This paper addresses the problems of minimizing nonnegative submodular functions under covering constraints, which generalize the vertex cover, edge cover, and set cover problems. We give approximation algorithms for these problems exploiting the discrete convexity of submodular functions. We first present a rounding 2-approximation algorithm for the submodular vertex cover problem based on the half-integrality of the continuous relaxation problem, and show that the rounding algorithm can be performed by one application of submodular function minimization on a ring family. We also show that a rounding algorithm and a primal-dual algorithm for the submodular cost set cover problem are both constant factor approximation algorithms if the maximum frequency is fixed. In addition, we give an essentially tight lower bound on the approximability of the submodular edge cover problem.

We consider an online combinatorial prediction problem where each combinatorial concept is represented as a vertex of a polyhedron described by a submodular function (base polyhedron). In general, there are exponentially many vertices in the base polyhedron. We propose polynomial time algorithms with regret bounds. In particular, for cardinalitybased submodular functions, we give O(n 2 )-time algorithms.

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Cognitive radio technology improves radio resource usage by reconfiguring the wireless connection settings according to the optimum decisions, which are made on the basis of the collected context information. This paper focuses on optimization algorithms for decision making to optimize radio resource usage in heterogeneous cognitive wireless networks. For networks with centralized management, we proposed a novel optimization algorithm whose solution is guaranteed to be exactly optimal. In order to avoid an exponential increase of computational complexity in largescale wireless networks, we model the target optimization problem as a minimum cost-flow problem and find the solution of the problem in polynomial time. For the networks with decentralized management, we propose a distributed algorithm using the distributed energy minimization dynamics of the Hopfield-Tank neural network. Our algorithm minimizes a given objective function without any centralized calculation.We derive the decision-making rule for each terminal to optimize the entire network. We demonstrate the validity of the proposed algorithms by several numerical simulations and the feasibility of the proposed schemes by designing and implementing them on experimental cognitive radio network systems.

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