved for ED on this class is much better than in the P-6-free case. Moreover, we show that squares of P-6-free graphs that have an e.d. are hole-free. Based on this result, we show that ED is solvable in polynomial time for (P-6, net)-free graphs;again, the time bound for ED on this class is much better than in the P-6-free case. (C) 2017 Elsevier B.V. All rights reserved.

关键词： Efficient domination P-6-free graphs Chordal graphs Hole-free graphs (House, hole, domino)-free graphs polynomial-time algorithm

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Fast solution of single-machine scheduling problem with embedded jobs

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THEORETICAL COMPUTER SCIENCE 2019年第0期782卷 91-106页

作者： Vakhania, Nodari UAEMor Ctr Invest Ciencias Cuernavaca Morelos Mexico

We present a fast polynomial-time algorithm for single-machine scheduling problem with release times (r(j)), processing times (p(j)) and due dates (d(j)) with the objective to minimize the maximum job lateness. The general setting is strongly NP-hard. We expose a particular embedded structure that typically possess a subset of jobs in an instance of the problem. We establish useful properties of the problem instances that tolerate this structure, i.e., in such an instance no subset of embedded jobs overlaps in time with the jobs that do not belong to that subset. Relying on these properties, our algorithm finds an optimal solution for the corresponding class of problem instances. Whether a given problem instance is tolerant of the revealed embedded structure becomes known during the execution of the algorithm. Nevertheless, some sufficient conditions in terms of job parameters can a priori be given. One such a condition is that for any pair of jobs j and i with r(i) > r(j) and d(i) < d(j), d(i) - r(j) - p(j) <= d(i) - r(i) - p(i), and if r(i) + p(i) >= r(j) + p(j), then d(i) >= d(j). As we show, our setting remains powerful enough to model important real-life applications. (C) 2019 Elsevier B.V. All rights reserved.

关键词： polynomial-time algorithm Single-machine scheduling Release time Due date Lateness time complexity

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Load Balancing User Association in Millimeter Wave MIMO Networks

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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 2019年第6期18卷 2932-2945页

作者： Alizadeh, Alireza Vu, Mai Tufts Univ Dept Elect & Comp Engn Medford MA 02155 USA

User association is necessary in dense millimeter wave (mmWave) networks to determine which base station a user connects to in order to balance base station loads and maximize a network utility. Given that mmWave connections are highly directional and vulnerable to small channel variations, user association changes these connections and hence significantly affects the network interference and consequently the users' instantaneous rates. In this paper, we introduce a new load balancing user association scheme for mmWave MIMO cellular networks which consider these dependencies. We formulate the user association problem as mixed integer nonlinear programming and design a polynomial-time algorithm, called worst connection swapping (WCS), to find a near-optimal solution. Simulation results confirm that the proposed user association scheme improves network performance significantly by adjusting the interference according to the association, and under the max-min fairness, also enhances cell-edge users' transmission rates. We also show how the proposed algorithm can be applied under mobility. Furthermore, the proposed WCS algorithm outperforms other generic algorithms for combinatorial programming such as the genetic algorithm in both accuracy and speed at several orders of magnitude faster, and for small networks, where exhaustive search is possible, it reaches the optimal solution.

关键词： User association mmWave MIMO systems load balancing MINLP polynomial-time algorithm

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