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Stable Set Problem: Branch & Cut Algorithms

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Encyclopedia of Optimization

Introduction

Let \( { G=(V,E) } \) be an undirected graph consisting of a nonempty finite set V, the node set; and a finite set E, the edge set, of unordered pairs of distinct elements of V. A stable set of graph G is defined as a set of nodes S with the property that the nodes of S are pairwise non adjacent; two nodes are called adjacent if there is an edge in E connecting them. In the literature, stable set is also called independent set, vertex packing, co-clique or anticlique. If each node v i of a graph G is assigned a weight c i , then the graph is called weighted. In this case, the maximum weighted stable set problem looks for a stable set S which maximizes the sum of the weights corresponding to the nodes in S, \( { \sum_{v_i \in S} c_i } \). In the case when G is not weighted, or all \( { c_i = 1 } \), we are interested in a stable set with the maximum number of nodes, which is called maximum cardinality stable set. The size of a maximum cardinality stable set is called...

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References

  1. Applegate D, Bixby R, Chvátal V, Cook W (2001) TSP Cuts Which Do Not Conform to the Template Paradigm. Comput Comb Optim LNCS, vol 2241:157–222

    Google Scholar 

  2. Alekseev VE (2003) On easy and hard hereditary classes of graphs with respect to the independent set problem. Discret Appl Math 132(1–3):17–26

    Article  MATH  Google Scholar 

  3. Arora S, Safra S (1992) Probabilistic Checking of Proofs; a new Characterization of NP. In Proceedings 33rd IEEE Symposium on Foundations of Computer Science, pp 2–13. IEEE Computer Society, Los Angeles

    Google Scholar 

  4. Babel L, Tinhofer G (1990) A branch and bound algorithm for the maximum clique problem. ZOR-Methods Models Oper Res 34:207–217

    Article  MathSciNet  MATH  Google Scholar 

  5. Balas E, Xue J (1996) Weighted and Unweighted Maximum Clique Algorithms with Upper Bounds from Fractional Coloring. Algorithmica 15(5):397–412

    Article  MathSciNet  MATH  Google Scholar 

  6. Balas E, Yu CS (1989) On graphs with polynomially solvable maximum-weight clique problem. Networks 19(2):247–253

    Article  MathSciNet  MATH  Google Scholar 

  7. Balas E, Yu CS (1986) Finding a Maximum Clique in an Arbitrary Graph. SIAM J 14(4):1054–1068

    Article  MathSciNet  Google Scholar 

  8. Balas E, Chvátala V, Nešetřil J (1987) On the Maximum Weight Clique Problem. Math Oper Res 12(3):522–535

    Article  MathSciNet  MATH  Google Scholar 

  9. Balas E, Padberg MW (1976) Set Partitioning: A Survey. SIAM Rev 18(4):710–761

    Article  MathSciNet  MATH  Google Scholar 

  10. Balasa E, Ceria S, Cornuejols G, Pataki G (1996) Polyhedral methods for the maximum clique problem. In: Johnson DS, Trick MA (eds) American Mathematical Society. DIMACS vol 26, pp 11–28

    Google Scholar 

  11. Balinski ML (1970) On Maximum Matching, Minimum Covering and their Connections. In: Kuhn HW (ed) Proceedings of the Princeton symposium on mathematical programming. Princeton University Press, Princeton, pp 303–312

    Google Scholar 

  12. Barnes ER (2000) A Branch-and-Bound Procedure for the Largest Clique in a Graph. Approximation and Complexity in Numerical Optimization: Continuous and Discrete Problems. Kluwer, Boston

    Google Scholar 

  13. BHOSLIB (2000) Benchmarks with Hidden Optimum Solutions for Graph Problems (Maximum Clique, Maximum Independent Set, Minimum Vertex Cover and Vertex Coloring) – Hiding Exact Solutions in Random Graphs. http://www.nlsde.buaa.edu.cn/~kexu/benchmarks/graph-benchmarks.htm

  14. Bhattacharya BK, Kaller D (1997) An O(m + n log n) Algorithm for the Maximum-Clique Problem in Circular-Arc Graphs. J Algorithms 25(3):336–358

    Article  MathSciNet  MATH  Google Scholar 

  15. Bomze IM, Budinich M, Pardalos PM, Pelillo M (1999) The Maximum Clique Problem. Handbook of Combinatorial Optimization. Kluwer, Boston

    Google Scholar 

  16. Bomze IM, Stix V (1999) Genetic engineering via negative fitness: Evolutionary dynamics for global optimization. Annals Oper Res 89:297–318

