We present a polynomial time algorithm for online maximization of @math -submodular maximization. For online (nonmonotone) @math -submodular maximization, our algorithm achieves a tight approximate factor in an approximate regret. For online monotone @math -submodular maximization, our approximate-regret matches to the best-known approximation ratio, which is tight asymptotically as @math tends to infinity. Our approach is based on the Blackwell approachability theorem and online linear optimization.
This paper presents a polynomial-time @math -approximation algorithm for maximizing nonnegative @math -submodular functions. This improves upon the previous @math -approximation by Ward and Z ivn 'y (SODA'14), where @math . We also show that for monotone @math -submodular functions there is a polynomial-time @math -approximation algorithm while for any @math a @math -approximation algorithm for maximizing monotone @math -submodular functions would require exponentially many queries. In particular, our hardness result implies that our algorithms are asymptotically tight. We also extend the approach to provide constant factor approximation algorithms for maximizing skew-bisubmodular functions, which were recently introduced as generalizations of bisubmodular functions. We consider the maximization problem in the value oracle model of functions defined on k-tuples of sets that are submodular in every orthant and r-wise monotone, where k g 2 and 1 l r l k. We give an analysis of a deterministic greedy algorithm that shows that any such function can be approximated to a factor of 1 (1 + r). For r e k, we give an analysis of a randomized greedy algorithm that shows that any such function can be approximated to a factor of 1 (1+sk 2. In the case of k e r e 2, the considered functions correspond precisely to bisubmodular functions, in which case we obtain an approximation guarantee of 1 2. We show that, as in the case of submodular functions, this result is the best possible both in the value query model and under the assumption that NP ≠ RP. Extending a result of , we show that for any k g 3, submodularity in every orthant and pairwise monotonicity (i.e., r e 2) precisely characterize k-submodular functions. Consequently, we obtain an approximation guarantee of 1 3 (and thus independent of k) for the maximization problem of k-submodular functions.  Bisubmodularity extends the concept of submodularity to set functions with two arguments. We show how bisubmodular maximization leads to richer value-of-information problems, using examples in sensor placement and feature selection. We present the first constant-factor approximation algorithm for a wide class of bisubmodular maximizations. Submodularity is one of the most important properties of combinatorial optimization, and k-submodularity is a generalization of submodularity. Maximization of a k-submodular function is NP-hard, and approximation algorithms have been studied. Most of algorithms use randomization and achieve the approximation ratio as the expected value. For unconstrained submodular maximization, [Buchbinder and Feldman 2016] gave a derandomization scheme, and showed that randomness is not necessary for obtaining optimal results. In this paper, we extend their scheme for unconstrained k-submodular maximization and give two deterministic algorithms for k-submodular maximization. The first is for monotone k-submodular functions. We derandomize (k (2k-1) )-approximation algorithm [Iwata, Tanigawa, and Yoshida 2016], and achieve the same approximation ratio. The second is for nonmonotone functions with (k 2 ). It achieves (k (3k-2) )-approximation, and it is better than a simple greedy algorithm [Ward and Živný 2016] which achieves 1 3-approximation for (k 3 ). A k-submodular function is a generalization of a submodular function, where the input consists of k disjoint subsets, instead of a single subset, of the domain. Many machine learning problems, including influence maximization with k kinds of topics and sensor placement with k kinds of sensors, can be naturally modeled as the problem of maximizing monotone k-submodular functions. In this paper, we give constant-factor approximation algorithms for maximizing monotone k-submodular functions subject to several size constraints. The running time of our algorithms are almost linear in the domain size. We experimentally demonstrate that our algorithms outperform baseline algorithms in terms of the solution quality.