Symmetry Handling in Binary Programs via Polyhedral Methods

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Symmetry Handling in Binary Programs via Polyhedral Methods"

Transcription

1 Symmetry Handling in Binary Programs via Polyhedral Methods Christopher Hojny joint work with Marc Pfetsch Technische Universität Darmstadt Department of Mathematics 26 International Conference on Mathematical Software ICMS 26 Christopher Hojny: Symmetry Handling in Binary Programs via Polyhedral Methods

2 Motivation Consider a necklace with n black () and white () beads. Associate with a necklace a vector in {, } n. necklaces x, x {, } n are equal cyclic shift γ such that γ(x) = x.? = ICMS 26 Christopher Hojny: Symmetry Handling in Binary Programs via Polyhedral Methods 2

3 Motivation Consider a necklace with n black () and white () beads. Associate with a necklace a vector in {, } n. necklaces x, x {, } n are equal cyclic shift γ such that γ(x) = x.? = shift ICMS 26 Christopher Hojny: Symmetry Handling in Binary Programs via Polyhedral Methods 2

4 Motivation Consider a necklace with n black () and white () beads. Associate with a necklace a vector in {, } n. necklaces x, x {, } n are equal cyclic shift γ such that γ(x) = x.? = 2 shifts ICMS 26 Christopher Hojny: Symmetry Handling in Binary Programs via Polyhedral Methods 2

5 Motivation Consider a necklace with n black () and white () beads. Associate with a necklace a vector in {, } n. necklaces x, x {, } n are equal cyclic shift γ such that γ(x) = x.? = 3 shifts ICMS 26 Christopher Hojny: Symmetry Handling in Binary Programs via Polyhedral Methods 2

6 Motivation Consider a necklace with n black () and white () beads. Associate with a necklace a vector in {, } n. necklaces x, x {, } n are equal cyclic shift γ such that γ(x) = x.? = 3 shifts Question: How can we characterize (the convex hull of) a representative system of distinct necklaces? ICMS 26 Christopher Hojny: Symmetry Handling in Binary Programs via Polyhedral Methods 2

7 Outline Symmetry Detection Representative Systems Symretopes and Symresacks Numerical Experiments ICMS 26 Christopher Hojny: Symmetry Handling in Binary Programs via Polyhedral Methods 3

8 Symmetry Detection Consider binary program (BP) max{c x : Ax b, x {, } n }. We can distinguish two types of permutation symmetries of BP: Problem Symmetry Group Contains all permutations γ S n with Aγ(x) b Ax b and c γ(x) = c x. Formulation Symmetry Group Contains all permutations γ S n for which there is σ S m s.t. γ(c) = c, σ(b) = b, and A σ(i),γ(j) = A i,j. ICMS 26 Christopher Hojny: Symmetry Handling in Binary Programs via Polyhedral Methods 4

9 Symmetry Detection Consider binary program (BP) max{c x : Ax b, x {, } n }. We can distinguish two types of permutation symmetries of BP: Problem Symmetry Group Contains all permutations γ S n with Aγ(x) b Ax b and c γ(x) = c x. NP-hard Formulation Symmetry Group Contains all permutations γ S n for which there is σ S m s.t. γ(c) = c, σ(b) = b, and A σ(i),γ(j) = A i,j. Graph-Isomorphism-hard ICMS 26 Christopher Hojny: Symmetry Handling in Binary Programs via Polyhedral Methods 4

10 How to Determine Formulation Symmetries? Build an auxillary colored graph and determine its color preserving automorphisms. max x +x 2 x 2 x 2x +2x x x 2 ICMS 26 Christopher Hojny: Symmetry Handling in Binary Programs via Polyhedral Methods 5

11 How to Determine Formulation Symmetries? Build an auxillary colored graph and determine its color preserving automorphisms. max x +x 2 x 2 x 2x +2x x x 2 ICMS 26 Christopher Hojny: Symmetry Handling in Binary Programs via Polyhedral Methods 5

12 How to Determine Formulation Symmetries? Build an auxillary colored graph and determine its color preserving automorphisms. max x +x 2 x 2 x 2 2x +2x Automorphism group can be determined, e.g., with bliss, nauty, and saucy. x x 2 ICMS 26 Christopher Hojny: Symmetry Handling in Binary Programs via Polyhedral Methods 5

13 Outline Symmetry Detection Representative Systems Symretopes and Symresacks Numerical Experiments ICMS 26 Christopher Hojny: Symmetry Handling in Binary Programs via Polyhedral Methods 6

14 Orbit Representative Systems Given a symmetry group Γ of a binary program (BP), how can we determine a representative system of the orbits where x is feasible for BP? Γ(x) = {γ(x) : γ Γ}, Idea: Add for each γ Γ to BP. a γ x := n (2 n γ(i) 2 n i ) x i i= ICMS 26 Christopher Hojny: Symmetry Handling in Binary Programs via Polyhedral Methods 7

15 Pros and Cons Pros a γ x cuts off symmetric solutions {x {, } n : Ax b} γ Γ {x Rn : a γ x } is lexmax representative system of BP, see [Friedman, 27] Cons coefficients of a γ x grow exponentially large possibly many inequalities, one for each γ Γ ICMS 26 Christopher Hojny: Symmetry Handling in Binary Programs via Polyhedral Methods 8

16 Outline Symmetry Detection Representative Systems Symretopes and Symresacks Numerical Experiments ICMS 26 Christopher Hojny: Symmetry Handling in Binary Programs via Polyhedral Methods 9

17 Symretopes Definition Given Γ S n, the symretope w.r.t. Γ is the polytope S(Γ) := conv ( {x {, } n : a γ x γ Γ} ). ICMS 26 Christopher Hojny: Symmetry Handling in Binary Programs via Polyhedral Methods

18 Symretopes Definition Given Γ S n, the symretope w.r.t. Γ is the polytope S(Γ) := conv ( {x {, } n : a γ x γ Γ} ). If Γ is given by generators, the optimization problem over symresacks is NP-hard. There is Γ S n such that the coefficients of facet inequalities grow exponentially. To avoid exponential coefficients, find tractable IP-formulation for symretopes with small coefficients. Idea: Consider /-knapsack polytope induced by a γ x. ICMS 26 Christopher Hojny: Symmetry Handling in Binary Programs via Polyhedral Methods

