Docstoc

of the TSP ZIB

Document Sample
of the TSP ZIB Powered By Docstoc
					   The Travelling Salesman Problem
            a brief survey

                       Martin Grötschel
                 Summary of Chapter 2
                       of the class
           Polyhedral Combinatorics (ADM III)
                      May 18, 2010

Martin Grötschel     Institute of Mathematics, Technische Universität Berlin (TUB)
                     DFG-Research Center “Mathematics for key technologies” (MATHEON)
                     Konrad-Zuse-Zentrum für Informationstechnik Berlin (ZIB)
groetschel@zib.de   http://www.zib.de/groetschel
2




            Contents
            1. Introduction
            2. The TSP and some of its history
            3. The TSP and some of its variants
            4. Some applications
            5. Modeling issues
            6. Heuristics
            7. How combinatorial optimizers do it


Martin
Grötschel
3




            Contents
            1. Introduction
            2. The TSP and some of its history
            3. The TSP and some of its variants
            4. Some applications
            5. Modeling issues
            6. Heuristics
            7. How combinatorial optimizers do it


Martin
Grötschel
4




            Combinatorial optimization
            Given a finite set E and a subset I of the power set of E (the set of
            feasible solutions). Given, moreover, a value (cost, length,…) c(e) for all
            elements e of E. Find, among all sets in I, a set I such that its total value
            c(I) (= sum of the values of all elements in I) is as small (or as large) as
            possible.


            The parameters of a combinatorial optimization problem are: (E, I, c).


                                            
                  min c(I)   c(e) | I  I  , where I  2E and E finite
                             eI            
            An important issue: How is I given?


Martin
Grötschel
5

            Special „simple“
            combinatorial optimization problems
            Finding a
             minimum spanning tree in a graph
             shortest path in a directed graph
             maximum matching in a graph
             a minimum capacity cut separating two given nodes of a
              graph or digraph
             cost-minimal flow through a network with capacities and
              costs on all edges
             …
            These problems are solvable in polynomial time.
Martin
Grötschel
6

            Special „hard“
            combinatorial optimization problems
             travelling salesman problem (the prototype problem)
             location und routing
             set-packing, partitioning, -covering
             max-cut
             linear ordering
             scheduling (with a few exceptions)
             node and edge colouring
             …
            These problems are NP-hard
              (in the sense of complexity theory).

Martin
Grötschel
7




            The travelling salesman problem
             Given n „cities“ and „distances“ between them.
             Find a tour (roundtrip) through all cities visiting
             every city exactly once such that the sum of all
             distances travelled is as small as possible. (TSP)


            The TSP is called symmetric (STSP) if, for every
             pair of cities i and j, the distance from i to j is
             the same as the one from j to i, otherwise the
             problem is called aysmmetric (ATSP).
Martin
Grötschel
http://www.tsp.gatech.edu/
9

            THE TSP
            book

            suggested reading for
            everyone interested
            in the TSP




Martin
Grötschel
10




            The travelling salesman problem
            Two mathematical formulations of the TSP
                1. Version :
                Let K n  (V , E ) be the complete graph digraph with n nodes
                and let ce be the length of e  E. Let H be the set of all
                hamiltonian cycles (tours ) in K n . Find
                               min{c(T ) | T  H }.

                2. Version :
                Find a cyclic permutation  of {1,..., n} such that
                                      n

                                    c
                                     i 1
                                            i (i )


                is as small as possible.


Martin
              Does that help solve the TSP?
Grötschel
11




            Contents
            1. Introduction
            2. The TSP and some of its history
            3. The TSP and some of its variants
            4. Some applications
            5. Modeling issues
            6. Heuristics
            7. How combinatorial optimizers do it


Martin
Grötschel
12




            Usually quoted as
            the forerunner of
            the TSP


            Usually quoted as
            the origin of
            the TSP



Martin
Grötschel
about 100
years
earlier
14




            By a proper choice and
            scheduling of the tour one
            can gain so much time
            that we have to make
            some suggestions


            The most important
            aspect is to cover as many
            locations as possible
            without visiting a
            location twice




Martin
Grötschel
15




            Ulysses roundtrip (an even older TSP ?)




