Deterministic (1/2 + ε)-Approximation for Submodular Maximization over a Matroid

We study the problem of maximizing a monotone submodular function subject to a matroid constraint and present a deterministic algorithm that achieves (1/2 + {\epsilon})-approximation for the problem. This algorithm is the first deterministic algorithm known to improve over the 1/2-approximation ratio of the classical greedy algorithm proved by Nemhauser, Wolsely and Fisher in 1978.

[1]  Laurence A. Wolsey,et al.  Best Algorithms for Approximating the Maximum of a Submodular Set Function , 1978, Math. Oper. Res..

[2]  Morteza Zadimoghaddam,et al.  Online Submodular Welfare Maximization: Greedy Beats 1/2 in Random Order , 2015, STOC.

[3]  Roy Schwartz,et al.  A Tight Approximation for Submodular Maximization with Mixed Packing and Covering Constraints , 2018, ICALP.

[4]  Alexander Schrijver,et al.  Combinatorial optimization. Polyhedra and efficiency. , 2003 .

[5]  Niv Buchbinder,et al.  Constrained Submodular Maximization via a Non-symmetric Technique , 2016, Math. Oper. Res..

[6]  M. L. Fisher,et al.  An analysis of approximations for maximizing submodular set functions—I , 1978, Math. Program..

[7]  Niv Buchbinder,et al.  Deterministic Algorithms for Submodular Maximization Problems , 2016, SODA.

[8]  Niv Buchbinder,et al.  Online submodular maximization: beating 1/2 made simple , 2018, IPCO.

[9]  Jan Vondrák,et al.  Maximizing a Monotone Submodular Function Subject to a Matroid Constraint , 2011, SIAM J. Comput..

[10]  Roy Schwartz,et al.  Comparing Apples and Oranges: Query Trade-off in Submodular Maximization , 2017, Math. Oper. Res..

[11]  Andreas Krause,et al.  Lazier Than Lazy Greedy , 2014, AAAI.

[12]  Jan Vondrák,et al.  Fast algorithms for maximizing submodular functions , 2014, SODA.

[13]  Vahab S. Mirrokni,et al.  Maximizing Nonmonotone Submodular Functions under Matroid or Knapsack Constraints , 2009, SIAM J. Discret. Math..

[14]  Yuval Filmus.  Monotone Submodular Maximization over a Matroid , 2013 .

[15]  Joseph Naor,et al.  A Tight Linear Time (1/2)-Approximation for Unconstrained Submodular Maximization , 2015, SIAM J. Comput..

[16]  Douglas R. Woodall.  An exchange theorem for bases of matroids , 1974 .

[17]  R. Brualdi.  Comments on bases in dependence structures , 1969, Bulletin of the Australian Mathematical Society.

[18]  Curtis Greene,et al.  A multiple exchange property for bases , 1973 .

[19]  Joseph Naor,et al.  Submodular Maximization with Cardinality Constraints , 2014, SODA.

[20]  U. Feige,et al.  Maximizing Non-monotone Submodular Functions , 2011 .

[21]  Jan Vondrák,et al.  Submodular maximization by simulated annealing , 2010, SODA '11.