19 Symresacks Definition Given γ S n, the symresack w.r.t. γ is the polytope P γ := conv ( {x {, } n : a γ x } ). Properties non-standard knapsack polytopes, feasible points can be completely characterized by minimal cover inequalities and box constraints, in general, there are exponential many minimal covers ICMS 26 Christopher Hojny: Symmetry Handling in Binary Programs via Polyhedral Methods

20 Separation Complexity Theorem The separation problem of minimal cover inequalities for P γ and x R n can be solved in time O(n 2 ). Consequences: Symmetry handling is possible with {, ±}-inequalities: separate minimal cover inequalities of P γ instead of adding a γ x, separation is possible in time O( Γ n 2 ), avoid exponential coefficients. ICMS 26 Christopher Hojny: Symmetry Handling in Binary Programs via Polyhedral Methods 2

21 Propagation Problem Input: Upper and lower bounds u, l {, } n, γ S n Questions: Is there a vertex x of P γ with l x u? Can we tighten some bounds? ICMS 26 Christopher Hojny: Symmetry Handling in Binary Programs via Polyhedral Methods 3

22 Propagation Problem Input: Upper and lower bounds u, l {, } n, γ S n Questions: Is there a vertex x of P γ with l x u? Can we tighten some bounds? x γ(x) l x i u x x 2 x 3 x 4 x 5 x 6 ICMS 26 Christopher Hojny: Symmetry Handling in Binary Programs via Polyhedral Methods 3

23 Propagation Problem Input: Upper and lower bounds u, l {, } n, γ S n Questions: Is there a vertex x of P γ with l x u? Can we tighten some bounds? Infeasibility Detection x γ(x) l x i u x x 2 x 3 x 4 x 5 x 6 ICMS 26 Christopher Hojny: Symmetry Handling in Binary Programs via Polyhedral Methods 3

24 Propagation Problem Input: Upper and lower bounds u, l {, } n, γ S n Questions: Is there a vertex x of P γ with l x u? Can we tighten some bounds? Infeasibility Detection x γ(x) l x i u x x 2 x 3 x 4 x 5 x 6 ICMS 26 Christopher Hojny: Symmetry Handling in Binary Programs via Polyhedral Methods 3

25 Propagation Problem Input: Upper and lower bounds u, l {, } n, γ S n Questions: Is there a vertex x of P γ with l x u? Can we tighten some bounds? Feasibility Detection x γ(x) l x i u x x 2 x 3 x 4 x 5 x 6 ICMS 26 Christopher Hojny: Symmetry Handling in Binary Programs via Polyhedral Methods 3

26 Propagation Problem Input: Upper and lower bounds u, l {, } n, γ S n Questions: Is there a vertex x of P γ with l x u? Can we tighten some bounds? Feasibility Detection x γ(x) l x i u x x 2 x 3 x 4 x 5 x 6 ICMS 26 Christopher Hojny: Symmetry Handling in Binary Programs via Polyhedral Methods 3

27 Propagation Problem Input: Upper and lower bounds u, l {, } n, γ S n Questions: Is there a vertex x of P γ with l x u? Can we tighten some bounds? Bound tightening x γ(x) l x i u x x 2 x 3 x 4 x 5 x 6 ICMS 26 Christopher Hojny: Symmetry Handling in Binary Programs via Polyhedral Methods 3

28 Propagation Problem Input: Upper and lower bounds u, l {, } n, γ S n Questions: Is there a vertex x of P γ with l x u? Can we tighten some bounds? Bound tightening x γ(x) l x i u x x 2 x 3 x 4 x 5 x 6 ICMS 26 Christopher Hojny: Symmetry Handling in Binary Programs via Polyhedral Methods 3

29 Propagation Problem Input: Upper and lower bounds u, l {, } n, γ S n Questions: Is there a vertex x of P γ with l x u? Can we tighten some bounds? Bound tightening x γ(x) l x i u x x 2 x 3 x 4 x 5 x 6 ICMS 26 Christopher Hojny: Symmetry Handling in Binary Programs via Polyhedral Methods 3

30 Propagation Problem Input: Upper and lower bounds u, l {, } n, γ S n Questions: Is there a vertex x of P γ with l x u? Can we tighten some bounds? Bound tightening x γ(x) l x i u x x 2 x 3 x 4 x 5 x 6 ICMS 26 Christopher Hojny: Symmetry Handling in Binary Programs via Polyhedral Methods 3

31 Propagation Problem Input: Upper and lower bounds u, l {, } n, γ S n Questions: Is there a vertex x of P γ with l x u? Can we tighten some bounds? Bound tightening x γ(x) l x i u x x 2 x 3 x 4 x 5 x 6 ICMS 26 Christopher Hojny: Symmetry Handling in Binary Programs via Polyhedral Methods 3

32 Propagation Problem Input: Upper and lower bounds u, l {, } n, γ S n Questions: Is there a vertex x of P γ with l x u? Can we tighten some bounds? Bound tightening x γ(x) l x i u x x 2 x 3 x 4 x 5 x 6 ICMS 26 Christopher Hojny: Symmetry Handling in Binary Programs via Polyhedral Methods 3

33 Propagation Problem Input: Upper and lower bounds u, l {, } n, γ S n Questions: Is there a vertex x of P γ with l x u? Can we tighten some bounds? Theorem The propagation problem for P γ can be solved in time O(n). ICMS 26 Christopher Hojny: Symmetry Handling in Binary Programs via Polyhedral Methods 3

34 Outline Symmetry Detection Representative Systems Symretopes and Symresacks Numerical Experiments ICMS 26 Christopher Hojny: Symmetry Handling in Binary Programs via Polyhedral Methods 4

35 Implementation implemented constraint handler for symresacks in SCIP 3.2. containing separation of minimal cover inequalities, propagation rules, resolution methods if propagation detected infeasibility, implemented constraint handler for orbisacks [Kaibel and Loos, 2] which are symresacks for γ = (, 2)(3, 4)... (2n, 2n) use automated symmetry detection methods of [Pfetsch and Rehn, 25], use CPLEX 2.6. as LP-solver ICMS 26 Christopher Hojny: Symmetry Handling in Binary Programs via Polyhedral Methods 5

36 Numerical Experiments tests were run on Linux cluster with Intel i3 3.2GHz dual core processors, 8GB memory, single thread time limit: 36 seconds test instances MIPLIB 2 benchmark SONET network design [Sherali and Smith, 2] Wagon Load Balancing [Ghoniem and Sherali, 2] ICMS 26 Christopher Hojny: Symmetry Handling in Binary Programs via Polyhedral Methods 6

