The maximum ratio clique problem (original) (raw)

Abstract

This paper introduces a fractional version of the classical maximum weight clique problem, the maximum ratio clique problem, which is to find a maximal clique that has the largest ratio of benefit and cost weights associated with the clique’s vertices. NP-completeness of the decision version of the problem is established, and three solution methods are proposed. The results of numerical experiments with standard graph instances, as well as with real-life instances arising in finance and energy systems, are reported.

Access this article

Log in via an institution

Subscribe and save

• Starting from 10 chapters or articles per month
• Access and download chapters and articles from more than 300k books and 2,500 journals
• Cancel anytime View plans

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

• Billionnet A (2002) Approximation algorithms for fractional knapsack problems. Oper Res Lett 30(5): 336–342
  Google Scholar
• Boginski V, Butenko S, Pardalos P (2006) Mining market data: a network approach. Comput Oper Res 33:3171–3184
  Article Google Scholar
• Bomze IM, Budinich M, Pardalos PM, Pelillo M (1999) The maximum clique problem. In: Du DZ, Pardalos PM (eds) Handbook of combinatorial optimization. Kluwer Academic Publishers, Dordrecht, pp 1–74
  Chapter Google Scholar
• Chandrasekaran R (1977) Minimal ratio spanning trees. Networks 7(4):335–342
  Article Google Scholar
• Chvátal V, Slater P (1993) A note on well-covered graphs. Ann Discrete Math 55:179–182
  Article Google Scholar
• Dantzig GB, Blattner WO, Rao MR (1967) Finding a cycle in a graph with minimum cost to time ratio with application to a ship routing problem. In: Rosentlehl P (ed) Theory of graphs. Gordon and Breach, New York, pp 77–84
  Google Scholar
• Dasdan A, Gupta R (1998) Faster maximum and minimum mean cycle algorithms for system-performance analysis. IEEE Trans Comput Aided Des Integr Circ Syst 17(10):889–899
  Article Google Scholar
• Dinkelbach W (1967) On nonlinear fractional programming. Manag Sci 13(7):492–498
  Article Google Scholar
• Elhedhli S (2005) Exact solution of a class of nonlinear knapsack problems. Oper Res Lett 33:615–624
  Article Google Scholar
• Fox B (1969) Finding minimal cost-time ratio circuits. Oper Res 17(3):546–551
  Article Google Scholar
• Garey MR, Johnson DS (1979) Computers and Intractability: a Guide to the theory of NP-completeness. W.H. Freeman and Company, New York
  Google Scholar
• Ibaraki T (1983) Parametric approaches to fractional programs. Math Program 26:345–362
  Article Google Scholar
• Isbell JR, Marlow WH (1956) Attrition games. Naval Res Logist Q 3(1–2):71–94
  Article Google Scholar
• Karp RM (1978) A characterization of the minimum cycle mean in a digraph. Discrete Math 23(3):309–311
  Article Google Scholar
• Lawler EL (1976) Combinatorial optimization: networks and matroids. Holt, Rinehart and Winston, New York
• Luce R, Perry A (1949) A method of matrix analysis of group structure. Psychometrika 14:95–116
  Article Google Scholar
• Megiddo N (1979) Combinatorial optimization with rational objective functions. Math Oper Res 4(4):414–424
  Article Google Scholar
• NREL: transmission grid integration - data and resources (2012). http://www.nrel.gov/electricity/transmission/data_resources.html. Accessed October 2013
• Orlin JB, Ahuja RK (1992) New scaling algorithms for the assignment and minimum mean cycle problems. Math Program 54:41–56
  Article Google Scholar
• Picard JC, Queyranne M (1982) A network flow solution to some nonlinear 0–1 programming problems, with applications to graph theory. Networks 12:141–159
  Article Google Scholar
• Prokopyev O, Meneses C, Oliveira C, Pardalos P (2005) On multiple-ratio hyperbolic 0–1 programming problems. Pac J Optim 1(2):327–345
  Google Scholar
• Prokopyev OA, Huang H, Pardalos PM (2005) On complexity of unconstrained hyperbolic 0–1 programming problems. Oper Res Lett 33(3):312–318
  Article Google Scholar
• Radzik T (1998) Fractional combinatorial optimization. In: Du DZ, Pardalos P (eds) Handbook of combinatorial optimization, vol 1. Kluwer Academic Publishers, pp 429–478
• Radzik T (1992) Newton’s method for fractional combinatorial optimization. In: Proceedings of 33rd annual symposium on foundations of computer science, pp 659–669
• Sankaranarayana R, Stewart L (1992) Complexity results for well-covered graphs. Networks 22:247–262
  Article Google Scholar
• Shigeno M, Saruwatari Y, Matsui T (1995) An algorithm for fractional assignment problems. Discrete Appl Math 56:333–343
  Article Google Scholar
• Skiscim C, Palocsay S (2004) The complexity of minimum ratio spanning tree problems. J Glob Optim 30:335–346
  Article Google Scholar
• Tawarmalani M, Ahmed S, Sahinidis NV (2002) Global optimization of 0–1 hyperbolic programs. J Glob Optim 24:385–416
  Article Google Scholar
• Ursulenko O, Butenko S, Prokopyev O (2013) A global optimization algorithm for solving the minimum multiple ratio spanning tree problem. J Glob Optim 56:1029–1043
  Article Google Scholar
• Wiser R, Lantz E, Bolinger M, Hand M (2012) Recent developments in the levelized cost of energy from us wind power projects. http://eetd.lbl.gov/ea/emp/reports/wind-energy-costs-2-2012.pdf. Accessed October 2013
• Wu T (1997) A note on a global approach for general 0–1 fractional programming. Eur J Oper Res 101(1):220–223
  Article Google Scholar

Download references

Acknowledgments

This research was partially supported by the US Department of Energy Grant DE-SC0002051 and US Air Force Office of Scientific Research Award No. FA9550-12-1-0103. The authors gratefully acknowledge the comments by two referees.

Author information

Authors and Affiliations

1. Department of Industrial and Systems Engineering, Texas A&M University, College Station, TX, 77843-3131, USA
   Samyukta Sethuraman & Sergiy Butenko

Authors

1. Samyukta Sethuraman
2. Sergiy Butenko

Corresponding author

Correspondence toSamyukta Sethuraman.

Rights and permissions

About this article

Cite this article

Sethuraman, S., Butenko, S. The maximum ratio clique problem.Comput Manag Sci 12, 197–218 (2015). https://doi.org/10.1007/s10287-013-0197-z

Download citation

• Received: 12 June 2013
• Accepted: 09 October 2013
• Published: 07 November 2013
• Issue date: January 2015
• DOI: https://doi.org/10.1007/s10287-013-0197-z

Keywords

Mathematics Subject Classification