Weighted Fractional and Integral K-matching in Hypergraphs

Abstract We consider the problem of finding polynomial-time approximations of maximal weighted k -matchings in a hypergraph and investigate the relationship between the integral and fractional maxima of the corresponding 0–1 integer linear program and its LP-relaxation. We extend results of Raghavan, who gave a deterministic approximation algorithm for unweighted k -matching, to the weighted case and compare the so obtained lower bound for the ratio of the integer and fractional maximum with a lower bound of Aharoni et al. (1985) and Alon et al. (1992).

[1]  N. Kuzyurin.  On the relationship between the optima of linear and integer linear programming , 1992 .

[2]  Anand Srivastav,et al.  Algorthmic Chernoff-Hoeffding Inequalitiers in Integer Programming , 1994, ISAAC.

[3]  Kenneth Steiglitz,et al.  Combinatorial Optimization: Algorithms and Complexity , 1981 .

[4]  Zoltán Füredi,et al.  On the fractional matching polytope of a hypergraph , 1993, Comb..

[5]  J. Spencer.  Ten lectures on the probabilistic method , 1987 .

[6]  Paul Erdös,et al.  Dual integer linear programs and the relationship between their optima , 1985, STOC '85.

[7]  László Lovász,et al.  On the ratio of optimal integral and fractional covers , 1975, Discret. Math..

[8]  Leslie G. Valiant,et al.  Fast probabilistic algorithms for hamiltonian circuits and matchings , 1977, STOC '77.

[9]  Anand Srivastav,et al.  Integer Multicommodity Flows with Reduced Demands , 1993, ESA.

[10]  Bonnie Berger,et al.  Simulating (logcn)-wise independence in NC , 1991, JACM.

[11]  Kurt Mehlhorn,et al.  Data Structures and Algorithms 1: Sorting and Searching , 2011, EATCS Monographs on Theoretical Computer Science.

[12]  Noga Alon,et al.  The Probabilistic Method , 2015, Fundamentals of Ramsey Theory.

[13]  Prabhakar Raghavan,et al.  Randomized rounding: A technique for provably good algorithms and algorithmic proofs , 1985, Comb..

[14]  Paul Erdös,et al.  On a Combinatorial Game , 1973, J. Comb. Theory A.

[15]  József Beck,et al.  An Algorithmic Approach to the Lovász Local Lemma. I , 1991, Random Struct. Algorithms.

[16]  Moni Naor,et al.  The Probabilistic Method Yields Deterministic Parallel Algorithms , 1994, J. Comput. Syst. Sci..

[17]  L. Lovász,et al.  Geometric Algorithms and Combinatorial Optimization , 1981 .

[18]  Zoltán Füredi,et al.  Maximum degree and fractional matchings in uniform hypergraphs , 1981, Comb..

[19]  A. Srivastav,et al.  Algorithmic Chernoo-hoeeding Inequalities in Integer Programming Zz , 1996 .

[20]  Noga Alon,et al.  A Parallel Algorithmic Version of the Local Lemma , 1991, Random Struct. Algorithms.