37 Numerical Experiments: MIPLIB 2 benchmark number of solved instances, default 6/87 prop sepa sepa+prob orbisacks symresacks speed up of solution time orbisacks symresacks reduction of nodes orbisacks symresacks ICMS 26 Christopher Hojny: Symmetry Handling in Binary Programs via Polyhedral Methods 7

38 Numerical Experiments: SONET number of solved instances, default 5/5 prop sepa sepa+prob orbisacks symresacks speed up of solution time orbisacks symresacks reduction of nodes orbisacks symresacks ICMS 26 Christopher Hojny: Symmetry Handling in Binary Programs via Polyhedral Methods 8

39 Numerical Experiments: Wagon Load Balancing number of solved instances, default 2/2 prop sepa sepa+prob orbisacks symresacks 2 2 speed up of solution time orbisacks..3. symresacks..2.2 reduction of nodes orbisacks..4. symresacks.95.. ICMS 26 Christopher Hojny: Symmetry Handling in Binary Programs via Polyhedral Methods 9

40 Thank you for your attention! ICMS 26 Christopher Hojny: Symmetry Handling in Binary Programs via Polyhedral Methods 2

41 Literature Friedman, E. J. (27). Fundamental domains for integer programs with symmetries. In Dress, A., Xu, Y., and Zhu, B., editors, Combinatorial Optimization and Applications, volume 466 of Lecture Notes in Computer Science, pages Springer Berlin Heidelberg. Ghoniem, A. and Sherali, H. D. (2). Defeating symmetry in combinatorial optimization via objective perturbations and hierarchical constraints. IIE Transactions, 43(8): Kaibel, V. and Loos, A. (2). Finding descriptions of polytopes via extended formulations and liftings. In Mahjoub, A. R., editor, Progress in Combinatorial Optimization. Wiley. Pfetsch, M. E. and Rehn, T. (25). A computational comparison of symmetry handling methods for mixed integer programs. Sherali, H. D. and Smith, J. C. (2). Improving discrete model representations via symmetry considerations. Management Science, 47(): ICMS 26 Christopher Hojny: Symmetry Handling in Binary Programs via Polyhedral Methods 2

Can linear programs solve NP-hard problems?

Can linear programs solve NP-hard problems? Can linear programs solve NP-hard problems? p. 1/9 Can linear programs solve NP-hard problems? Ronald de Wolf Linear programs Can linear programs solve NP-hard problems? p. 2/9 Can linear programs solve

More information

arxiv:cs/0106002v2 [cs.dm] 21 Aug 2001

arxiv:cs/0106002v2 [cs.dm] 21 Aug 2001 Solving Assembly Line Balancing Problems by Combining IP and CP Alexander Bockmayr and Nicolai Pisaruk arxiv:cs/0106002v2 [cs.dm] 21 Aug 2001 Université Henri Poincaré, LORIA B.P. 239, F-54506 Vandœuvre-lès-Nancy,

More information

In this paper we present a branch-and-cut algorithm for

In this paper we present a branch-and-cut algorithm for SOLVING A TRUCK DISPATCHING SCHEDULING PROBLEM USING BRANCH-AND-CUT ROBERT E. BIXBY Rice University, Houston, Texas EVA K. LEE Georgia Institute of Technology, Atlanta, Georgia (Received September 1994;

More information

3. Linear Programming and Polyhedral Combinatorics

3. Linear Programming and Polyhedral Combinatorics Massachusetts Institute of Technology Handout 6 18.433: Combinatorial Optimization February 20th, 2009 Michel X. Goemans 3. Linear Programming and Polyhedral Combinatorics Summary of what was seen in the

More information

Scheduling Home Health Care with Separating Benders Cuts in Decision Diagrams

Scheduling Home Health Care with Separating Benders Cuts in Decision Diagrams Scheduling Home Health Care with Separating Benders Cuts in Decision Diagrams André Ciré University of Toronto John Hooker Carnegie Mellon University INFORMS 2014 Home Health Care Home health care delivery

More information

Branch and Cut for TSP

Branch and Cut for TSP Branch and Cut for TSP jla,jc@imm.dtu.dk Informatics and Mathematical Modelling Technical University of Denmark 1 Branch-and-Cut for TSP Branch-and-Cut is a general technique applicable e.g. to solve symmetric

More information

Linear Programming. Widget Factory Example. Linear Programming: Standard Form. Widget Factory Example: Continued.

Linear Programming. Widget Factory Example. Linear Programming: Standard Form. Widget Factory Example: Continued. Linear Programming Widget Factory Example Learning Goals. Introduce Linear Programming Problems. Widget Example, Graphical Solution. Basic Theory:, Vertices, Existence of Solutions. Equivalent formulations.

More information

A Reference Point Method to Triple-Objective Assignment of Supporting Services in a Healthcare Institution. Bartosz Sawik

A Reference Point Method to Triple-Objective Assignment of Supporting Services in a Healthcare Institution. Bartosz Sawik Decision Making in Manufacturing and Services Vol. 4 2010 No. 1 2 pp. 37 46 A Reference Point Method to Triple-Objective Assignment of Supporting Services in a Healthcare Institution Bartosz Sawik Abstract.

More information

Transportation Polytopes: a Twenty year Update

Transportation Polytopes: a Twenty year Update Transportation Polytopes: a Twenty year Update Jesús Antonio De Loera University of California, Davis Based on various papers joint with R. Hemmecke, E.Kim, F. Liu, U. Rothblum, F. Santos, S. Onn, R. Yoshida,

More information

Branch-and-Price Approach to the Vehicle Routing Problem with Time Windows

Branch-and-Price Approach to the Vehicle Routing Problem with Time Windows TECHNISCHE UNIVERSITEIT EINDHOVEN Branch-and-Price Approach to the Vehicle Routing Problem with Time Windows Lloyd A. Fasting May 2014 Supervisors: dr. M. Firat dr.ir. M.A.A. Boon J. van Twist MSc. Contents

More information

The Line Connectivity Problem

The Line Connectivity Problem Konrad-Zuse-Zentrum für Informationstechnik Berlin Takustraße 7 D-14195 Berlin-Dahlem Germany RALF BORNDÖRFER MARIKA NEUMANN MARC E. PFETSCH The Line Connectivity Problem Supported by the DFG Research