            The paper „The Optimized Odyssey“ by Martin Grötschel
            and Manfred Padberg is downloadable from
            http://www.zib.de/groetschel/pubnew/paper/groetschelpadberg2001a.pdf
Martin
Grötschel
16




            Ulysses




              The distance table


Martin
Grötschel
17




            Ulysses roundtrip




                 optimal „Ulysses tour“
Martin
Grötschel
18

            Malen nach Zahlen
            TSP in art ?




             When was this invented?
Martin
Grötschel
19




            Survey Books
            Literature: more than 1000 entries in Zentralblatt/Math


            Zbl 0562.00014 Lawler, E.L.(ed.); Lenstra, J.K.(ed.); Rinnooy
               Kan, A.H.G.(ed.); Shmoys, D.B.(ed.)
               The traveling salesman problem. A guided tour of
               combinatorial optimization. Wiley-Interscience Series in Discrete
               Mathematics. A Wiley-Interscience publication. Chichester etc.: John
               Wiley \& Sons. X, 465 p. (1985). MSC 2000: *00Bxx 90-06


            Zbl 0996.00026 Gutin, Gregory (ed.); Punnen, Abraham P.(ed.)
               The traveling salesman problem and its variations.
               Combinatorial Optimization. 12. Dordrecht: Kluwer Academic
               Publishers. xviii, 830 p. (2002). MSC 2000: *00B15 90-06 90Cxx
Martin
Grötschel
20




            Contents
            1. Introduction
            2. The TSP and some of its history
            3. The TSP and some of its variants
            4. Some applications
            5. Modeling issues
            6. Heuristics
            7. How combinatorial optimizers do it


Martin
Grötschel
                The Travelling Salesman Problem
21




                    and Some of its Variants
             The symmetric TSP
             The asymmetric TSP
             The TSP with precedences or time windows
             The online TSP
             The symmetric and asymmetric m-TSP
             The price collecting TSP
             The Chinese postman problem
              (undirected, directed, mixed)
             Bus, truck, vehicle routing
             Edge/arc & node routing with capacities
             Combinations of these and more

Martin
Grötschel
22

            http://www.densis.fee.unicamp.br/~
            moscato/TSPBIB_home.html




Martin
Grötschel
23




            Contents
            1. Introduction
            2. The TSP and some of its history
            3. The TSP and some of its variants
            4. Some applications
            5. Modeling issues
            6. Heuristics
            7. How combinatorial optimizers do it


Martin
Grötschel
24




            Production of ICs and PCBs




            Integrated Circuit (IC)       Printed Circuit Board (PCB)


                 Problems: Logical Design, Physical Design
                  Correctness, Simulation, Placement of
                     Components, Routing, Drilling,...
Martin
Grötschel
25

            Correct modelling of a
            printed circuit board drilling problem
                                            length of a
                                            move of the
                                            drilling head:
                                            Euclidean norm,
                                            Max norm,
                                            Manhatten norm?




                 2103 holes to be drilled
Martin
Grötschel
26




            Drilling 2103 holes into a PCB

                                  Significant Improvements
                                            via TSP

                                  (due to Padberg & Rinaldi)




Martin
Grötschel
              industry solution       optimal solution
                          Siemens-Problem
                                  PCB da4
         Martin Grötschel, Michael Jünger, Gerhard Reinelt,
         Optimal Control of Plotting and Drilling Machines:
         A Case Study, Zeitschrift für Operations Research,
         35:1 (1991) 61-84
         http://www.zib.de/groetschel/pubnew/paper/groetscheljuengerreinelt1991.pdf




before                                                  after
          Siemens-Problem
                  PCB da1

