In this paper, we consider the fault-tolerant facility location problem with submodular penalties that is a common generalization of the fault-tolerant facility location problem and the facility location problem with submodular penalties. For this problem, we present a combinatorial 3 ·HR-approximation algorithm, where R is the maximum connectivity requirement and HR is the R-th harmonic number...

Submodular functions are the discrete ananlogue of convexity and generalize many known and well studied settings. They capture several economic principles such as decreasing marginal cost and correlated utilities. Multi-agent allocation problems over submodular functions arise in a variety of natural scenarios and have been the subject of extensive study over the last decade. In this proposal I...

Recently, the first combinatorial strongly polynomial algorithms for submodular function minimization have been devised independently by Iwata, Fleischer, and Fujishige and by Schrijver. In this paper, we improve the running time of Schrijver’s algorithm by designing a push-relabel framework for submodular function minimization (SFM). We also extend this algorithm to carry out parametric minimi...

This paper presents the current fastest known weakly polynomial algorithm for the submodular flow problem when the costs are not too big. It combines Röck’s or Bland and Jensen’s cost scaling algorithms, Cunningham and Frank’s primal-dual algorithm for submodular flow, and Fujishige and Zhang’s push-relabel algorithm for submodular maximum flow to get a running time of O(nh logC), where n is th...

All the functions we consider are set functions defined over subsets of a ground set N . Definition 1. A function f : 2 → R is monotone iff: ∀S ⊆ T ⊆ N, f(S) ≤ f(T ) Definition 2. For f : 2 → R and S ⊆ N , the marginal contribution to S is the function fS defined by: ∀T ⊆ N, fS(T ) = f(S ∪ T )− f(S) When there is no ambiguity, we write fS(e) instead of fS({e}) for e ∈ N , S + e instead of S ∪ {...

We show that any submodular minimization (SM) problem defined on linear constraint set with constraints having up to two variables per inequality, are 2-approximable in polynomial time. If the constraints are monotone (the two variables appear with opposite sign coefficients) then the problems of submodular minimization or supermodular maximization are polynomial time solvable. The key idea is ...

Many problems in Machine Learning can be modeled as submodular optimization problems. Recent work has focused on stochastic or adaptive versions of these problems. We consider the Scenario Submodular Cover problem, which is a counterpart to the Stochastic Submodular Cover problem studied by Golovin and Krause (2011). In Scenario Submodular Cover, the goal is to produce a cover with minimum expe...

We consider the submodular function minimization (SFM) and the quadratic minimization problems regularized by the Lovász extension of the submodular function. These optimization problems are intimately related; for example, min-cut problems and total variation denoising problems, where the cut function is submodular and its Lovász extension is given by the associated total variation. When a qua...

Submodular set-functions have many applications in combinatorial optimization, as they can be minimized and approximately maximized in polynomial time. A key element in many of the algorithms and analyses is the possibility of extending the submodular set-function to a convex function, which opens up tools from convex optimization. Submodularity goes beyond set-functions and has naturally been ...

A. Huber and V. Kolmogorov (ISCO 2012) introduced a concept of k-submodular function as a generalization of ordinary submodular (set) functions and bisubmodular functions. They presented a min-max relation for the k-submodular function minimization by considering l1 norm, which requires a non-convex set of feasible solutions associated with the k-submodular function. Our approach overcomes the ...