Journal of Combinatorial Optimization

, Volume 20, Issue 4, pp 335–360 | Cite as

Separator-based data reduction for signed graph balancing

  • Falk Hüffner
  • Nadja Betzler
  • Rolf Niedermeier


Polynomial-time data reduction is a classical approach to hard graph problems. Typically, particular small subgraphs are replaced by smaller gadgets. We generalize this approach to handle any small subgraph that has a small separator connecting it to the rest of the graph. The problem we study is the NP-hard Balanced Subgraph problem, which asks for a 2-coloring of a graph that minimizes the inconsistencies with given edge labels. It has applications in social networks, systems biology, and integrated circuit design. The data reduction scheme unifies and generalizes a number of previously known data reductions, and can be applied to a large number of graph problems where a coloring or a subset of the vertices is sought. To solve the instances that remain after reduction, we use a fixed-parameter algorithm based on iterative compression with a very effective heuristic speedup. Our implementation can solve biological real-world instances exactly for which previously only approximations were known. In addition, we present experimental results for financial networks and random networks.


Preprocessing Exact algorithm Parameterized algorithmics Algorithm engineering Gene-regulatory network Financial network 


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  1. Abu-Khzam FN, Collins RL, Fellows MR, Langston MA, Suters WH, Symons CT (2004) Kernelization algorithms for the vertex cover problem: Theory and experiments. In: Proceedings of the 6th workshop on algorithm engineering and experiments (ALENEX ’04). SIAM, pp 62–69 Google Scholar
  2. Abu-Khzam FN, Fellows MR, Langston MA, Suters WH (2007) Crown structures for vertex cover kernelization. Theory Comput Syst 41(3):411–430 zbMATHCrossRefMathSciNetGoogle Scholar
  3. Agarwal A, Charikar M, Makarychev K, Makarychev Y (2005) \(O(\sqrt{\log n})\) approximation algorithms for min UnCut, min 2CNF deletion, and directed cut problems. In: Proceedings of the 37th ACM symposium on theory of computing (STOC ’05). ACM, pp 573–581 Google Scholar
  4. Antal T, Krapivsky PL, Redner S (2006) Social balance on networks: The dynamics of friendship and enmity. Physica D: Nonlinear Phenom 224(1–2):130–136 zbMATHCrossRefMathSciNetGoogle Scholar
  5. Avidor A, Langberg M (2007) The multi-multiway cut problem. Theor Comput Sci 377(1–3):35–42 zbMATHCrossRefMathSciNetGoogle Scholar
  6. Barahona F (1980) On the complexity of max cut. Tech Rep 186, IMAG, Université Joseph Fourier, Grenoble, France Google Scholar
  7. Barahona F (1982) On the computational complexity of Ising spin glass models. J Phys A: Math Gen 15(10):3241–3253 CrossRefMathSciNetGoogle Scholar
  8. Barahona F, Ridha Mahjoub A (1989) Facets of the balanced (acyclic) induced subgraph polytope. Math Program 45(1–3):21–33 zbMATHCrossRefGoogle Scholar
  9. Barvinok AI, Woods K (2003) Short rational generating functions for lattice point problems. J Am Math Soc 16(4):957–979 zbMATHCrossRefMathSciNetGoogle Scholar
  10. Bodlaender HL, Koster AMCA (2008) Combinatorial optimization on graphs of bounded treewidth. Comput J 51:255–269 CrossRefMathSciNetGoogle Scholar
  11. Boginski V, Butenko S, Pardalos PM (2005) Statistical analysis of financial networks. Comput Stat Data Anal 48(2):431–443 zbMATHCrossRefMathSciNetGoogle Scholar
  12. Boginski V, Butenko S, Pardalos PM (2006) Mining market data: A network approach. Comput Oper Res 33(11):3171–3184 zbMATHCrossRefGoogle Scholar
  13. Boros E, Hammer PL (1991) The max-cut problem and quadratic 0–1 optimization; polyhedral aspects, relaxations and bounds. Ann Oper Res 33(3):151–180 zbMATHCrossRefMathSciNetGoogle Scholar
  14. Chen J, Kanj IA, Jia W (2001) Vertex cover: Further observations and further improvements. J Algorithms 41(2):280–301 zbMATHCrossRefMathSciNetGoogle Scholar
