Abstract
We consider the problem of finding a fixed number of vertex-disjoint cliques of given sizes in a complete undirected weighted graph so that the total weight of vertices and edges in the cliques would be minimal. We show that the problem is strongly NP-hard both in the general case and in two subclasses, which have important applications. An approximation algorithm for this problem is presented. We show that the algorithm finds a solution with a bounded approximation ratio for the considered subclasses of the problem, and the bound is attainable. In the case when the number of cliques to be found is fixed in advance (i.e., is a parameter), the time complexity of the algorithm is polynomial.
Similar content being viewed by others
References
R. Burkard, M. Dell’Amico, and S. Martello, Assignment Problems (SIAM, Philadelphia, 2009).
J. B. J. M. De Kort, “Lower bounds for symmetric K-peripatetic salesman problems,” Optimization 22(1), 113–122 (1991).
E. A. Dinits and M. A. Kronrod, “One algorithm for solving the assignment problem,” Dokl. Akad Nauk SSSR 189(1), 23–25 (1969).
M. R. Garey and D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness (Freeman, San Fransisco, 1979).
E. Kh. Gimadi, “Approximation efficient algorithms with performance guarantees for some hard routing problems,” in Optimization and Applications: Proceedings of the Second International Conference, Petrovac, Montenegro, 2011, pp. 98–101.
J. Håstad, “Clique is hard to approximate within n 1 − ε,” Acta Math. 182(1), 105–142 (1999).
P. Kleinschmidt and H. Schannath, “A strongly polynomial algorithm for the transportation problem,” Math. Programming 68(1), Ser. A, 1–13 (1995).
J. Krarup, “The peripatetic salesman and some related unsolved problems,” in Combinatorial Programming: Methods and Applications, Ed. by B. Roy (Reidel, Dordrecht, 1975), NATO Advanced Study Institutes Series, Vol. 19, pp. 173–178.
K. Park, K. Lee, and S. Park, “An extended formulation approach to the edge-weighted maximal clique problem,” European J. Oper. Res. 95, 671–682 (1996).
R. C. Prim, “Shortest connection networks and some generalizations,” Bell System Technical J. 36(6), 1389–1401 (1957).
J. Roskind and R. E. Tarjan, “A note on finding minimum-cost edge-disjoint spanning trees,” Math. Oper. Res. 10(4), 701–708 (1985).
F. C. R. Spieksma, “Multi-index assignment problems: Complexity, approximation, applications,” in Nonlinear Assignment Problems: Algorithms and Applications, Ed. by L. Pitsoulis and P. Pardalos (Kluwer, Dordrecht, 2000), Ser. Combinatorial Optimization, Vol. 7, pp. 1–12.
The Traveling Salesman Problem and Its Variations, Ed. by G. Gutin and A. P. Punnen (Kluwer, Dordrecht, 2002), Ser. Combinatorial Optimization, Vol. 12.
A. E. Baburin and E. Kh. Gimadi, “On the asymptotic optimality of an algorithm for solving the maximum m-PSP in a multidimensional Euclidean space,” Proc. Steklov Inst. Math. 272(Suppl. 1), S1–S13 (2011).
A. E. Galashov and A. V. Kel’manov, “A 2-approximate algorithm to solve one problem of the family of disjoint vector subsets,” Autom. Remote Control 75(4), 595–606 (2014).
E. Kh. Gimadi, Yu. V. Glazkov, and A. N. Glebov, “Approximation algorithms for solving the 2-peripatetic salesman problem on a complete graph with edge weights 1 and 2,” J. Appl. Ind. Math. 3(1), 46–60 (2007).
I. I. Eremin, E. Kh. Gimadi, A. V. Kel’manov, A. V. Pyatkin, and M. Yu. Khachai, “2-Approximation algorithm for finding a clique with minimum weight of vertices and edges,” Proc. Steklov Inst. Math. 284(Suppl. 1), S87–S95 (2014).
A. V. Kel’manov and A. V. Pyatkin, “NP-completeness of some problems of choosing a vector subset,” J. Appl. Ind. Math. 5(3), 352–357 (2011).
A. V. Kel’manov and S. M. Romanchenko, “An approximation algorithm for solving a problem of search for a vector subset,” J. Appl. Ind. Math. 6(4), 90–96 (2012).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © E.Kh.Gimadi, A.V.Kel’manov, A.V. Pyatkin, M.Yu. Khachai, 2014, published in Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2014, Vol. 20, No. 2.
Rights and permissions
About this article
Cite this article
Gimadi, E.K., Kel’manov, A.V., Pyatkin, A.V. et al. Efficient algorithms with performance guarantees for some problems of finding several cliques in a complete undirected weighted graph. Proc. Steklov Inst. Math. 289 (Suppl 1), 88–101 (2015). https://doi.org/10.1134/S0081543815050089
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0081543815050089