         Grötschel, Jünger, Reinelt




before                      after
29




Martin
Grötschel
30

            Leiterplatten-Bohrmaschine
            Printed Circuit Board Drilling Machine




Martin
Grötschel
31

            Foto einer Flachbaugruppe
            (Leiterplatte)




Martin
Grötschel
32

            Foto einer Flachbaugruppe
            (Leiterplatte) - Rückseite




Martin
Grötschel
33




            442 holes to be drilled




Martin
Grötschel
34

                   Typical PCB drilling problems at
                              Siemens

                                   da1             da2       da3       da4




            Number of holes       2457             423      2203      2104
            Number of drills         7               7         6        10
            Tour length        3518728         1049956   1958161   4347902




                                     Table 4




Martin
Grötschel
35




                                 Fast heuristics

                                      da1          da2       da3       da4




            CPU time (min:sec)       1:58         0:05      1:43      1:43
            Tour length           1695042       984636   1642027   1928371


            Improvement in %        56.87        14.60     26.94     58.38


                                      Table 5




Martin
Grötschel
36

            Optimizing the stacker cranes of a
            Siemens-Nixdorf warehouse




Martin
Grötschel
37




            Herlitz at Falkensee (Berlin)




Martin
Grötschel
            Example: Control of the stacker
38




             cranes in a Herlitz warehouse




Martin
Grötschel
39

            Logistics of collecting
              electronics garbage

               Andrea Grötschel
                 Diplomarbeit (2004)




Martin
Grötschel
40


            Location plus tour planning (m-TSP)




Martin
Grötschel
41

            The Dispatching Problem at ADAC:
                    an online m-TSP



                                     Dispatching Center (Pannenzentrale)




                                  Data Transm.




                                                 Dispatcher
                 „Gelber Engel“
Martin
Grötschel
          Online-TSP (in a metric space)
Instance:   r1 , r2 ,   , rn where ri  (ti , xi )
              x1                              x1       x2

                   0                               0
 t  t1                        t  t2

Goal:     Find fastest tour serving all requests
                (starting and ending in 0)
  Algorithm ALG is c-competitive if
                   ALG    c  OPT  
  for all request sequences 
43




            Implementation competitions




Martin
Grötschel
44




            Contents
            1. Introduction
            2. The TSP and some of its history
            3. The TSP and some of its variants
            4. Some applications
            5. Modeling issues
            6. Heuristics
            7. How combinatorial optimizers do it


Martin
Grötschel
45




                    LP Cutting Plane Approach

                            Even MODELLING is not easy!

                          What is the „right“ LP relaxation?

                   N. Ascheuer, M. Fischetti, M. Grötschel,
                „Solving the Asymmetric Travelling Salesman
               Problem with time windows by branch-and-cut“,
                  Mathematical Programming A (2001), see
            http://www.zib.de/groetschel/pubnew/paper/ascheuerfischettigroetschel2001.pdf



Martin
Grötschel
46




            IP formulation of the asymmetric TSP

            min cT x
            x( (i ))
                
                            1    i  V  0
            x(  (i ))     1    i  V  0
            x( A(W )) | W | 1   W  V  0 , 2 | W | n
            xij  0,1           (i, j )  A.




Martin
Grötschel
47




            Time Windows
             This is a typical situation in delivery problems.
             Customers must be served during a certain
              period of time, usually a time interval is given.
                access to pedestrian areas
                opening hours of a customer
                delivery to assembly lines
                just in time processes




Martin
Grötschel
48




                                 Model 1
                   T
            min c x
            x(  (i))                  1      i  V  0
            x(  (i))                  1      i  V  0
            ti  ij  (1  xij )  M    tj    i, j  A, j  0
            ri        ti                di    i  V
            ti                          N      i  V  0
            xij                          0,1 i, j  A.
Martin
Grötschel
49



                                   Model 2
            min cT x
            x(  (i ))      1      i  V  0
            x(  (i ))      1      i  V  0
            x( A(W ))  | W | 1     W  V  0 , 2 | W | n
            x( P )  | P | 1  k  2 infeasible path P  (v1 , v2 ,   , vk )
            xij  0,1              (i, j )  A.




