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Separators in Graphs

  • Reference work entry
  • First Online:
Encyclopedia of Algorithms

  • 184 Accesses

Years and Authors of Summarized Original Work

  • 1998; Leighton, Rao

  • 1999; Leighton, Rao

Problem Definition

The (balanced) separator problem asks for a cut of minimum (edge)-weight in a graph, such that the two shores of the cut have approximately equal (node)-weight.

Formally, given an undirected graph \( G=(V,E) \), with a nonnegative edge-weight function \( c:E\to\mathbb{R}_+ \), a nonnegative node-weight function \( \pi:V\to\mathbb{R}_+ \), and a constant \( b\leq 1/2 \), a cut \( (S:V\setminus S) \) is said to be b-balanced, or a \( (b,1-b) \)-separator, if \( b\pi(V)\leq \pi(S)\leq (1-b)\pi(V) \)\( ( \)where \( \pi(S) \) stands for \( \sum \nolimits_{v\in S} \pi(v) \)\( ) \).

Problem 1 (b-balanced separator)

Input: Edge- and node-weighted graph \( G=(V,E,c,\pi) \), constant \( b\leq 1/2 \).

Output: A b-balanced cut \( (S:V\setminus S) \). Goal: minimize the edge weight \( c(\delta(S)) \).

Closely related is the product sparsest cut problem.

Problem 2 ((Product) Sparsest cut)

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Recommended Reading

  1. Agrawal A, Klein PN, Ravi R (1993) Cutting down on fill using nested dissection: provably good elimination orderings. In: Brualdi RA, Friedland S, Klee V (eds) Graph theory and sparse matrix computation. IMA volumes in mathematics and its applications. Springer, New York, pp 31–55

    Chapter  Google Scholar 

  2. Arora S, Hazan E, Kale S (2004) O (√logn) approximation to sparsest cut in Õ (n2) time. In: FOCS '04: proceedings of the 45th annual IEEE symposium on foundations of computer science (FOCS'04). IEEE Computer Society, Washington, pp 238–247

    Google Scholar 

  3. Arora S, Kale S (2007) A combinatorial, primal-dual approach to semidefinite programs. In: STOC '07: proceedings of the 39th annual ACM symposium on theory of computing. ACM, pp 227–236

    Google Scholar 

  4. Arora S, Lee JR, Naor A (2005) Euclidean distortion and the sparsest cut. In: STOC '05: proceedings of the thirty-seventh annual ACM symposium on theory of computing. ACM, New York, pp 553–562

    Google Scholar 

  5. Arora S, Rao S, Vazirani U (2004) Expander flows, geometric embeddings and graph partitioning. In: STOC '04: proceedings of the thirty-sixth annual ACM symposium on theory of computing. ACM, New York, pp 222–231

    Google Scholar 

  6. Aumann Y, Rabani Y (1998) An (log) approximate min-cut maxflow theorem and approximation algorithm. SIAM J Comput 27(1):291–301

    Article  MathSciNet  MATH  Google Scholar 

  7. Benczúr AA, Karger DR (1996) Approximating s-t minimum cuts in Õ (n2) time. In: STOC '96: proceedings of the twenty-eighth annual ACM symposium on theory of computing. ACM, New York, pp 47–55

    Google Scholar 

  8. Bhatt SN, Leighton FT (1984) A framework for solving vlsi graph layout problems. J Comput Syst Sci 28(2):300–343

    Article  MathSciNet  MATH  Google Scholar 

  9. Bodlaender HL, Gilbert JR, Hafsteinsson H, Kloks T (1995) Approximating treewidth, pathwidth, frontsize, and shortest elimination tree. J Algorithms 18(2):238–255

    Article  MathSciNet  MATH  Google Scholar 

  10. Bourgain J (1985) On Lipshitz embedding of finite metric spaces in Hilbert space. Isr J Math 52:46–52

    Article  MathSciNet  MATH  Google Scholar 

  11. Devanur NR, Khot SA, Saket R, Vishnoi NK (2006) Integrality gaps for sparsest cut and minimum linear arrangement problems. In: STOC '06: proceedings of the thirty-eighth annual ACM symposium on theory of computing. ACM, New York, pp 537–546

    Google Scholar 

  12. Even G, Naor JS, Rao S, Schieber B (2000) Divide-and-conquer approximation algorithms via spreading metrics. J ACM 47(4):585–616