More information

Noncommercial Software for Mixed-Integer Linear Programming

Noncommercial Software for Mixed-Integer Linear Programming Noncommercial Software for Mixed-Integer Linear Programming J. T. Linderoth T. K. Ralphs December, 2004. Revised: January, 2005. Abstract We present an overview of noncommercial software tools for the

More information

CHAPTER 9. Integer Programming

CHAPTER 9. Integer Programming CHAPTER 9 Integer Programming An integer linear program (ILP) is, by definition, a linear program with the additional constraint that all variables take integer values: (9.1) max c T x s t Ax b and x integral

More information

Dynamic programming. Doctoral course Optimization on graphs - Lecture 4.1. Giovanni Righini. January 17 th, 2013

Dynamic programming. Doctoral course Optimization on graphs - Lecture 4.1. Giovanni Righini. January 17 th, 2013 Dynamic programming Doctoral course Optimization on graphs - Lecture.1 Giovanni Righini January 1 th, 201 Implicit enumeration Combinatorial optimization problems are in general NP-hard and we usually

More information

Hub Cover and Hub Center Problems

Hub Cover and Hub Center Problems Hub Cover and Hub Center Problems Horst W. Hamacher, Tanja Meyer Department of Mathematics, University of Kaiserslautern, Gottlieb-Daimler-Strasse, 67663 Kaiserslautern, Germany Abstract Using covering

More information

5 INTEGER LINEAR PROGRAMMING (ILP) E. Amaldi Fondamenti di R.O. Politecnico di Milano 1

5 INTEGER LINEAR PROGRAMMING (ILP) E. Amaldi Fondamenti di R.O. Politecnico di Milano 1 5 INTEGER LINEAR PROGRAMMING (ILP) E. Amaldi Fondamenti di R.O. Politecnico di Milano 1 General Integer Linear Program: (ILP) min c T x Ax b x 0 integer Assumption: A, b integer The integrality condition

More information

Logic Cuts Generation in a Branch and Cut Framework for Location Problems Mara A. Osorio Lama School of Computer Science Autonomous University of Puebla, Puebla 72560 Mexico Rosalba Mujica Garca Abstract

More information

Cloud Branching. Timo Berthold. joint work with Domenico Salvagnin (Università degli Studi di Padova)

Cloud Branching. Timo Berthold. joint work with Domenico Salvagnin (Università degli Studi di Padova) Cloud Branching Timo Berthold Zuse Institute Berlin joint work with Domenico Salvagnin (Università degli Studi di Padova) DFG Research Center MATHEON Mathematics for key technologies 21/May/13, CPAIOR

More information

Linear Programming I

Linear Programming I Linear Programming I November 30, 2003 1 Introduction In the VCR/guns/nuclear bombs/napkins/star wars/professors/butter/mice problem, the benevolent dictator, Bigus Piguinus, of south Antarctica penguins

More information

A Branch and Bound Algorithm for Solving the Binary Bi-level Linear Programming Problem

A Branch and Bound Algorithm for Solving the Binary Bi-level Linear Programming Problem A Branch and Bound Algorithm for Solving the Binary Bi-level Linear Programming Problem John Karlof and Peter Hocking Mathematics and Statistics Department University of North Carolina Wilmington Wilmington,

More information

Mathematical finance and linear programming (optimization)

Mathematical finance and linear programming (optimization) Mathematical finance and linear programming (optimization) Geir Dahl September 15, 2009 1 Introduction The purpose of this short note is to explain how linear programming (LP) (=linear optimization) may

More information

2.3 Scheduling jobs on identical parallel machines

2.3 Scheduling jobs on identical parallel machines 2.3 Scheduling jobs on identical parallel machines There are jobs to be processed, and there are identical machines (running in parallel) to which each job may be assigned Each job = 1,,, must be processed

More information

CHARACTERIZING MULTI-DRUG TREATMENTS ON METABOLIC NETWORKS

CHARACTERIZING MULTI-DRUG TREATMENTS ON METABOLIC NETWORKS CHARACTERIZING MULTI-DRUG TREATMENTS ON METABOLIC NETWORKS Claudio Altafini Linköping University, Sweden SISSA, Trieste First and foremost The real authors of this work: Giuseppe Facchetti Mattia Zampieri

More information

Some representability and duality results for convex mixed-integer programs.

Some representability and duality results for convex mixed-integer programs. Some representability and duality results for convex mixed-integer programs. Santanu S. Dey Joint work with Diego Morán and Juan Pablo Vielma December 17, 2012. Introduction About Motivation Mixed integer

More information

A Constraint Programming based Column Generation Approach to Nurse Rostering Problems

A Constraint Programming based Column Generation Approach to Nurse Rostering Problems Abstract A Constraint Programming based Column Generation Approach to Nurse Rostering Problems Fang He and Rong Qu The Automated Scheduling, Optimisation and Planning (ASAP) Group School of Computer Science,

More information

Proximal mapping via network optimization

Proximal mapping via network optimization L. Vandenberghe EE236C (Spring 23-4) Proximal mapping via network optimization minimum cut and maximum flow problems parametric minimum cut problem application to proximal mapping Introduction this lecture:

More information

Definition of a Linear Program

Definition of a Linear Program Definition of a Linear Program Definition: A function f(x 1, x,..., x n ) of x 1, x,..., x n is a linear function if and only if for some set of constants c 1, c,..., c n, f(x 1, x,..., x n ) = c 1 x 1

More information

Throughput constraint for Synchronous Data Flow Graphs

Throughput constraint for Synchronous Data Flow Graphs Throughput constraint for Synchronous Data Flow Graphs *Alessio Bonfietti Michele Lombardi Michela Milano Luca Benini!"#$%&'()*+,-)./&0&20304(5 60,7&-8990,.+:&;/&."!?@A>&"'&=,0B+C. !"#$%&'()* Resource

More information

Actually Doing It! 6. Prove that the regular unit cube (say 1cm=unit) of sufficiently high dimension can fit inside it the whole city of New York.