Martin
Grötschel
50



                                            Model 3
                        T
            min c x
            x(  (i ))                             1          i  V  0
            x(  (i ))                             1          i  V  0
             n               n                         n

             y  
            i 1
                   ij
                            i 0
                                   ij    xij         y jk j  V
                                                      k 1
            i j            i j                      k j

            ri  xij  yij  di  xij               i, j  0,     , n, i  j , i  0
            xij                            0,1               (i, j )  A
            yij                            0,1, 2,...        (i, j )  A
Martin
Grötschel
                                    Model 1, 2, 3
min cT x                                               min cT x
x(  (i))                  1      i  V  0       x(  (i ))                            1          i  V  0
x(  (i))                  1      i  V  0       x(  (i ))                            1          i  V  0
ti  ij  (1  xij )  M    tj    i, j  A, j  0    n              n                         n

ri       ti                 di    i  V              y  
                                                       i 1
                                                               ij
                                                                      i 0
                                                                             ij    xij         y jk j  V
                                                                                                k 1
                                    i  V  0
                                                       i j           i j                      k j
ti                          N
                                                       ri  xij  yij  di  xij              i, j  0,     , n, i  j , i  0
xij                          0,1 i, j  A.
                                                       xij                           0,1               (i, j )  A
                                                       yij                           0,1, 2,...        (i, j )  A

min cT x
x(  (i ))           1     i  V  0
x(  (i ))           1     i  V  0
x( A(W )) | W | 1          W  V  0 , 2 | W | n
x( P ) | P | 1  k  2     infeasible path P  (v1 , v2 ,        , vk )
xij  0,1                  (i, j )  A.
52


                 Cutting Planes Used for all Three Models
                          (Separation Routines)
                Subtour Elimination Constraints (SEC)
                2-Matching Constraints
                , ,( , )-Inequalities
                "Special“ Inequalities and PCB-Inequalities
                Dk-Inequalities
                Infeasible Path Elimination Constraints (IPEC)
                Strengthened  ,  -Inequalities
                Two-Job Cuts
                Pool Separation
                SD-Inequalities
                + various strengthenings/liftings

Martin
Grötschel
53




                Further Implementation Details
             Preprocessing
                    Tightening Time Windows
                    Release and Due Date Adjustment
                    Construction of Precedences
                    Elimination of Arcs
                Branching (only on x-variables)
                Enumeration Strategy (DFS, Best-FS)
                Pricing Frequency (every 5th iteration)
                Tailing Off
                LP-exploitation Heuristics (after a new feasible LP
                 solution is found),
                they outperform the other heuristics
Martin
Grötschel
54




                                Results
             Very uneven performance
             Model 1 is really bad in general
             Model 2 is best on the average (winner in 16
              of 22 test cases)
             Model 3 is better when few time windows are
              active (6 times winner, last in all other cases,
              severe numerical problems, very difficult LPs)


                   How could you have guessed?
Martin
Grötschel
               Unevenness of Computational
55




                        Results

            problem   #nodes      gap    #cutting     #LPs      time
                                          planes
            rbg041a        43   9.16%    > 1 mio    109,402 > 5 h


            rbg067a        69      0%        176          2     6 sec


              Largest problem solved to optimality: 127 nodes
              Largest problem not solved optimally: 43 nodes

Martin
Grötschel
56




            Contents
            1. Introduction
            2. The TSP and some of its history
            3. The TSP and some of its variants
            4. Some applications
            5. Modeling issues
            6. Heuristics
            7. How combinatorial optimizers do it


Martin
Grötschel
57




            Need for Heuristics
             Many real-world instances of hard combinatorial
              optimization problems are (still) too large for exact
              algorithms.
             Or the time limit stipulated by the customer for the
              solution is too small.
             Therefore, we need heuristics!
             Exact algorithms usually also employ heuristics.
             What is urgently needed is a decision guide:

              Which heuristic will most likely work well on what
              problem ?
Martin
Grötschel
58




            Primal and Dual Heuristics
             Primal Heuristic: Finds a (hopefully) good feasible solution.
             Dual Heuristic: Finds a bound on the optimum solution value
              (e.g., by finding a feasible solution of the LP-dual of an LP-
              relaxation of a combinatorial optimization problem).



