In recent years, the issue of maximizing submodular functions has attracted much interest from research communities. However, most are specified in a set function. Meanwhile, advancements have been studied for diminishing return (DR-submodular) function on integer lattice. Because plenty publications show that DR-submodular wide applications optimization problems such as sensor placement impose...

Some situations concerning cost allocation are formulated as combinatorial optimization games. We consider a minimum coloring game and a minimum vertex cover game. For a minimum coloring game, Deng{Ibaraki{Nagamochi 1] showed that deciding the core nonemptiness of a given minimum coloring game is NP-complete, which implies that a good characterization of balanced minimum coloring games is unlik...

We present an optimal, combinatorial 1−1/e approximation algorithm for monotone submodular optimization over a matroid constraint. Compared to the continuous greedy algorithm (Calinescu, Chekuri, Pál and Vondrák, 2008), our algorithm is extremely simple and requires no rounding. It consists of the greedy algorithm followed by local search. Both phases are run not on the actual objective functio...

Submodular functions are a broad class of set functions, which naturally arise in diverse areas such as economics, operations research and game theory. Many algorithms have been suggested for the maximization of these functions, achieving both strong theoretical guarantees and good practical performance. Unfortunately, once the function deviates from submodularity (even slightly), the known alg...

“Discrete Convex Analysis” is aimed at establishing a novel theoretical framework for solvable discrete optimization problems by means of a combination of the ideas in continuous optimization and combinatorial optimization. The theoretical framework of convex analysis is adapted to discrete settings and the mathematical results in matroid/submodular function theory are generalized. Viewed from ...

We study discrete optimization problems with a submodular mean-risk minimization objective. For 0-1 problems a linear characterization of the convex lower envelope is given. For mixed 0-1 problems we derive an exponential class of conic quadratic inequalities. We report computational experiments on risk-averse capital budgeting problems with uncertain returns.

Consider a situation in which one has a suboptimal solution S to a maximization problem which only constitutes a weak approximation to the problem. Suppose that even though the value of S is small compared to an optimal solution OPT to the problem, S happens to be structurally similar to OPT . A natural question to ask in this scenario is whether there is a way of improving the value of S based...

With the extensive application of submodularity, its generalizations are constantly being proposed. However, most of them are tailored for special problems. In this paper, we focus on quasi-submodularity, a universal generalization, which satisfies weaker properties than submodularity but still enjoys favorable performance in optimization. Similar to the diminishing return property of submodula...

We study the problem of maximizing a monotone submodular function subject to Multiple Knapsack constraint. The input is set I items, each has non-negative weight, and bins arbitrary capacities. Also, we are given submodular, f over subsets items. objective find packing subset items A⊆I in such that f(A) maximized. Our main result an almost optimal polynomial time (1−e−1−ε)-approximation algorit...

We consider the problem of learning sparse representations of data sets, where the goal is to reduce a data set in manner that optimizes multiple objectives. Motivated by applications of data summarization, we develop a new model which we refer to as the two-stage submodular maximization problem. This task can be viewed as a combinatorial analogue of representation learning problems such as dic...