Actually Doing It! 6. Prove that the regular unit cube (say 1cm=unit) of sufficiently high dimension can fit inside it the whole city of New York. 1: 1. Compute a random 4-dimensional polytope P as the convex hull of 10 random points using rand sphere(4,10). Run VISUAL to see a Schlegel diagram. How many 3-dimensional polytopes do you see? How many

More information

Optimization in R n Introduction

Optimization in R n Introduction Optimization in R n Introduction Rudi Pendavingh Eindhoven Technical University Optimization in R n, lecture Rudi Pendavingh (TUE) Optimization in R n Introduction ORN / 4 Some optimization problems designing

More information

A Robust Formulation of the Uncertain Set Covering Problem

A Robust Formulation of the Uncertain Set Covering Problem A Robust Formulation of the Uncertain Set Covering Problem Dirk Degel Pascal Lutter Chair of Management, especially Operations Research Ruhr-University Bochum Universitaetsstrasse 150, 44801 Bochum, Germany

More information

Using diversification, communication and parallelism to solve mixed-integer linear programs

Using diversification, communication and parallelism to solve mixed-integer linear programs Using diversification, communication and parallelism to solve mixed-integer linear programs R. Carvajal a,, S. Ahmed a, G. Nemhauser a, K. Furman b, V. Goel c, Y. Shao c a Industrial and Systems Engineering,

More information

5.1 Bipartite Matching

5.1 Bipartite Matching CS787: Advanced Algorithms Lecture 5: Applications of Network Flow In the last lecture, we looked at the problem of finding the maximum flow in a graph, and how it can be efficiently solved using the Ford-Fulkerson

More information

A Column-Generation and Branch-and-Cut Approach to the Bandwidth-Packing Problem

A Column-Generation and Branch-and-Cut Approach to the Bandwidth-Packing Problem [J. Res. Natl. Inst. Stand. Technol. 111, 161-185 (2006)] A Column-Generation and Branch-and-Cut Approach to the Bandwidth-Packing Problem Volume 111 Number 2 March-April 2006 Christine Villa and Karla

More information

AN INTEGER PROGRAMMING APPROACH TO EQUITABLE COLORING PROBLEMS. Laura Bahiense. Clicia Valladares Peixoto Friedman FFP/UERJ, cliciavp@terra.com.

AN INTEGER PROGRAMMING APPROACH TO EQUITABLE COLORING PROBLEMS. Laura Bahiense. Clicia Valladares Peixoto Friedman FFP/UERJ, cliciavp@terra.com. AN INTEGER PROGRAMMING APPROACH TO EQUITABLE COLORING PROBLEMS Laura Bahiense COPPE-Produção/UFRJ, laura@pep.ufrj.br Clicia Valladares Peixoto Friedman FFP/UERJ, cliciavp@terra.com.br Samuel Jurkiewicz

More information

Completely Positive Cone and its Dual

Completely Positive Cone and its Dual On the Computational Complexity of Membership Problems for the Completely Positive Cone and its Dual Peter J.C. Dickinson Luuk Gijben July 3, 2012 Abstract Copositive programming has become a useful tool

More information

Nan Kong, Andrew J. Schaefer. Department of Industrial Engineering, Univeristy of Pittsburgh, PA 15261, USA

Nan Kong, Andrew J. Schaefer. Department of Industrial Engineering, Univeristy of Pittsburgh, PA 15261, USA A Factor 1 2 Approximation Algorithm for Two-Stage Stochastic Matching Problems Nan Kong, Andrew J. Schaefer Department of Industrial Engineering, Univeristy of Pittsburgh, PA 15261, USA Abstract We introduce

More information

Lecture 3: Linear Programming Relaxations and Rounding

Lecture 3: Linear Programming Relaxations and Rounding Lecture 3: Linear Programming Relaxations and Rounding 1 Approximation Algorithms and Linear Relaxations For the time being, suppose we have a minimization problem. Many times, the problem at hand can

More information

An Optimization Approach for Cooperative Communication in Ad Hoc Networks

An Optimization Approach for Cooperative Communication in Ad Hoc Networks An Optimization Approach for Cooperative Communication in Ad Hoc Networks Carlos A.S. Oliveira and Panos M. Pardalos University of Florida Abstract. Mobile ad hoc networks (MANETs) are a useful organizational

More information

Discrete Optimization

Discrete Optimization Discrete Optimization [Chen, Batson, Dang: Applied integer Programming] Chapter 3 and 4.1-4.3 by Johan Högdahl and Victoria Svedberg Seminar 2, 2015-03-31 Todays presentation Chapter 3 Transforms using

More information

26 Linear Programming

26 Linear Programming The greatest flood has the soonest ebb; the sorest tempest the most sudden calm; the hottest love the coldest end; and from the deepest desire oftentimes ensues the deadliest hate. Th extremes of glory

More information

Integer factorization is in P

Integer factorization is in P Integer factorization is in P Yuly Shipilevsky Toronto, Ontario, Canada E-mail address: yulysh2000@yahoo.ca Abstract A polynomial-time algorithm for integer factorization, wherein integer factorization

More information

On the binary solitaire cone

On the binary solitaire cone On the binary solitaire cone David Avis a,1 Antoine Deza b,c,2 a McGill University, School of Computer Science, Montréal, Canada b Tokyo Institute of Technology, Department of Mathematical and Computing

More information

24. The Branch and Bound Method

24. The Branch and Bound Method 24. The Branch and Bound Method It has serious practical consequences if it is known that a combinatorial problem is NP-complete. Then one can conclude according to the present state of science that no

More information

Using the analytic center in the feasibility pump

Using the analytic center in the feasibility pump Using the analytic center in the feasibility pump Daniel Baena Jordi Castro Dept. of Stat. and Operations Research Universitat Politècnica de Catalunya daniel.baena@upc.edu jordi.castro@upc.edu Research

More information

. P. 4.3 Basic feasible solutions and vertices of polyhedra. x 1. x 2

. P. 4.3 Basic feasible solutions and vertices of polyhedra. x 1. x 2 4. Basic feasible solutions and vertices of polyhedra Due to the fundamental theorem of Linear Programming, to solve any LP it suffices to consider the vertices (finitely many) of the polyhedron P of the

More information

Degrees of freedom in (forced) symmetric frameworks. Louis Theran (Aalto University / AScI, CS)

Degrees of freedom in (forced) symmetric frameworks. Louis Theran (Aalto University / AScI, CS) Degrees of freedom in (forced) symmetric frameworks Louis Theran (Aalto University / AScI, CS) Frameworks Graph G = (V,E); edge lengths l(ij); ambient dimension d Length eqns. pi - pj 2 = l(ij) 2 The p

More information

Some Optimization Fundamentals

Some Optimization Fundamentals ISyE 3133B Engineering Optimization Some Optimization Fundamentals Shabbir Ahmed E-mail: sahmed@isye.gatech.edu Homepage: www.isye.gatech.edu/~sahmed Basic Building Blocks min or max s.t. objective as

More information

! Solve problem to optimality. ! Solve problem in poly-time. ! Solve arbitrary instances of the problem. !-approximation algorithm.

