Abstract
We investigate a natural combinatorial optimization problem called the Label Cut problem. Given an input graph G with a source s and a sink t, the edges of G are classified into different categories, represented by a set of labels. The labels may also have weights. We want to pick a subset of labels of minimum cardinality (or minimum total weight), such that the removal of all edges with these labels disconnects s and t. We give the first non-trivial approximation and hardness results for the Label Cut problem. Firstly, we present an \(O(\sqrt{m})\) -approximation algorithm for the Label Cut problem, where m is the number of edges in the input graph. Secondly, we show that it is NP-hard to approximate Label Cut within \(2^{\log ^{1-1/\log\log^{c}n}n}\) for any constant c<1/2, where n is the input length of the problem. Thirdly, our techniques can be applied to other previously considered optimization problems. In particular we show that the Minimum Label Path problem has the same approximation hardness as that of Label Cut, simultaneously improving and unifying two known hardness results for this problem which were previously the best (but incomparable due to different complexity assumptions).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Agarwal A, Alon N, Charikar M (2007) Improved approximation for directed cut problems. In: Proceedings of the 39th annual ACM symposium on theory of computing (STOC), pp 671–680
Arora S, Lund C (1997) Hardness of approximation. In: Hochbaum D (ed) Approximation algorithms for NP-hard problems. PWS, Boston, pp 399–446
Arora S, Babai L, Stern J, Sweedyk Z (1997) The hardness of approximate optima in lattices, codes, and systems of linear equations. J Comput Syst Sci 54(2):317–331
Broersma H, Li X (1997) Spanning trees with many or few colors in edge-colored graphs. Discuss Math Graph Theory 17(2):259–269
Broersma H, Li X, Woeginger G, Zhang S (2005) Paths and cycles in colored graphs. Austral J Comb 31:299–311
Bruggemann T, Monnot J, Woeginger GJ (2003) Local search for the minimum label spanning tree problem with bounded color classes. Oper Res Lett 31(3):195–201
Carr R, Doddi S, Konjevod G, Marathe M (2000) On the red-blue set cover problem. In: Proceedings of the 11th annual ACM-SIAM symposium on discrete algorithms (SODA), pp 345–353
Chang R-S, Leu S-J (1997) The minimum labeling spanning trees. Inf Process Lett 63(5):277–282
Chawla S, Krauthgamer R, Kumar R, Rabani Y, Sivakumar D (2005) On the hardness of approximating multicut and sparsest-cut. In: Proceedings of IEEE conference on computational complexity, pp 144–153
Chuzhoy J, Khanna S (2007) Polynomial flow-cut gaps and hardness of directed cut problems. In: Proceedings of the 39th annual ACM symposium on theory of computing (STOC), pp 179–188
Coudert D, Datta P, Perennes S, Rivano H, Voge M-E (2007) Shared risk resource group: complexity and approximability issues. Parallel Process Lett 17(2):169–184
Couetoux B, Gourves L, Monnot J, Telelis O (2008) On labeled traveling salesman problems. In: Proceedings of the 19th international symposium on algorithms and computation (ISAAC). LNCS, vol 5369. Springer, Berlin, pp 776–787
Dinur I, Safra S (2004) On the hardness of approximating label cover. Inf Process Lett 89(5):247–254
Dinur I, Fischer E, Kindler G, Raz R, Safra S (1999) PCP characterizations of NP: towards a polynomially-small error-probability. In: Proceedings of the 31st annual ACM symposium on theory of computing (STOC), pp 29–40
Garg N, Vazirani V, Yannakakis M (1996) Approximate max-flow min-(multi)cut theorems and their applications. SIAM J Comput 25:235–251
Garg N, Vazirani V, Yannakakis M (1997) Primal-dual approximation algorithms for integral flow and multicut in trees. Algorithmica 18:3–20
Hassin R, Monnot J, Segev D (2007) Approximation algorithms and hardness results for labeled connectivity problems. J Comb Optim 14(4):437–453
Jha S, Sheyner O, Wing JM (2002) Two formal analyses of attack graphs. In: Proceedings of the 15th IEEE computer security foundations workshop, Nova Scotia, Canada, June 2002, pp 49–63
Karger D, Klein P, Stein C, Thorup M, Young N (1999) Rounding algorithms for a geometric embedding of minimum multiway cut. In: Proceedings of the 31st annual ACM symposium on theory of computing (STOC), pp 668–678
Khot S, Vishnoi N (2005) The unique games conjecture, integrability gap for cut problems and the embeddability of negative type metrics into l 1. In: Proceedings of the 46th annual IEEE symposium on foundations of computer science (FOCS), pp 53–62
Khuller S, Moss A, Naor J (1999) The budgeted maximum coverage problem. Inf Process Lett 70(1):39–45
Krumke S, Wirth H (1998) On the minimum label spanning tree problem. Inf Process Lett 66(2):81–85
Lund C, Yannakakis M (1994) On the hardness of approximating minimization problems. J ACM 41(5):960–981
Maffioli F, Rizzi R, Benati S (2007) Least and most colored bases. Discrete Appl Math 155(15):1958–1970
Monnot J (2005) The labeled perfect matching in bipartite graphs. Inf Process Lett 96(3):81–88
Saran H, Vazirani V (1995) Finding k-cuts within twice the optimal. SIAM J Comput 24:101–108
Sheyner O, Wing JM (2004) Tools for generating and analyzing attack graphs. In: Proceedings of workshop on formal methods for components and objects, pp 344–371
Sheyner O, Haines J, Jha S, Lippmann R, Wing JM (2002) Automated generation and analysis of attack graphs. In: Proceedings of the IEEE symposium on security and privacy, Oakland, CA, May 2002, pp 273–284
Wirth H (2001) Multicriteria approximation of network design and network upgrade problems. PhD thesis, Department of Computer Science, Würzburg University, 2001
Xiong Y, Golden B, Wasil E (2005) Worst-case behavior of the MVCA heuristic for the minimum labeling spanning tree problem. Oper Res Lett 33(1):77–80
Author information
Authors and Affiliations
Corresponding author
Additional information
Peng Zhang is supported by NSFC 60325206 and China Postdoctoral Science Foundation No. 20080441144. Jin-Yi Cai is supported by NSF CCF-0511679. Lin-Qing Tang is supported by NSFC 60325206 and NSFC 60310213.
Rights and permissions
About this article
Cite this article
Zhang, P., Cai, JY., Tang, LQ. et al. Approximation and hardness results for label cut and related problems. J Comb Optim 21, 192–208 (2011). https://doi.org/10.1007/s10878-009-9222-0
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10878-009-9222-0