            Minimization:

            dual heuristic value ≤ optimum value ≤ primal heuristic value


                              quality guarantee
                              in practice and theory
Martin
Grötschel
59




            Heuristics: A Survey
             Greedy Algorithms
             Exchange & Insertion Algorithms
             Neighborhood/Local Search
             Variable Neighborhood Search, Iterated Local Search
             Random sampling
             Simulated Annealing
             Taboo search
             Great Deluge Algorithms
             Simulated Tunneling
             Neural Networks
             Scatter Search
             Greedy Randomized Adaptive Search Procedures
Martin
Grötschel
60




            Heuristics: A Survey
             Genetic, Evolutionary, and similar Methods
             DNA-Technology
             Ant and Swarm Systems
             (Multi-) Agents
             Population Heuristics
             Memetic Algorithms (Meme are the “missing links” gens and
              mind)
             Fuzzy Genetics-Based Machine Learning
             Fast and Frugal Method (Psychology)
             Method of Devine Intuition (Psychologist Thorndike)


             …..
Martin
Grötschel
61




            Heuristics: A Survey
            Currently best heuristic with respect to worst-case guarantee:
               Christofides heuristic
             compute shortest spanning tree
             compute minimum perfect 1-matching of graph induced by the odd
              nodes of the minimum spanning tree
             the union of these edge sets is a connected Eulerian graph
             turn this graph into a tour by making short-cuts.
            For distance functions satisfying the triangle inequality, the resulting tour
               is at most 50% above the optimum value




Martin
Grötschel
62

            Understanding Heuristics,
            Approximation Algorithms
             worst case analysis
                 There is no polynomial time approx. algorithm for STSP/ATSP.
                 Christofides algorithm for the STSP with triangle inequality
             average case analysis
                 Karp‘s analysis of the patching algorithm for the ATSP
             probabilistic problem analysis
                 for Euclidean STSP in unit square, TSP constant 1.714.. n
             polynomial time approximation schemes (PAS)
                 Arora‘s polynomial-time approximation schemes for
                  Euclidean STSPs
             fully-polynomial time approximation schemes (FPAS)
                 not for TSP/ATSP but, e.g., for knapsack (Ibarra&Kim)

             These concepts – unfortunately – often do not really help to guide
              practice.
             experimental evaluation
                 Lin-Kernighan for STSP (DIMACS challenges))
Martin
Grötschel
63




            Contents
            1. Introduction
            2. The TSP and some of its history
            3. The TSP and some of its variants
            4. Some applications
            5. Modeling issues
            6. Heuristics
            7. How combinatorial optimizers do it


Martin
Grötschel
64




               Polyhedral Theory (of the TSP)
            STSP-, ATSP-,TSP-with-side-constraints-
            Polytope:= Convex hull of all incidence
                       vectors of feasible tours
            To be investigated:
             Dimension
             Equation system defining the affine hull
             Facets
             Separation algorithms
Martin
Grötschel
65




            The symmetric travelling salesman polytope

            QTn : conv{ T  Z E | T tour in K n }   (  ij  1 if ij  T , else  0)
                                                          T


                {x  R E | x( (i ))  2         i  V
                            x( E (W )) | W | 1 W  V \ 1 ,3 | W | n  3
                            0  xij  1           ij  E}


            min cT x
            x( (i ))  2        i  V
            x( E (W )) | W | 1 W  V \ 1 ,3 | W | n  3
            xij  0,1           ij  E