! Solve problem to optimality. ! Solve problem in poly-time. ! Solve arbitrary instances of the problem. !-approximation algorithm. Approximation Algorithms Chapter Approximation Algorithms Q Suppose I need to solve an NP-hard problem What should I do? A Theory says you're unlikely to find a poly-time algorithm Must sacrifice one of

More information

Robust Geometric Programming is co-np hard

Robust Geometric Programming is co-np hard Robust Geometric Programming is co-np hard André Chassein and Marc Goerigk Fachbereich Mathematik, Technische Universität Kaiserslautern, Germany Abstract Geometric Programming is a useful tool with a

More information

Approximation Algorithms

Approximation Algorithms Approximation Algorithms or: How I Learned to Stop Worrying and Deal with NP-Completeness Ong Jit Sheng, Jonathan (A0073924B) March, 2012 Overview Key Results (I) General techniques: Greedy algorithms

More information

Practical Guide to the Simplex Method of Linear Programming

Practical Guide to the Simplex Method of Linear Programming Practical Guide to the Simplex Method of Linear Programming Marcel Oliver Revised: April, 0 The basic steps of the simplex algorithm Step : Write the linear programming problem in standard form Linear

More information

Minimally Infeasible Set Partitioning Problems with Balanced Constraints

Minimally Infeasible Set Partitioning Problems with Balanced Constraints Minimally Infeasible Set Partitioning Problems with alanced Constraints Michele Conforti, Marco Di Summa, Giacomo Zambelli January, 2005 Revised February, 2006 Abstract We study properties of systems of

More information

How to speed-up hard problem resolution using GLPK?

How to speed-up hard problem resolution using GLPK? How to speed-up hard problem resolution using GLPK? Onfroy B. & Cohen N. September 27, 2010 Contents 1 Introduction 2 1.1 What is GLPK?.......................................... 2 1.2 GLPK, the best one?.......................................

More information

Routing in Line Planning for Public Transport

Routing in Line Planning for Public Transport Konrad-Zuse-Zentrum für Informationstechnik Berlin Takustraße 7 D-14195 Berlin-Dahlem Germany MARC E. PFETSCH RALF BORNDÖRFER Routing in Line Planning for Public Transport Supported by the DFG Research

More information

954 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 51, NO. 3, MARCH 2005

954 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 51, NO. 3, MARCH 2005 954 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 51, NO. 3, MARCH 2005 Using Linear Programming to Decode Binary Linear Codes Jon Feldman, Martin J. Wainwright, Member, IEEE, and David R. Karger, Associate

More information

Scheduling Shop Scheduling. Tim Nieberg

Scheduling Shop Scheduling. Tim Nieberg Scheduling Shop Scheduling Tim Nieberg Shop models: General Introduction Remark: Consider non preemptive problems with regular objectives Notation Shop Problems: m machines, n jobs 1,..., n operations

More information

Minkowski Sum of Polytopes Defined by Their Vertices

Minkowski Sum of Polytopes Defined by Their Vertices Minkowski Sum of Polytopes Defined by Their Vertices Vincent Delos, Denis Teissandier To cite this version: Vincent Delos, Denis Teissandier. Minkowski Sum of Polytopes Defined by Their Vertices. Journal

More information

An Introduction on SemiDefinite Program

An Introduction on SemiDefinite Program An Introduction on SemiDefinite Program from the viewpoint of computation Hayato Waki Institute of Mathematics for Industry, Kyushu University 2015-10-08 Combinatorial Optimization at Work, Berlin, 2015

More information

Applied Algorithm Design Lecture 5

Applied Algorithm Design Lecture 5 Applied Algorithm Design Lecture 5 Pietro Michiardi Eurecom Pietro Michiardi (Eurecom) Applied Algorithm Design Lecture 5 1 / 86 Approximation Algorithms Pietro Michiardi (Eurecom) Applied Algorithm Design

More information

New Exact Solution Approaches for the Split Delivery Vehicle Routing Problem

New Exact Solution Approaches for the Split Delivery Vehicle Routing Problem New Exact Solution Approaches for the Split Delivery Vehicle Routing Problem Gizem Ozbaygin, Oya Karasan and Hande Yaman Department of Industrial Engineering, Bilkent University, Ankara, Turkey ozbaygin,

More information

Dantzig-Wolfe bound and Dantzig-Wolfe cookbook

Dantzig-Wolfe bound and Dantzig-Wolfe cookbook Dantzig-Wolfe bound and Dantzig-Wolfe cookbook thst@man.dtu.dk DTU-Management Technical University of Denmark 1 Outline LP strength of the Dantzig-Wolfe The exercise from last week... The Dantzig-Wolfe

More information

A Weighted-Sum Mixed Integer Program for Bi-Objective Dynamic Portfolio Optimization

A Weighted-Sum Mixed Integer Program for Bi-Objective Dynamic Portfolio Optimization AUTOMATYKA 2009 Tom 3 Zeszyt 2 Bartosz Sawik* A Weighted-Sum Mixed Integer Program for Bi-Objective Dynamic Portfolio Optimization. Introduction The optimal security selection is a classical portfolio

More information

Chapter 11. 11.1 Load Balancing. Approximation Algorithms. Load Balancing. Load Balancing on 2 Machines. Load Balancing: Greedy Scheduling

Chapter 11. 11.1 Load Balancing. Approximation Algorithms. Load Balancing. Load Balancing on 2 Machines. Load Balancing: Greedy Scheduling Approximation Algorithms Chapter Approximation Algorithms Q. Suppose I need to solve an NP-hard problem. What should I do? A. Theory says you're unlikely to find a poly-time algorithm. Must sacrifice one

More information

Optimization Modeling for Mining Engineers

Optimization Modeling for Mining Engineers Optimization Modeling for Mining Engineers Alexandra M. Newman Division of Economics and Business Slide 1 Colorado School of Mines Seminar Outline Linear Programming Integer Linear Programming Slide 2

More information

Cost Efficient Network Synthesis from Leased Lines

Cost Efficient Network Synthesis from Leased Lines Konrad-Zuse-Zentrum für Informationstechnik Berlin Takustraße 7 D-14195 Berlin-Dahlem Germany DIMITRIS ALEVRAS ROLAND WESSÄLY MARTIN GRÖTSCHEL Cost Efficient Network Synthesis from Leased Lines Preprint

More information

Discuss the size of the instance for the minimum spanning tree problem.