    Article  MathSciNet  MATH  Google Scholar 

  13. Feige U, Krauthgamer R (2002) A polylogarithmic approximation of the minimum bisection. SIAM J Comput 31(4):1090–1118

    Article  MathSciNet  MATH  Google Scholar 

  14. Fleischer L (2000) Approximating fractional multicommodity flow independent of the number of commodities. SIAM J Discret Math 13(4):505–520

    Article  MathSciNet  MATH  Google Scholar 

  15. Garg N, Könemann J (1998) Faster and simpler algorithms for multicommodity flow and other fractional packing problems. In: FOCS '98: proceedings of the 39th annual symposium on foundations of computer science. IEEE Computer Society, Washington, p 300

    Google Scholar 

  16. Karakostas G (2002) Faster approximation schemes for fractional multicommodity flow problems. In: SODA '02: proceedings of the thirteenth annual ACM-SIAM symposium on discrete algorithms. Society for Industrial and Applied Mathematics, Philadelphia, pp 166–173

    Google Scholar 

  17. Khandekar R, Rao S, Vazirani U (2006) Graph partitioning using single commodity flows. In: STOC '06: proceedings of the thirty-eighth annual ACM symposium on theory of computing. ACM, New York, pp 385–390

    Google Scholar 

  18. Khot S, Vishnoi NK (2005) The unique games conjecture, integrality gap for cut problems and embeddability of negative type metrics into l1. In: FOCS '07: proceedings of the 46th annual IEEE symposium on foundations and computer science. IEEE Computer Society, pp 53–62

    Google Scholar 

  19. Klein PN, Plotkin SA, Stein C, Tardos É (1994) Faster approximation algorithms for the unit capacity concurrent flow problem with applications to routing and finding sparse cuts. SIAM J Comput 23(3):466–487

    Article  MathSciNet  MATH  Google Scholar 

  20. Krauthgamer R, Rabani Y (2006) Improved lower bounds for embeddings into l1. In: SODA '06: proceedings of the seventeenth annual ACM-SIAM symposium on discrete algorithm. ACM, New York, pp 1010–1017

    Google Scholar 

  21. Lang K, Rao S (1993) Finding near-optimal cuts: an empirical evaluation. In: SODA '93: proceedings of the fourth annual ACM SIAM symposium on discrete algorithms. Society for Industrial and Applied Mathematics, Philadelphia, pp 212–221

    Google Scholar 

  22. Leighton FT, Makedon F, Plotkin SA, Stein C, Stein É, Tragoudas S (1995) Fast approximation algorithms for multicommodity flow problems. J Comput Syst Sci 50(2):228–243

    Article  MathSciNet  MATH  Google Scholar 

  23. Leighton T, Rao S (1988) An approximate max-flow min-cut theorem for uniform multicommodity flow problems with applications to approximation algorithms. In: Proceedings of the 29th annual symposium on foundations of computer science. IEEE Computer Society, Washington, DC, pp 422–431

    Google Scholar 

  24. Leighton T, Rao S (1999) Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms. J ACM 46(6):787–832

    Article  MathSciNet  MATH  Google Scholar 

  25. Leong T, Shor P, Stein C (1991) Implementation of a combinatorial multicommodity flow algorithm. In: Johnson DS, McGeoch CC (eds) Network flows and matching. DIMACS series in discrete mathematics and theoretical computer science, vol 12, AMS, Providence, pp 387–406

    Google Scholar 

  26. Linial N, London E, Rabinovich Y (1995) The geometry of graphs and some of its algorithmic applications. Combinatorica 15(2):215–245

    Article  MathSciNet  MATH  Google Scholar 

  27. Shahrokhi F, Matula DW (1990) The maximum concurrent flow problem. J ACM 37(2):318–334

    Article  MathSciNet  MATH  Google Scholar 

  28. Shmoys DB (1997) Cut problems and their applications to divide-and-conquer. In: Hochbaum DS (ed) Approximation algorithms for NP-hard problems. PWS Publishing Company, pp 192–235

    Google Scholar 

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Authors and Affiliations

  1. Department of Computer Science and Engineering, Arizona State University, Tempe, AZ, USA

    Goran Konjevod

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  1. Goran Konjevod
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Correspondence to Goran Konjevod .

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Editors and Affiliations

  1. Department of Electrical Engineering and Computer Science, Northwestern University, Evanston, IL, USA

    Ming-Yang Kao

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Konjevod, G. (2016). Separators in Graphs. In: Kao, MY. (eds) Encyclopedia of Algorithms. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-2864-4_362

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