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...tion algorithm for the original integral problem using the solution for the fractional relaxation (Section 3.3). Let F :[0, 1] n → R+, thefractional relaxation of f, be the ‘extension-by-expectation’ =-=[7]-=-, F (x) = X f(S) · Πi∈Sxi · Πj∈S(1 − xj). S⊆V Note that F is a multi-linear polynomial in variables x1, ··· ,xn, and has continuous derivatives of all orders. Furthermore, ∂ asshowninVondrák [51], fo...

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..., Wolsey and Fisher [3], which has shown an ee−1 -approximation for maximizing monotone submodular set functions over a uniform matroid; and this bound is known to be tight even for a general matroid =-=[47]-=-. Our max-sum diversification problem with monotone submodular set functions can be viewed as an extension of that problem: the objective function now not only contains a submodular part, but also has...

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...in our problem is that no constraint can be violated at all. There is a large body of work on constrained maximization of submodular functions; we only cite the relevant papers here. Calinescu et al. =-=[10]-=- introduced a continuous relaxation (called the multi-linear extension or extensionby-expectation) of submodular functions and subsequently Vondrák [29] gave e e−1 an elegant -approximation algorithm ...

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.... Another line of work deals with maximizing normalized monotone submodular functions, again, subject to various combinatorial constraints. A continuous greedy algorithm was given by Calinescu et al. =-=[1]-=- for maximizing a normalized monotone submodular function subject to a matroid constraint. Later, Lee et al. [12] gave a local search algorithm achieving 1/p − ε approximation for maximizing such func...

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...rithm terminates without exceeding any knapsack constraint, we have fSρ(O≥ρ) ≤ ((1 + )p+ ) · f(Sρ). The proof of this claim is essentially the analysis of the greedy algorithm for a p-system, as in =-=[2]-=-. We supply the proof for completeness. Proof. Consider the elements of Sρ in the order they are added by the algorithm: denote them by e1, e2, . . . , er. For i ∈ {1, . . . , r}, we define a set Ai ⊆...

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... maximization of a monotone submodular function under various combinatorial constraints (such as a matroid, the intersection of several matroids and knapsack constraints) has also been widely studied =-=[7, 8, 21, 23, 25]-=-. Thus, it comes as no surprise that recent works have combined the secretary problem with submodular optimization. Gupta et al. [17] were the first to consider this combination. For the variant where...

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...transparent and results in a better bound. Our approach can also be used to get a slightly better ratio than [15] for the Steiner forest problem on trees. Constrained submodular maximization problems =-=[17,7,23,3,25]-=- appear very relevant at first sight: e.g., the k-max-min version of min-cut (“find the k sources whose separation from the sink costs the most”) is precisely submodular maximization under a cardinali...