The submodular system k-partition problem is a problem of partitioning a given finite set V into k non-empty subsets V1, V2, . . . , V k so thatis minimized where f is a non-negative submodular function on V , and k is a fixed integer. This problem contains the hypergraph k-cut problem. In this paper, we design the first exact algorithm for k = 3 and approximation algorithms for k ≥ 4. We also analyze the approximation factor for the hypergraph k-cut problem.

Given an undirected graph on a node set V and positive integers k and m, a k-connected m-dominating set ((k, m)-CDS) is defined as a subset S of V such that each node in V \ S has at least m neighbors in S, and a k-connected subgraph is induced by S. The weighted (k, m)-CDS problem is to find a minimum weight (k, m)-CDS in a given node-weighted graph. The problem is called the unweighted (k, m)-CDS problem if the objective is to minimize the cardinality of a (k, m)-CDS. These problems have been actively studied for unit disk graphs, motivated by the application of constructing a virtual backbone in a wireless ad hoc network. In this paper, we consider the case in which k ≤ m, and we present a simple O(k5 k )-approximation algorithm for the unweighted (k, m)-CDS problem, and a primal-dual O(k 2 log k)-approximation algorithm for the weighted (k, m)-CDS problem.A CDS does not give a fault-tolerant virtual backbone network. This is because a CDS is only required to be connected, and each node outside a CDS is required to have only one neighbor in the CDS. Hence, if a backbone node fails, the virtual backbone network may be disconnected, or a nonbackbone node may lose access to the virtual backbone network. To overcome this disadvantage, Dai and Wu [10] proposed replacing a CDS by a k-connected k-dominating set, and they addressed the problem of finding a minimum k-connected k-dominating set in a unit disk graph. For a graph with the node set V , a subset S of V is called k-connected if the subgraph induced by S is kconnected (i.e., it is connected even if any k − 1 nodes are removed), and is called k-dominating if each node v ∈ V \ S has k neighbors in S. Triggered by their study, much attention has been paid to this problem, extending the notion of a k-connected k-dominating set to a more-general k-connected m-dominating set ((k, m)-CDS).The problem of finding a minimum cardinality (k, m)-CDS in a unit disk graph is called the unweighted (k, m)-CDS problem. If each node is given a nonnegative weight, and the objective is to minimize the weight of a (k, m)-CDS, then this is called the weighted (k, m)-CDS problem. As for the unweighted (k, m)-CDS problem, several constant-approximation algorithms were given for k ≤ 3 [21,22,26,27,30]. As for the weighted (k, m)-CDS problem, there are several constantapproximation algorithms for k = m = 1 [2, 31], but no approximation algorithm was known for the case of (k, m) = (1, 1) before our study (see Section 2 for more literature reviews).After these previous studies, a natural question arises as to whether there is a constantapproximation algorithm for the unweighted (k, m)-CDS problem with k ≥ 4, and for the weighted problem with (k, m) = (1, 1). For the unweighted problem, this question has been already addressed in both [26] and [27].

Abstract. We consider the problem of finding a minimum edge cost subgraph of a graph satisfying both given node-connectivity requirements and degree upper bounds on nodes. We present an iterative rounding algorithm of the biset LP relaxation for this problem. For directed graphs and k-out-connectivity requirements from a root, our algorithm computes a solution that is a 2-approximation on the cost, and the degree of each node v in the solution is at most 2b(v) + O(k) where b(v) is the degree upper bound on v. For undirected graphs and element-connectivity requirements with maximum connectivity requirement k, our algorithm computes a solution that is a 4-approximation on the cost, and the degree of each node v in the solution is at most 4b(v) + O(k). These ratios improve the previous O(log k)-approximation on the cost and O(2 k b(v)) approximation on the degrees. Our algorithms can be used to improve approximation ratios for other nodeconnectivity problems such as undirected k-out-connectivity, directed and undirected k-connectivity, and undirected rooted k-connectivity and subset k-connectivity.

We formulate a new stochastic submodular maximization problem by introducing the performance-dependent costs of items. In this problem, we consider selecting items for the case where the performance of each item (i.e., how much an item contributes to the objective function) is decided randomly, and the cost of an item depends on its performance. The goal of the problem is to maximize the objective function subject to a budget constraint on the costs of the selected items. We present an adaptive algorithm for this problem with a theoretical guaran-√ tee that its expected objective value is at least (1−1/ 4 e)/2 times the maximum value attained by any adaptive algorithms. We verify the performance of the algorithm through numerical experiments.

Abstract. We consider the problem of finding a minimum edge cost subgraph of a graph satisfying both given node-connectivity requirements and degree upper bounds on nodes. We present an iterative rounding algorithm of the biset LP relaxation for this problem. For directed graphs and k-out-connectivity requirements from a root, our algorithm computes a solution that is a 2-approximation on the cost, and the degree of each node v in the solution is at most 2b(v) + O(k) where b(v) is the degree upper bound on v. For undirected graphs and element-connectivity requirements with maximum connectivity requirement k, our algorithm computes a solution that is a 4-approximation on the cost, and the degree of each node v in the solution is at most 4b(v) + O(k). These ratios improve the previous O(log k)-approximation on the cost and O(2 k b(v)) approximation on the degrees. Our algorithms can be used to improve approximation ratios for other nodeconnectivity problems such as undirected k-out-connectivity, directed and undirected k-connectivity, and undirected rooted k-connectivity and subset k-connectivity.