Discuss the size of the instance for the minimum spanning tree problem. 3.1 Algorithm complexity The algorithms A, B are given. The former has complexity O(n 2 ), the latter O(2 n ), where n is the size of the instance. Let n A 0 be the size of the largest instance that can

More information

Minimum Makespan Scheduling

Minimum Makespan Scheduling Minimum Makespan Scheduling Minimum makespan scheduling: Definition and variants Factor 2 algorithm for identical machines PTAS for identical machines Factor 2 algorithm for unrelated machines Martin Zachariasen,

More information

Lecture 5: Conic Optimization: Overview

Lecture 5: Conic Optimization: Overview EE 227A: Conve Optimization and Applications January 31, 2012 Lecture 5: Conic Optimization: Overview Lecturer: Laurent El Ghaoui Reading assignment: Chapter 4 of BV. Sections 3.1-3.6 of WTB. 5.1 Linear

More information

FE670 Algorithmic Trading Strategies. Stevens Institute of Technology

FE670 Algorithmic Trading Strategies. Stevens Institute of Technology FE670 Algorithmic Trading Strategies Lecture 6. Portfolio Optimization: Basic Theory and Practice Steve Yang Stevens Institute of Technology 10/03/2013 Outline 1 Mean-Variance Analysis: Overview 2 Classical

More information

17.3.1 Follow the Perturbed Leader

17.3.1 Follow the Perturbed Leader CS787: Advanced Algorithms Topic: Online Learning Presenters: David He, Chris Hopman 17.3.1 Follow the Perturbed Leader 17.3.1.1 Prediction Problem Recall the prediction problem that we discussed in class.

More information

arxiv:1203.1525v1 [math.co] 7 Mar 2012

arxiv:1203.1525v1 [math.co] 7 Mar 2012 Constructing subset partition graphs with strong adjacency and end-point count properties Nicolai Hähnle haehnle@math.tu-berlin.de arxiv:1203.1525v1 [math.co] 7 Mar 2012 March 8, 2012 Abstract Kim defined

More information

2.3 Convex Constrained Optimization Problems

2.3 Convex Constrained Optimization Problems 42 CHAPTER 2. FUNDAMENTAL CONCEPTS IN CONVEX OPTIMIZATION Theorem 15 Let f : R n R and h : R R. Consider g(x) = h(f(x)) for all x R n. The function g is convex if either of the following two conditions

More information

Solving NP Hard problems in practice lessons from Computer Vision and Computational Biology

Solving NP Hard problems in practice lessons from Computer Vision and Computational Biology Solving NP Hard problems in practice lessons from Computer Vision and Computational Biology Yair Weiss School of Computer Science and Engineering The Hebrew University of Jerusalem www.cs.huji.ac.il/ yweiss

More information

A REMARK ON ALMOST MOORE DIGRAPHS OF DEGREE THREE. 1. Introduction and Preliminaries

A REMARK ON ALMOST MOORE DIGRAPHS OF DEGREE THREE. 1. Introduction and Preliminaries Acta Math. Univ. Comenianae Vol. LXVI, 2(1997), pp. 285 291 285 A REMARK ON ALMOST MOORE DIGRAPHS OF DEGREE THREE E. T. BASKORO, M. MILLER and J. ŠIRÁŇ Abstract. It is well known that Moore digraphs do

More information

Scheduling of Mixed Batch-Continuous Production Lines

Scheduling of Mixed Batch-Continuous Production Lines Université Catholique de Louvain Faculté des Sciences Appliquées Scheduling of Mixed Batch-Continuous Production Lines Thèse présentée en vue de l obtention du grade de Docteur en Sciences Appliquées par

More information

Classification of Cartan matrices

Classification of Cartan matrices Chapter 7 Classification of Cartan matrices In this chapter we describe a classification of generalised Cartan matrices This classification can be compared as the rough classification of varieties in terms

More information

The Quadratic Assignment Problem

The Quadratic Assignment Problem The Quadratic Assignment Problem Rainer E. Burkard Eranda Çela Panos M. Pardalos Leonidas S. Pitsoulis Abstract This paper aims at describing the state of the art on quadratic assignment problems (QAPs).

More information

Recovery of primal solutions from dual subgradient methods for mixed binary linear programming; a branch-and-bound approach

Recovery of primal solutions from dual subgradient methods for mixed binary linear programming; a branch-and-bound approach MASTER S THESIS Recovery of primal solutions from dual subgradient methods for mixed binary linear programming; a branch-and-bound approach PAULINE ALDENVIK MIRJAM SCHIERSCHER Department of Mathematical

More information

Branch, Cut, and Price: Sequential and Parallel

Branch, Cut, and Price: Sequential and Parallel Branch, Cut, and Price: Sequential and Parallel T.K. Ralphs 1, L. Ladányi 2, and L.E. Trotter, Jr. 3 1 Department of Industrial and Manufacturing Systems Engineering, Lehigh University, Bethlehem, PA 18017,

More information

! Solve problem to optimality. ! Solve problem in poly-time. ! Solve arbitrary instances of the problem. #-approximation algorithm.

! Solve problem to optimality. ! Solve problem in poly-time. ! Solve arbitrary instances of the problem. #-approximation algorithm. Approximation Algorithms 11 Approximation Algorithms Q Suppose I need to solve an NP-hard problem What should I do? A Theory says you're unlikely to find a poly-time algorithm Must sacrifice one of three

More information

A Mathematical Programming Solution to the Mars Express Memory Dumping Problem

A Mathematical Programming Solution to the Mars Express Memory Dumping Problem A Mathematical Programming Solution to the Mars Express Memory Dumping Problem Giovanni Righini and Emanuele Tresoldi Dipartimento di Tecnologie dell Informazione Università degli Studi di Milano Via Bramante

More information

FRACTIONAL COLORINGS AND THE MYCIELSKI GRAPHS

FRACTIONAL COLORINGS AND THE MYCIELSKI GRAPHS FRACTIONAL COLORINGS AND THE MYCIELSKI GRAPHS By G. Tony Jacobs Under the Direction of Dr. John S. Caughman A math 501 project submitted in partial fulfillment of the requirements for the degree of Master