    Article  MathSciNet  MATH  Google Scholar 

  17. Bonomo F, Durán G, Lin MC, Szwarcfiter JL (2005) On Balanced Graphs. Math Program 105(2–3):233–250

    Google Scholar 

  18. Bourjolly J-M, Laporte G, Mercure H (1997) A combinatorial column generation algorithm for the maximum stable set problem. Oper Res Lett 20(1):21–29

    Article  MathSciNet  MATH  Google Scholar 

  19. Bron C, Kerbosch J (1973) Algorithm 457: Finding all cliques on an undirected graph. Commun ACM 16:575–57

    Article  MATH  Google Scholar 

  20. Brønsted A (1983) An introduction to Convex Polytopes. Graduate Texts in Mathematics, vol 90. Springer, New York

    Google Scholar 

  21. Burer S, Monteiro RDC, Zhang Y (2002) Maximum stable set formulations and heuristics based on continuous optimization. Math Program 94(1):137–166

    Article  MathSciNet  MATH  Google Scholar 

  22. Butenko S (2003) Maximum Independent Set and Related Problems, with Applications. PhD thesis, University of Florida

    Google Scholar 

  23. Caprara A, Fiscetti M, Letchford AN (2000) On the Separation of Maximally Violated mod-k Cuts. Math Program 87(1):37–56

    MathSciNet  MATH  Google Scholar 

  24. Carr RD, Lancia G, Istrail S (2000) Branch-and-Cut Algorithms for Independent Set Problems: Integrality Gap and An Application to Protein Structure Alignment. Technical report, Sandia National Laboratories, Albuquerque, US; Sandia National Laboratories, Livermore

    Google Scholar 

  25. Carraghan R, Pardalos PM (1990) An exact algorithm for the maximum clique problem. Oper Res Lett 9(6):375–382

    Article  MATH  Google Scholar 

  26. Cheng E, Cunningham WH (1995) Separation problems for the stable set polytope. In: Balas E, Clausen J (eds) The 4th Integer Programming and Combinatorial Optimization Conference Proceedings. pp 65–79

    Google Scholar 

  27. Cheng E, Cunningham WH (1997) Wheel Inequalities for Stable Set Polytopes. Math Program 77:389–421

    MathSciNet  Google Scholar 

  28. Cheng E, de Vries S (2002) Antiweb-wheel inequalities and their separation problems over the stable set polytopes. Math Program 92(1):153–175

    Article  MathSciNet  MATH  Google Scholar 

  29. Chiba N, Nishizeki T (1985) Arboricity and subgraph listing algorithms. SIAM J 14:210–223

    Article  MathSciNet  MATH  Google Scholar 

  30. Chudnovsky M, Cornuéjols G, Liu X, Seymour P, Vuškoviċ K (2005) Recognizing Berge Graphs. Comb 25(2):143–186

    Google Scholar 

  31. Chudnovsky M, Robertson N, Seymour P, Thomas R (2004) The strong perfect graph theorem. Ann Math 164:51229

    MathSciNet  Google Scholar 

  32. Cogisa O, Thierry E (2005) Computing maximum stable sets for distance-hereditary graphs. Discret Optim 2(3):185–188

    Article  Google Scholar 

  33. Diestel R (2000) Graph Theory. Electronic Edition 2000. Springer, New York

    Google Scholar 

  34. Second DIMACS Challenge, 1992/1993. http://mat.gsia.cmu.edu/challenge.html

  35. Fahle T (2002) Simple and Fast: Improving a Branch-And-Bound Algorithm for Maximum Clique, vol 2461/2002 Lecture Notes in Computer Science pp 485–498

    Google Scholar 

  36. Fujisawa K, Morito S, Kubo M (1995) Experimental Analyses of the Life Span Method for the Maximum Stable Set Problem. The Institute of Statistical Mathematics Cooperative Research Report 75:135–165

    Google Scholar 

  37. Garey MR, Johnson DS (1979) Computers and Intractability, A guide to the Theory of NP-Completeness. In: Klee V (ed) A series of books in the mathematical sciences. Freeman WH and Company, New York

    Google Scholar 

  38. Gerards AMH, Schrijver A (1986) Matrices with the Edmonds-Johnson property. Comb 6(4):365–379

    Google Scholar 

  39. Giandomenico M, Letchford AN (2006) Exploring the Relationship Between Max-Cut and Stable Set Relaxations. Math Program 106(1):159–175