                The LP relaxation is solvable in polynomial time
Martin
Grötschel
66




            Relation between IP and LP-relaxation

            Open Problem:
             If costs satisfy the triangle inequality, then
                        IP-OPT <= 4/3 LP-SEC
                        IP-OPT <= 3/2 LP-SEC (Wolsey)




Martin
Grötschel
67


            General cutting plane theory:
            Gomory Mixed-Integer Cut
             Given y, x j  ¢  and
                                 ,
                   y   aij x j  d  d   f , f  0
                                        
             Rounding: Where aij  aij   f j , define
                                              
                                                              
                 t  y    aij  x j : f j  f    aij  x j : f j  f  ¢
                                                                                   
             Then
                    f x j    j   : f j  f     f j  1x j : f j  f  d  t
             Disjunction:
                    t  d     f j x j : f j  f   f
                         
                                       
                 t   d    1  f j  x j : f j  f  1  f   
             Combining


               f   j                                                                
                              f  x j : f j  f   1  f j  1  f   x j : f j  f  1
                                                                        
Martin
Grötschel
68




            clique trees
             A clique tree is a connected graph C=(V,E), composed of
              cliques satisfying the following properties




Martin
Grötschel
69




              Polyhedral Theory of the TSP
            Comb inequality

                                   2-matching
                                   constraint




            handle
                          tooth
Martin
Grötschel
70



            Clique Tree Inequalities




Martin
Grötschel
71



                       Clique Tree Inequalities
            http://www.zib.de/groetschel/pubnew/paper/groetschelpulleyblank1986.pdf

             h                t                h

             x (  ( H ))   x (  (T ))   | H
                      i               j                i
                                                               |  h  2t
            i 1             j 1             i 1

             h                t                    h                 t
                                                                                              t 1
             x ( E ( H ))   x ( E (T ))   | H
                       i                  j                i
                                                               |    (| T   j
                                                                                 | t j ) 
            i 1             j 1              i 1                 i 1                       2


             Hi, i=1,…,h are the handles
             Tj, j=1,…,t are the teeth
             tj is the number of handles
                 that tooth Tj intersects

Martin
Grötschel
72




                  Valid Inequalities for STSP
               Trivial inequalities
               Degree constraints
               Subtour elimination constraints
               2-matching constraints, comb inequalities
               Clique tree inequalities (comb)
               Bipartition inequalities (clique tree)
               Path inequalities (comb)
               Star inequalities (path)
               Binested Inequalities (star, clique tree)
               Ladder inequalities (2 handles, even # of teeth)
               Domino inequalities
               Hypohamiltonian, hypotraceable inequalities
Martin
             etc.
Grötschel
73




            A very special case

                                Petersen graph, G = (V, F),
                                the smallest hypohamiltonian graph




             x( F )  9 defines a facet of QT
                                            10


                      but not a facet of QT , n  11
                                          n


            M. Grötschel & Y. Wakabayashi


Martin
Grötschel
74




            Hypotraceable graphs and the STSP
            On the right is the smallest
            known hypotraceable graph
            (Thomassen graph, 34 nodes).
            Such graphs have no
            hamiltonian path, but when
            any node is deleted, the
            remaining graph has a
            hamiltonian path.
            How do such graphs induce
            inequalities valid for the
            symmetric travelling salesman
            polytope?

            For further information see:
Martin
Grötschel
            http://www.zib.de/groetschel/pubnew/paper/groetschel1980b.pdf
            “Wild facets of the asymmetric
75




            travelling salesman polytope”
             Hypohamiltonian and hypotraceable directed graphs also exist and
              induce facets of the polytopes associated with the asymmetric TSP.




