Guess Free Maximization of Submodular and Linear Sums

Abstract

We consider the problem of maximizing the sum of a monotone submodular function and a linear function subject to a general solvable polytope constraint. Recently, Sviridenko et al. [16] described an algorithm for this problem whose approximation guarantee is optimal in some intuitive and formal senses. Unfortunately, this algorithm involves a guessing step which makes it less clean and significantly affects its time complexity. In this work we describe a clean alternative algorithm that uses a novel weighting technique in order to avoid the problematic guessing step while keeping the same approximation guarantee as the algorithm of [16].

An Efficient Framework for Balancing Submodularity and Cost

“…Existing hardness results imply that no multiplicative approximation guarantees are possible in polynomial time for maximizing a potentially negative submodular function with or without constraints [13,31]. 1 Nevertheless, the objective function we consider has some structure that has been exploited in previous works [14,17,34]. These works have shown that in this case we should aim for a weaker notion of approximation and find a solution Q satisfying…”

“…One of the main downsides of these algorithms is that the running time can be prohibitive. The works [14,34] propose algorithms based on the continuous greedy algorithm that maximizes the multilinear extension, a continuous function extending the submodular function to the domain [0, 1] n . The multilinear extension is expensive to evaluate and the continuous greedy algorithm requires many iterations to converge.…”

“…The algorithm of [14] is based on the standard discrete greedy algorithm, which is much more efficient, but it applies the greedy approach to a distorted objective that changes throughout the algorithm and we cannot use techniques such as lazy evaluations to speed up the algorithm. Thus the running time of the algorithm of [14] is Θ(n 2 ), where n is the size of the ground set, whereas the standard greedy algorithm can be implemented to run in nearly-linear time using approximate lazy evaluations. Moreover, the implementation of the standard greedy algorithm using exact lazy evaluations achieves significant speedups in practice without affecting the approximation guarantee [27].…”

“…Our main insight is very simple but very effective: instead of maximizing the original objective f (Q) − c(Q), we maximize a scaled objective f (Q) − s • c(Q), where s > 1 is an absolute constant. Unlike the approach of [14], our objective does not change throughout the algorithm and thus we can use lazy evaluations. Moreover, we can leverage a wide-range of existing algorithmic approaches, such as the standard greedy algorithm in the offline setting and variants of greedy in the online setting.…”

“…The constant addition of data in online platforms makes this setting increasingly relevant. Algorithmic contributions: The structure of the combined objective in Equation (1) in the offline setting and under the cardinality constraint has been addressed by previous works [14,17] as well. What distinguishes us from these works is that we intentionally design algorithms with slightly weaker theoretical approximation guarantees that allow the use of runtime acceleration techniques, such as lazy greedy evaluations.…”