More information

COMBINATORIAL PROPERTIES OF THE HIGMAN-SIMS GRAPH. 1. Introduction

COMBINATORIAL PROPERTIES OF THE HIGMAN-SIMS GRAPH. 1. Introduction COMBINATORIAL PROPERTIES OF THE HIGMAN-SIMS GRAPH ZACHARY ABEL 1. Introduction In this survey we discuss properties of the Higman-Sims graph, which has 100 vertices, 1100 edges, and is 22 regular. In fact

More information

Polytope Examples (PolyComp Fukuda) Matching Polytope 1

Polytope Examples (PolyComp Fukuda) Matching Polytope 1 Polytope Examples (PolyComp Fukuda) Matching Polytope 1 Matching Polytope Let G = (V,E) be a graph. A matching in G is a subset of edges M E such that every vertex meets at most one member of M. A matching

More information

Robust Adaptive Resource Allocation in Container Terminals

Robust Adaptive Resource Allocation in Container Terminals Robust Adaptive Resource Allocation in Container Terminals Marco Laumanns IBM Research - Zurich, 8803 Rueschlikon, Switzerland, mlm@zurich.ibm.com Rico Zenklusen, Kaspar Schüpbach Institute for Operations

More information

NP-Hardness Results Related to PPAD

NP-Hardness Results Related to PPAD NP-Hardness Results Related to PPAD Chuangyin Dang Dept. of Manufacturing Engineering & Engineering Management City University of Hong Kong Kowloon, Hong Kong SAR, China E-Mail: mecdang@cityu.edu.hk Yinyu

More information

Two-Stage Stochastic Linear Programs

Two-Stage Stochastic Linear Programs Two-Stage Stochastic Linear Programs Operations Research Anthony Papavasiliou 1 / 27 Two-Stage Stochastic Linear Programs 1 Short Reviews Probability Spaces and Random Variables Convex Analysis 2 Deterministic

More information

M.S. Ibrahim 1*, N. Maculan 2 and M. Minoux 3

M.S. Ibrahim 1*, N. Maculan 2 and M. Minoux 3 Pesquisa Operacional (2014) 34(1): 117-124 2014 Brazilian Operations Research Society Printed version ISSN 0101-7438 / Online version ISSN 1678-5142 www.scielo.br/pope A NOTE ON THE NP-HARDNESS OF THE

More information

Autodesk 3ds Max 2010 Boot Camp FAQ

Autodesk 3ds Max 2010 Boot Camp FAQ Autodesk 3ds Max 2010 Boot Camp Frequently Asked Questions (FAQ) Frequently Asked Questions and Answers This document provides questions and answers about using Autodesk 3ds Max 2010 software with the

More information

LINEAR PROGRAMMING WITH ONLINE LEARNING

LINEAR PROGRAMMING WITH ONLINE LEARNING LINEAR PROGRAMMING WITH ONLINE LEARNING TATSIANA LEVINA, YURI LEVIN, JEFF MCGILL, AND MIKHAIL NEDIAK SCHOOL OF BUSINESS, QUEEN S UNIVERSITY, 143 UNION ST., KINGSTON, ON, K7L 3N6, CANADA E-MAIL:{TLEVIN,YLEVIN,JMCGILL,MNEDIAK}@BUSINESS.QUEENSU.CA

More information

An Integer Programming Model for the School Timetabling Problem

An Integer Programming Model for the School Timetabling Problem An Integer Programming Model for the School Timetabling Problem Geraldo Ribeiro Filho UNISUZ/IPTI Av. São Luiz, 86 cj 192 01046-000 - República - São Paulo SP Brazil Luiz Antonio Nogueira Lorena LAC/INPE

More information

Adaptive Linear Programming Decoding

Adaptive Linear Programming Decoding Adaptive Linear Programming Decoding Mohammad H. Taghavi and Paul H. Siegel ECE Department, University of California, San Diego Email: (mtaghavi, psiegel)@ucsd.edu ISIT 2006, Seattle, USA, July 9 14, 2006

More information

A Branch-Cut-and-Price Approach to the Bus Evacuation Problem with Integrated Collection Point and Shelter Decisions

A Branch-Cut-and-Price Approach to the Bus Evacuation Problem with Integrated Collection Point and Shelter Decisions A Branch-Cut-and-Price Approach to the Bus Evacuation Problem with Integrated Collection Point and Shelter Decisions Marc Goerigk, Bob Grün, and Philipp Heßler Fachbereich Mathematik, Technische Universität

More information

An interval linear programming contractor

An interval linear programming contractor An interval linear programming contractor Introduction Milan Hladík Abstract. We consider linear programming with interval data. One of the most challenging problems in this topic is to determine or tight

More information

Random graphs with a given degree sequence

Random graphs with a given degree sequence Sourav Chatterjee (NYU) Persi Diaconis (Stanford) Allan Sly (Microsoft) Let G be an undirected simple graph on n vertices. Let d 1,..., d n be the degrees of the vertices of G arranged in descending order.

More information

MOSEK modeling manual

MOSEK modeling manual MOSEK modeling manual August 12, 2014 Contents 1 Introduction 1 2 Linear optimization 3 2.1 Introduction....................................... 3 2.1.1 Basic notions.................................. 3

More information

Online Algorithm for Servers Consolidation in Cloud Data Centers

Online Algorithm for Servers Consolidation in Cloud Data Centers 1 Online Algorithm for Servers Consolidation in Cloud Data Centers Makhlouf Hadji, Paul Labrogere Technological Research Institute - IRT SystemX 8, Avenue de la Vauve, 91120 Palaiseau, France. Abstract

More information

Generalized DCell Structure for Load-Balanced Data Center Networks

Generalized DCell Structure for Load-Balanced Data Center Networks Generalized DCell Structure for Load-Balanced Data Center Networks Markus Kliegl, Jason Lee,JunLi, Xinchao Zhang, Chuanxiong Guo,DavidRincón Swarthmore College, Duke University, Fudan University, Shanghai

More information

A Study of Local Optima in the Biobjective Travelling Salesman Problem

A Study of Local Optima in the Biobjective Travelling Salesman Problem A Study of Local Optima in the Biobjective Travelling Salesman Problem Luis Paquete, Marco Chiarandini and Thomas Stützle FG Intellektik, Technische Universität Darmstadt, Alexanderstr. 10, Darmstadt,

More information