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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) [email protected] 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 eI 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 Grötschel Does that help solve the TSP? 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 and Some of its Variants 21 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 Martin Grötschel http://www.densis.fee.unicamp.br/~ moscato/TSPBIB_home.html 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 Martin Grötschel Foto einer Flachbaugruppe (Leiterplatte) 32 Martin Grötschel Foto einer Flachbaugruppe (Leiterplatte) - Rückseite 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 3518728 1049956 1958161 4347902 Tour length Table 4 Martin Grötschel 35 Fast heuristics CPU time (min:sec) Tour length Improvement in % da1 da2 da3 da4 1:58 0:05 1:43 1:43 1695042 984636 1642027 1928371 56.87 14.60 26.94 58.38 Table 5 Martin Grötschel 36 Martin Grötschel Optimizing the stacker cranes of a Siemens-Nixdorf warehouse 37 Herlitz at Falkensee (Berlin) Martin Grötschel 38 Martin Grötschel Example: Control of the stacker cranes in a Herlitz warehouse 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. „Gelber Engel“ Martin Grötschel Dispatcher Online-TSP (in a metric space) Instance: r1 , r2 , , rn where ri (ti , xi ) x1 x1 t t1 Goal: 0 t t2 x2 0 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 Martin Grötschel 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. 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 Martin Grötschel x( (i )) 1 i V 0 x( (i )) 1 i V 0 ti ij (1 xij ) M tj i, j A, j 0 ri di i V ti i V 0 ti N xij 0,1 i, j A. 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 , xij 0,1 Martin Grötschel (i, j ) A. , vk ) 50 Model 3 T min c x x( (i )) 1 i V 0 x( (i )) 1 i V 0 n n y i 1 i j ij i 0 i j ij xij ri xij yij di xij Martin Grötschel n y jk j V k 1 k j i, j 0, , n, i j , i 0 xij 0,1 (i, j ) A yij 0,1, 2,... (i, j ) A Model 1, 2, 3 min cT 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 di i V ti i V 0 ti N xij 0,1 i, j A. min cT x x( (i )) 1 i V 0 x( (i )) 1 i V 0 n n y i 1 i j ij i 0 i j 1 i V 0 x( (i )) 1 i V 0 x( A(W )) | W | 1 k 1 k j i, j 0, , n, i j , i 0 0,1 (i, j ) A yij 0,1, 2,... (i, j ) A W V 0 , 2 | W | n (i, j ) A. y jk j V xij x( P) | P | 1 k 2 infeasible path P (v1 , v2 , xij 0,1 xij ri xij yij di xij min cT x x( (i )) ij n , vk ) 52 Cutting Planes Used for all Three Models (Separation Routines) Martin Grötschel 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 53 Further Implementation Details Preprocessing Martin Grötschel 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 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 55 Unevenness of Computational Results problem #nodes gap #cutting planes rbg041a 43 9.16% > 1 mio rbg067a 69 0% 176 #LPs 109,402 > 5 h 2 Largest problem solved to optimality: 127 nodes Largest problem not solved optimally: 43 nodes Martin Grötschel time 6 sec 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 LPrelaxation 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-constraintsPolytope:= 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 } {x R E | x( (i)) 2 ( ijT 1 if ij T , else 0) i V x( E (W )) | W | 1 W V \ 1 ,3 | W | n 3 0 xij 1 min cT x x( (i )) 2 ij E} 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 1x 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 Martin Grötschel j f x j : f j f 1 f j 1 f x j : f j f 1 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 1 j 1 h t i 1 h x ( E ( H )) x ( E (T )) | H i i 1 j j 1 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 | h 2t i i 1 t i | (| T i 1 j | t j ) t 1 2 72 Valid Inequalities for STSP Martin Grötschel 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 etc. 73 A very special case Petersen graph, G = (V, F), the smallest hypohamiltonian graph x( F ) 9 defines a facet of QT10 but not a facet of QTn , n 11 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? Martin Grötschel For further information see: http://www.zib.de/groetschel/pubnew/paper/groetschel1980b.pdf 75 “Wild facets of the asymmetric 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 Martin Grötschel 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 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 Root Solve LP relaxation: v=0.5 (fractional) Upper Bound G A P Lower Bound Integer 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 Martin Grötschel Cuts: Separation astronomical CORE LP Column generation: Pricing. ~ 3|V| variables ~1.5|V| constraints 90 Martin Grötschel A Pictorial History of Some TSP World Records Some TSP World Records 91 2006 pla 85,900 solved 3,646,412,050 variables number of cities 2000x increase 4,000,000 times problem size increase in 52 years 2005 Martin Grötschel year authors # cities # variables 1954 DFJ 42/49 820/1,146 1977 G 120 7,140 1987 PR 532 141,246 1988 GH 666 221,445 1991 PR 2,392 2,859,636 1992 ABCC 3,038 4,613,203 1994 ABCC 7,397 27,354,106 1998 ABCC 13,509 91,239,786 2001 ABCC 15,112 114,178,716 2004 ABCC 24,978 311,937,753 W. Cook, D. Epsinoza, M. Goycoolea 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: Martin Grötschel 4 cpu years 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/ Martin Grötschel 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 1,904,711 cities. The Travelling Salesman Problem a brief survey Martin Grötschel Summary of Chapter 2 of the class Polyhedral Combinatorics (ADM III) May 18, 2010 The END Martin Grötschel Institute of Mathematics, Technische Universität Berlin (TUB) DFG-Research Center “Mathematics for key technologies” (MATHEON) Konrad-Zuse-Zentrum für Informationstechnik Berlin (ZIB) [email protected] http://www.zib.de/groetschel