             Information “hypohamiltonian” and “hypotraceable” inequalities can
              be found in
              http://www.zib.de/groetschel/pubnew/paper/groetschelwakabayashi1981a.pdf
              http://www.zib.de/groetschel/pubnew/paper/groetschelwakabayashi1981b.pdf

Martin
Grötschel
76


            Valid and facet defining inequalities for
                     STSP: Survey articles


             M. Grötschel, M. W. Padberg (1985 a, b)

             M. Jünger, G. Reinelt, G. Rinaldi (1995)

             D. Naddef (2002)

             The TSP book (ABCC, 2006)




Martin
Grötschel
77




                 Counting Tours and Facets
            n    # tours     # different facets   # facet classes
            3         1                      0                 0
            4         3                      3                 1
            5        12                     20                 2
            6        60                    100                 4
            7       360                  3,437                 6
            8      2520               194,187                 24
            9     20,160           42,104,442                192
            10   181,440   >= 52,043,900,866          >=15,379
Martin
Grötschel
78




                     Separation Algorithms
             Given a system of valid inequalities (possibly
              of exponential size).
             Is there a polynomial time algorithm (or a
              good heuristic) that,
               given a point,
               checks whether the point satisfies all inequalities
                of the system, and
               if not, finds an inequality violated by the given
                point?



Martin
Grötschel
79




            Separation




                K




                         Grötschel, Lovász, Schrijver (GLS):
                         “Separation and optimization
                         are polynomial time equivalent.”
Martin
Grötschel
80




                        Separation Algorithms
             There has been great success in finding exact
              polynomial time separation algorithms, e.g.,
                for subtour-elimination constraints
                for 2-matching constraints (Padberg&Rao, 1982)

             or fast heuristic separation algorithms, e.g.,
                for comb constraints
                for clique tree inequalities

             and these algorithms are practically efficient




Martin
Grötschel
81




                  Polyhedral Combinatorics
             This line of research has resulted in
              powerful cutting plane algorithms for
              combinatorial optimization problems.
             They are used in practice to solve
              exactly or approximately (including
              branch & bound) large-scale real-world
              instances.




Martin
Grötschel
82

                  Deutschland
                       15,112
            D. Applegate, R.Bixby,
             V. Chvatal, W. Cook
                   15,112
                     cities
                 114,178,716
                   variables


                    2001




Martin
Grötschel
83




            How do we solve a TSP like this?

             Upper bound:               Lower bound:
              Heuristic search             Linear programming

               Chained Lin-Kernighan      Divide-and-conquer
                                           Polyhedral combinatorics
                                           Parallel computation
                                           Algorithms & data structures




                 The LOWER BOUND is the mathematically and
                 algorithmically hard part of the work
Martin
Grötschel
84


            Work on LP relaxations of the
            symmetric travelling salesman polytope

             QTn : conv{ T  Z E | T tour in K n }

                    T
              min c x
              x( (i ))  2       i  V
              x( E (W )) | W | 1 W  V \ 1 ,3 | W | n  3
             0  xij  1          ij  E
              xij  0,1          ij  E
                    Integer Programming Approach
Martin
Grötschel
85

            cutting plane technique for integer and
            mixed-integer programming

            Feasible
            integer
            solutions
            Objective
            function
            Convex
            hull
            LP-based
            relaxation
            Cutting
            planes
Martin
Grötschel
86




            Clique-tree cut for pcb442 from B. Cook




Martin
Grötschel
87




            LP-based Branch & Bound
                                     Solve LP relaxation:
                             Root     v=0.5 (fractional)     Upper Bound

                                                                G
                                                                A
                                                                P
             Integer                                         Lower Bound




                            Infeas




                 Integer

                           Remark: GAP = 0  Proof of optimality


Martin
Grötschel
88

            A Branching
            Tree
            Applegate
            Bixby
            Chvátal
            Cook




Martin
Grötschel
89




            Managing the LPs of the TSP

                                                      |V|(|V|-1)/2

                            CORE LP              Column generation: Pricing.