  15. Chiang C, Kahng AB, Sinha S, Xu X, Zelikovsky AZ (2007) Fast and efficient bright-field AAPSM conflict detection and correction. IEEE Trans Comput-Aided Des Integr Circ Syst 26(1):115–126 CrossRefGoogle Scholar
  16. Coleman T, Saunderson J, Wirth A (2008) A local-search 2-approximation for 2-correlation-clustering. In: Proceedings of the 16th annual European symposium on algorithms (ESA ’08). LNCS, vol 5193. Springer, Berlin, pp 308–319 Google Scholar
  17. DasGupta B, Enciso GA, Sontag ED, Zhang Y (2007) Algorithmic and complexity results for decompositions of biological networks into monotone subsystems. Biosystems 90(1):161–178 CrossRefGoogle Scholar
  18. Downey RG, Fellows MR (1999) Parameterized complexity. Springer, Berlin Google Scholar
  19. Estivill-Castro V, Fellows MR, Langston MA, Rosamond FA (2006) FPT is P-time extremal structure I. In: Proceedings of the 1st algorithms and complexity in Durham workshop (ACiD ’06). Texts in Algorithmics, vol 4. College Publications, pp 1–41 Google Scholar
  20. Feist AM, Scholten JCM, Palsson BØ, Brockman FJ, Ideker T (2006) Modeling methanogenesis with a genome-scale metabolic reconstruction of Methanosarcina barkeri. Mol Syst Biol 2:2006 .0004 CrossRefGoogle Scholar
  21. Fellows MR, Langston MA (1989) An analogue of the Myhill–Nerode theorem and its use in computing finite-basis characterizations. In: Proceedings of the 30th annual IEEE symposium on foundations of computer science (FOCS ’89). IEEE, pp 520–525 Google Scholar
  22. Flum J, Grohe M (2006) Parameterized complexity theory. Springer, Berlin Google Scholar
  23. Gabow HN (2000) Path-based depth-first search for strong and biconnected components. Inf Process Lett 74(3–4):107–114 CrossRefMathSciNetGoogle Scholar
  24. Goemans MX, Williamson DP (1995) Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J ACM 42(6):1115–1145 zbMATHCrossRefMathSciNetGoogle Scholar
  25. Grötschel M, Pulleyblank WR (1981) Weakly bipartite graphs and the max-cut problem. Oper Res Lett 1(1):23–27 zbMATHCrossRefMathSciNetGoogle Scholar
  26. Guo J, Niedermeier R (2007) Invitation to data reduction and problem kernelization. ACM SIGACT News 38(1):31–45 CrossRefGoogle Scholar
  27. Guo J, Hüffner F, Niedermeier R (2004) A structural view on parameterizing problems: Distance from triviality. In: Proceedings of the 1st international workshop on parameterized and exact computation (IWPEC ’04). LNCS, vol 3162. Springer, Berlin, pp 162–173 CrossRefGoogle Scholar
  28. Guo J, Gramm J, Hüffner F, Niedermeier R, Wernicke S (2006) Compression-based fixed-parameter algorithms for feedback vertex set and edge bipartization. J Comput Syst Sci 72(8):1386–1396 zbMATHCrossRefGoogle Scholar
  29. Guo J, Moser H, Niedermeier R (2008) Iterative compression for exactly solving NP-hard minimization problems. In: Proceedings of the DFG SPP 1126 “algorithmics of large and complex networks”. LNCS. Springer, Berlin, to appear Google Scholar
  30. Gutwenger C, Mutzel P (2000) A linear time implementation of SPQR-trees. In: Proceedings of the 8th international symposium on graph drawing (GD ’00). LNCS, vol 1984. Springer, Berlin, pp 77–90 Google Scholar
  31. Harary F (1953) On the notion of balance of a signed graph. Michigan Math J 2(2):143–146 CrossRefMathSciNetGoogle Scholar
  32. Harary F (1959) On the measurement of structural balance. Behav Sci 4(4):316–323 CrossRefGoogle Scholar
  33. Harary F, Lim MH, Wunsch DC (2002) Signed graphs for portfolio analysis in risk management. IMA J Manag Math 13(3):201–210 zbMATHCrossRefGoogle Scholar
  34. Henzinger MR, Rao S, Gabow HN (2000) Computing vertex connectivity: New bounds from old techniques. J Algorithms 43(2):222–250 CrossRefMathSciNetGoogle Scholar
  35. Hicks IV, Koster AMCA, Kolotoğlu E (2005) Branch and tree decomposition techniques for discrete optimization. In: TutORials 2005, tutorials in operations research, INFORMS, pp 1–29 Google Scholar
  36. Hopcroft JE, Tarjan RE (1973) Dividing a graph into triconnected components. SIAM J Comput 2(3):135–158 CrossRefMathSciNetGoogle Scholar
