Submodular Function Maximization in Parallel via the Multilinear Relaxation

Abstract

Balkanski and Singer [4] recently initiated the study of adaptivity (or parallelism) for constrained submodular function maximization, and studied the setting of a cardinality constraint. Subsequent improvements for this problem by Balkanski, Rubinstein, and Singer [6] and Ene and Nguyen [21] resulted in a near-optimal (1−1/e−ǫ)-approximation in O(log n/ǫ 2 ) rounds of adaptivity. Partly motivated by the goal of extending these results to more general constraints, we describe parallel algorithms for approximately maximizing the multilinear relaxation of a monotone submodular function subject to packing constraints. Formally our problem is to maximize F (x) over x ∈ [0, 1] n subject to Ax ≤ 1 where F is the multilinear relaxation of a monotone submodular function. Our algorithm achieves a near-optimal (1 − 1/e − ǫ)-approximation in O(log 2 m log n/ǫ 4 ) rounds where n is the cardinality of the ground set and m is the number of packing constraints. For many constraints of interest, the resulting fractional solution can be rounded via known randomized rounding schemes that are oblivious to the specific submodular function. We thus derive randomized algorithms with poly-logarithmic adaptivity for a number of constraints including partition and laminar matroids, matchings, knapsack constraints, and their intersections.Our algorithm takes a continuous view point and combines several ideas ranging from the continuous greedy algorithm of [38,13], its adaptation to the MWU framework for packing constraints [19], and parallel algorithms for packing LPs [31,41]. For the basic setting of cardinality constraints, this viewpoint gives rise to an alternative, simple to understand algorithm that matches recent results [6,21]. Our algorithm to solve the multilinear relaxation is deterministic if it is given access to a value oracle for the multilinear extension and its gradient; this is possible in some interesting cases such as the coverage function of an explicitly given set system.

Parallelizing greedy for submodular set function maximization in matroids and beyond

“…The first condition, (1.A), says we do not sample past the point where the margins are decreasing by a lot. A simpler form appears in previous work in the monotone cardinality setting [21]. The second condition (1.B), is a more significant departure from the cardinality setting, and strikes a balance between trying to span many elements, and not having to prune too many of the sampled elements.…”

“…We can both search for the appropriate value of δ and sample with probability δ in parallel. The basic idea of greedy sampling is directly inspired by a much simpler greedy sampling procedure in the cardinality setting in our previous work [21]. 1 We now iterate along greedy blocks, where each iteration is w/r/t the residual system induced by previously selected greedy blocks.…”

“…A number of follow up works have extended these results to knapsack and packing constraints [21,30] and nonnegative submodular functions [5,33,30], or improved other aspects such as the total number of oracle calls [32]. One could argue that all of these papers essentially build upon the understanding and analysis for the cardinality constraint (even the ones which address significantly more general packing constraints [21,30]). One classical setting that is important to address is submodular set function maximization subject to an arbirary matroid constraint.…”

An optimal approximation for submodular maximization under a matroid constraint in the adaptive complexity model

“…Functions with bounded curvature have also been studied using adaptive sampling under a cardinality constraint [BS18b]. The more general family of packing constraints, which includes partition and laminar matroids, has been considered in [CQ19]. In particular, under m packing constraints, a 1 − 1/e − ǫ approximation was obtained in O(log 2 m log n) rounds using a combination of continuous optimization and multiplicative weight update techniques.…”

Unconstrained submodular maximization with constant adaptive complexity

“…The approximation guarantee of [4] was very quickly improved in several independent works [3,20,23] to 1 − 1 /e − ε (using O(ε −2 log n) adaptive rounds), which almost matches an impossibility result by [42] showing that no polynomial time algorithm can achieve (1 − 1 /e + ε)-approximation for the problem, regardless of the amount of adaptivity it uses. It should be noted also that [23] manages to achieve the above parameters while keeping the query complexity linear in n. An even more recent line of work studies algorithms with low adaptivity for more general submodular maximization problems, which includes problems with non-monotone objective functions and/or constraints beyond the cardinality constraint [15,21,22]. Since all these results achieve constant approximation for problems generalizing the maximization of a monotone submodular function subject to a cardinality constraint, they all inherit the impossibility result of [4], and thus, use at least Ω(log n) adaptive rounds.…”