    Article  MathSciNet  MATH  Google Scholar 

  40. Grötschel M, Jünger M, Reinelt G (1984) A Cutting Plane Algorithm for the Linear Ordering Problem. Oper Res 32:1195–1220

    Article  MathSciNet  MATH  Google Scholar 

  41. Grötschel M, Pulleyblank WR (1981) Weakly Bipartite Graphs and the Max-cut Problem. Oper Res Lett 1(1):23–27

    Article  MathSciNet  MATH  Google Scholar 

  42. Grötschel M, Lovász L, Schrijver A (1988) Geometric Algorithms and Combinatorial Optimization. Algorithms and Combinatorics 2. Springer, Berlin

    MATH  Google Scholar 

  43. Grötschel M, Lovász L, Schrijver A (1981) The Ellipsoid Method and Its Consequences in Combinatorial Optimization. Comb 1:169–197

    Google Scholar 

  44. Harary F, Ross IC (1957) A procedure for clique detection using the group matrix. Sociom 20:205–215

    Google Scholar 

  45. Hasselberg J, Pardalos PM, Vairaktarakis G (1993) Test case generators and computational results for the maximum clique problem. J Glob Optim 3(4):463–482

    Article  MathSciNet  MATH  Google Scholar 

  46. Kallrath J, Wilson JM (1997) Business Optimization using Mathematical Programming. Macmillan, New York

    Google Scholar 

  47. Lawler E (2001) Combinatorial Optimization: Networks and Matroids. Reprint of the 1976 original. Dover Publications, Inc., Mineola

    Google Scholar 

  48. Lehmann KA, Kaufmann M, Steigele S, Nieselt K (2006) On the maximal cliques in c-max-tolerance graphs and their application in clustering molecular sequences. Algorithm Molecular Biol 1:9:1–17

    Google Scholar 

  49. Loukakis E, Tsouros C (1981) A depth first serach algorithm to generate the family of maximal independet sets of a graph lexicographically. Comput 27:249–266

    MathSciNet  Google Scholar 

  50. Lovász L (1979) On the Shannon capacity of a graph. IEEE Trans Inform Theory 25(1):1–7

    Article  MathSciNet  MATH  Google Scholar 

  51. Mannino C, Sassano A (2005) An exact algorithm for the maximum stable set problem. Comput Optim Appl 3(3):243–258

    Article  MathSciNet  Google Scholar 

  52. Mannino C, Sassano A (1996) Edge Projection and the Maximum Cardinality Stable Set Problem. DIMACS Series Discret Math Theor Comput Sci 26:249–261

    MathSciNet  Google Scholar 

  53. Mannino C, Stefanutti E (1999) An augmentation algorithm for the maximum weighted stable set problem. Comput Optim Appl 14(3):367–381

    Article  MathSciNet  MATH  Google Scholar 

  54. Masuda S, Nakajima K, Kashiwabara T, Fujisawa T (1990) Efficient algorithms for finding maximum cliques of an overlap graph. Networks 20(2):157–171

    Article  MathSciNet  MATH  Google Scholar 

  55. Mosca R (1997) Polynomial algorithms for the maximum stable set problem on particular classes of p5-free graphs. Inf Process Lett 61(3):137–143

    Article  MathSciNet  Google Scholar 

  56. Nemhauser GL, Trotter LE Jr (1975) Vertex Packings: Structural Properties and Algorithms. Math Program 8:232–248

    Article  MathSciNet  MATH  Google Scholar 

  57. Nemhauser GL, Trotter LE Jr (1974) Properties of Vertex Packing and Independence System Polyhedra. Math Program 6:48–61

    Article  MathSciNet  MATH  Google Scholar 

  58. Nemhauser GL, Wolsey LA (1988) Integer and Combinatorial Optimization. Wiley-Interscience Series in Discrete Mathematics and Optimization. Wiley, New York

    Google Scholar 

  59. Olariu S (1989) Weak bipolarizable graphs. Discret Math 74(1–2):159–171

    Article  MathSciNet  MATH  Google Scholar 

  60. Östergård PRJ (2002) A fast algorithm for the maximum clique problem. Discret Appl Math 120(1–3):197–207

    Article  MATH  Google Scholar 

  61. Padberg MW, Rinaldi G (1987) Optimization of a 532 City Symmetric Traveling Salesman Problem by Branch and Cut. Oper Res Lett 6:1–7