  37. Hüffner F (2005) Algorithm engineering for optimal graph bipartization. In: Proceedings of the 4th international workshop on experimental and efficient algorithms (WEA ’05). LNCS, vol 3503. Springer, Berlin, pp 240–252. Extended version to appear in J Graph Algorithms Appl CrossRefGoogle Scholar
  38. Hüffner F (2007) Algorithms and experiments for parameterized approaches to hard graph problems. PhD thesis, Institut für Informatik, Friedrich-Schiller-Universität Jena Google Scholar
  39. Hüffner F, Komusiewicz C, Moser H, Niedermeier R (2008) Fixed-parameter algorithms for cluster vertex deletion. In: Proceedings of the 8th Latin American theoretical informatics symposium (LATIN ’08). LNCS, vol 4598. Springer, Berlin, pp 711–722. Extended version to appear in Theory Comput Syst CrossRefGoogle Scholar
  40. Khot S (2002) On the power of unique 2-prover 1-round games. In: Proceedings of the 34th ACM symposium on theory of computing (STOC ’02). ACM, pp 767–775 Google Scholar
  41. Kőnig D (1936) Theorie der endlichen und unendlichen Graphen. Akademische Verlagsgesellschaft, Leipzig, English translation: Theory of finite and infinite graphs, Birkhäuser, 1990 Google Scholar
  42. Leroy X, Vouillon J, Doligez D et al (1996) The objective caml system. Available on the web,
  43. Makhorin A (2004) GNU linear programming kit reference manual version 4.8. Department of Applied Informatics, Moscow Aviation Institute Google Scholar
  44. Mi H, Lazareva-Ulitsky B, Loo R, Kejariwal A, Vandergriff J, Rabkin S, Guo N, Muruganujan A, Doremieux O, Campbell MJ, Kitano H, Thomas PD (2005) The PANTHER database of protein families, subfamilies, functions and pathways. Nucleic Acids Res 33(Supplement 1):284–288 Google Scholar
  45. Niedermeier R (2006) Invitation to fixed-parameter algorithms. Oxford University Press, Oxford zbMATHCrossRefGoogle Scholar
  46. Oda K, Kitano H (2006) A comprehensive map of the toll-like receptor signaling network. Mol Syst Biol 2:2006.0015 CrossRefGoogle Scholar
  47. Oda K, Kimura T, Matsuoka Y, Funahashi A, Muramatsu M, Kitano H (2004) Molecular interaction map of a macrophage. AfCS Res Rep 2:14 Google Scholar
  48. Papadimitriou CH, Yannakakis M (1991) Optimization, approximation, and complexity classes. J Comput Syst Sci 43(3):425–440 zbMATHCrossRefMathSciNetGoogle Scholar
  49. Polzin T, Vahdati Daneshmand S (2006) Practical partitioning-based methods for the Steiner problem. In: Proceedings of the 4th international workshop on experimental and efficient algorithms (WEA ’06). LNCS, vol 4007. Springer, Berlin, pp 241–252 Google Scholar
  50. Razgon I, O’Sullivan B (2008) Almost 2-SAT is fixed-parameter tractable. In: Proceedings of the 35th international colloquium on automata, languages and programming (ICALP ’08). LNCS, vol 5125. Springer, Berlin, pp 551–562 CrossRefGoogle Scholar
  51. Reed B, Smith K, Vetta A (2004) Finding odd cycle transversals. Oper Res Lett 32(4):299–301 zbMATHCrossRefMathSciNetGoogle Scholar
  52. Sturmfels B (1996) Gröbner bases and convex polytopes. University lecture series, vol 8. American Mathematical Society, Providence zbMATHGoogle Scholar
  53. Thagard P, Verbeurgt K (1998) Coherence as constraint satisfaction. Cogn Sci 22(1):1–24 CrossRefGoogle Scholar
  54. Volz E (2004) Random networks with tunable degree distribution and clustering. Phys Rev E 70(5):056115 CrossRefGoogle Scholar
  55. Wernicke S (2003) On the algorithmic tractability of single nucleotide polymorphism (SNP) analysis and related problems. Diplomarbeit, Wilhelm-Schickard-Institut für Informatik, Universität Tübingen Google Scholar
  56. Yannakakis M (1981) Edge-deletion problems. SIAM J Comput 10(2):297–309 zbMATHCrossRefMathSciNetGoogle Scholar
  57. Zaslavsky T (1998) Bibliography of signed and gain graphs. Electronic Journal of Combinatorics DS8, updated version available at

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Falk Hüffner
    • 1
  • Nadja Betzler
    • 1
  • Rolf Niedermeier
    • 1
  1. 1.Institut für InformatikFriedrich-Schiller-Universität JenaJenaGermany

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