                                                     ~ 3|V| variables
             astronomical


                              Cuts: Separation




                                                     ~1.5|V| constraints




Martin
Grötschel
90

            A Pictorial History of Some
                TSP World Records




Martin
Grötschel
                      Some TSP World Records
91




           2006            year     authors       # cities     # variables
       pla 85,900
          solved          1954           DFJ       42/49       820/1,146
     3,646,412,050        1977              G        120            7,140
        variables
                          1987            PR         532         141,246
     number of cities     1988            GH         666         221,445
         2000x
        increase          1991            PR       2,392        2,859,636

        4,000,000         1992         ABCC        3,038        4,613,203
           times          1994         ABCC         7,397      27,354,106
       problem size
         increase         1998         ABCC       13,509       91,239,786

            in 52         2001         ABCC       15,112      114,178,716
            years         2004         ABCC       24,978      311,937,753
Martin  2005 W. Cook, D. Epsinoza, M. Goycoolea
Grötschel
                                                  33,810     571,541,145
92




            The current champions
            ABCC stands for
            D. Applegate, B. Bixby, W. Cook, V. Chvátal


             almost 15 years of code development
             presentation at ICM’98 in Berlin, see proceedings
             have made their code CONCORDE available in
              the Internet


Martin
Grötschel
93




                               USA 49




                                                        49 cities
                                                     1,146 variables

                                                          1954



            G. Dantzig, D.R. Fulkerson, S. Johnson
Martin
Grötschel
94


            West-Deutschland und Berlin

                                   120 Städte
                                  7140 Variable

                                 1975/1977/1980




                                  M. Grötschel




Martin
Grötschel
95




                      A tour around the world
                                                                      666 cities
                                                                  221,445 variables

                                                                      1987/1991




            M. Grötschel, O. Holland, see
Martin
Grötschel
            http://www.zib.de/groetschel/pubnew/paper/groetschelholland1991.pdf
96




             USA cities with population >500

                                                           13,509
                                                            cities
                                                         91,239,786
                                                          Variables

                                                           1998




            D. Applegate, R.Bixby, V. Chvátal, W. Cook
Martin
Grötschel
97




            usa13509: The branching tree




                                    0.01%
                                    initial gap




Martin
Grötschel
98




            Summary: usa13509
              9539 nodes branching tree
              48 workstations (Digital Alphas, Intel
               Pentium IIs, Pentium Pros, Sun
               UntraSparcs)

              Total CPU time:   4 cpu years




Martin
Grötschel
99



              Overlay of
              3 Optimal
              Germany
              tours



                       from
                     ABCC 2001

            http://www.math.princeton.edu/
                tsp/d15sol/dhistory.html




Martin
Grötschel
100




            Optimal Tour of Sweden


                            311,937,753
                             variables

                               ABCC
                                 plus
                            Keld Helsgaun
                            Roskilde Univ.
                              Denmark.




Martin
Grötschel
101


            World Tour, current status
                                    http://www.tsp.gatech.edu/world/




            We give links to several images of the World TSP tour
            of length 7,516,353,779 found by Keld Helsgaun in
            December 2003. A lower bound provided by the
            Concorde TSP code shows that this tour is at most
            0.076% longer than an optimal tour through the
Martin      1,904,711 cities.
Grötschel
   The Travelling Salesman Problem
            a brief survey

                       Martin Grötschel
                 Summary of Chapter 2
                           The END
                       of the class
           Polyhedral Combinatorics (ADM III)
                      May 18, 2010

Martin Grötschel     Institute of Mathematics, Technische Universität Berlin (TUB)
                     DFG-Research Center “Mathematics for key technologies” (MATHEON)
                     Konrad-Zuse-Zentrum für Informationstechnik Berlin (ZIB)
groetschel@zib.de   http://www.zib.de/groetschel

				
DOCUMENT INFO
Shared By:
Categories:
Tags:
Stats:
views:4
posted:11/8/2012
language:Unknown
pages:102