    Article  MathSciNet  MATH  Google Scholar 

  62. Padberg MW (1973) On the Facial Structure of Set Packing Polyhedra. Math Prog 5:199–215

    Article  MathSciNet  MATH  Google Scholar 

  63. Papadimitriou CH, Yannakakis M (1981) The clique problem for planar graphs. Inf Process Lett 13(4–5):131–133

    Article  MathSciNet  Google Scholar 

  64. Pardalos PM, Phillips AT (1990) A global optimization approach for solving the maximum clique problem. Int J Comput Math 33(3–4):209–216

    Article  MATH  Google Scholar 

  65. Pardalos PM, Rodgers GP (1992) A branch and bound algorithm for the maximum clique problem. Comput Oper Res 19(5):363–375

    Article  MATH  Google Scholar 

  66. Ramírez-Alfonsín JL, Reed BA (eds) (2001) Perfect Graphs, Wiley‐Interscience Series in Discrete Mathematics and Optimization. Wiley, New York

    Google Scholar 

  67. Rebennack S (2006) Maximum Stable Set Problem: A Branch & Cut Solver. Diplomarbeit, Ruprecht–Karls Universität Heidelberg, Heidelberg, Germany

    Google Scholar 

  68. Régin J-C (2003) Using constraint Programming to Solve the Maximum Clique Problem. Lecture Notes in Computer Science. Springer, Berlin, pp 634–648

    Google Scholar 

  69. Rossi F, Smriglio S (2001) A Branch-and-Cut Algorithm for the Maximum Cardinality Stable Set Problem. Oper Res Lett 28:63–74

    Article  MathSciNet  MATH  Google Scholar 

  70. Schrijver A (2003) Combinatorial Optimization: Polyhedra and Efficiency, vol 24 of Algorithms and Combinatorics. Springer, Berlin

    Google Scholar 

  71. Sewell EC (1998) A Branch and Bound Algorithm for the Stability Number of a Sparse Graph. INFORMS J Comput 10(4):438–447

    Article  MathSciNet  Google Scholar 

  72. Strijk T, Verweij B, Aardal K (2000) Algorithms for maximum independent set applied to map labelling. Technical Report UU-CS-2000–22, http://citeseer.ist.psu.edu/article/strijk00algorithms.html

  73. Tomita E, Tanaka A, Takahashi H (1988) The worst-time complexity for finding all the cliques. Technical report, University of Electro-Communications, Tokyo, Japan

    Google Scholar 

  74. Trotter LE (1975) A class of facet producing graphs for vertex packing polyhedra. Discret Math 12(4):373–388

    Article  MathSciNet  MATH  Google Scholar 

  75. Verweij B, Aardal K (1999) An Optimisation Algorithm for Maximum Independent Set with Applications in Map Labelling, vol 1643/1999 Lecture Notes in Computer Science, pp 426–437

    Google Scholar 

  76. Warren JS, Hicks IV (2006) Combinatorial Branch-and-Bound for the Maximum Weight Independent Set Problem. working paper, August 7

    Google Scholar 

  77. Warrier D, Wilhelm WE, Warren JS, Hicks IV (2005) A branch-and-price approach for the maximum weight independent set problem. Network 46(4):198–209

    Article  MathSciNet  MATH  Google Scholar 

  78. West DB (2000) Introduction to Graph Theory, 2nd edn. Prentice Hall

    Google Scholar 

  79. Wolsey LA (1998) Integer Programming. Wiley‐Interscience Series in Discrete Mathematics and Optimization. Wiley‐Interscience, New York

    Google Scholar 

  80. Wood DR (1997) An algorithm for finding a maximum clique in a graph. Oper Res Lett 21(5):211–217

    Article  MathSciNet  MATH  Google Scholar 

  81. Yildirim EA, Fan-Orzechowski X (2006) On Extracting Maximum Stable Sets in Perfect Graphs Using Lovász's Theta Function. Comput Optim Appl 33(2–3):229–247

    Article  MathSciNet  MATH  Google Scholar 

  82. Ziegler GM (1995) Lecture on Polytopes. Graduate Texts in Mathematics. Springer, New York

    Google Scholar 

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  1. Center of Applied Optimization, University of Florida, Gainesville, USA

    Steffen Rebennack

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  1. Department of Chemical Engineering, Princeton University, Princeton, NJ, 08544-5263, USA

    Christodoulos A. Floudas

  2. Center for Applied Optimization, Department of Industrial and Systems Engineering, University of Florida, Gainesville, FL, 32611-6595, USA

    Panos M. Pardalos

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Rebennack, S. (2008). Stable Set Problem: Branch & Cut Algorithms . In: Floudas, C., Pardalos, P. (eds) Encyclopedia of Optimization. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-74759-0_634

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