Maximizing Symmetric Submodular Functions

Symmetric submodular functions are an important family of submodular functions capturing many interesting cases including cut functions of graphs and hypergraphs. Maximization of such functions subject to various constraints receives little attention by current research, unlike similar minimization problems which have been widely studied. In this work, we identify a few submodular maximization problems for which one can get a better approximation for symmetric objectives than the state of the art approximation for general submodular functions. We first consider the problem of maximizing a non-negative symmetric submodular function $f\colon 2^\mathcal{N} \to \mathbb{R}^+$ subject to a down-monotone solvable polytope $\mathcal{P} \subseteq [0, 1]^\mathcal{N}$. For this problem we describe an algorithm producing a fractional solution of value at least $0.432 \cdot f(OPT)$, where $OPT$ is the optimal integral solution. Our second result considers the problem $\max \{f(S) : |S| = k\}$ for a non-negative symmetric submodular function $f\colon 2^\mathcal{N} \to \mathbb{R}^+$. For this problem, we give an approximation ratio that depends on the value $k / |\mathcal{N}|$ and is always at least $0.432$. Our method can also be applied to non-negative non-symmetric submodular functions, in which case it produces $1/e - o(1)$ approximation, improving over the best known result for this problem. For unconstrained maximization of a non-negative symmetric submodular function we describe a deterministic linear-time $1/2$-approximation algorithm. Finally, we give a $[1 - (1 - 1/k)^{k - 1}]$-approximation algorithm for Submodular Welfare with $k$ players having identical non-negative submodular utility functions, and show that this is the best possible approximation ratio for the problem.Comment: 31 pages, an extended abstract appeared in ESA 201

Randomized Composable Core-sets for Distributed Submodular Maximization

An effective technique for solving optimization problems over massive data sets is to partition the data into smaller pieces, solve the problem on each piece and compute a representative solution from it, and finally obtain a solution inside the union of the representative solutions for all pieces. This technique can be captured via the concept of {\em composable core-sets}, and has been recently applied to solve diversity maximization problems as well as several clustering problems. However, for coverage and submodular maximization problems, impossibility bounds are known for this technique \cite{IMMM14}. In this paper, we focus on efficient construction of a randomized variant of composable core-sets where the above idea is applied on a {\em random clustering} of the data. We employ this technique for the coverage, monotone and non-monotone submodular maximization problems. Our results significantly improve upon the hardness results for non-randomized core-sets, and imply improved results for submodular maximization in a distributed and streaming settings. In summary, we show that a simple greedy algorithm results in a $1/3$-approximate randomized composable core-set for submodular maximization under a cardinality constraint. This is in contrast to a known $O({\log k\over \sqrt{k}})$ impossibility result for (non-randomized) composable core-set. Our result also extends to non-monotone submodular functions, and leads to the first 2-round MapReduce-based constant-factor approximation algorithm with $O(n)$ total communication complexity for either monotone or non-monotone functions. Finally, using an improved analysis technique and a new algorithm $\mathsf{PseudoGreedy}$, we present an improved $0.545$-approximation algorithm for monotone submodular maximization, which is in turn the first MapReduce-based algorithm beating factor $1/2$ in a constant number of rounds

Constrained Non-Monotone Submodular Maximization: Offline and Secretary Algorithms

Constrained submodular maximization problems have long been studied, with near-optimal results known under a variety of constraints when the submodular function is monotone. The case of non-monotone submodular maximization is less understood: the first approximation algorithms even for the unconstrainted setting were given by Feige et al. (FOCS '07). More recently, Lee et al. (STOC '09, APPROX '09) show how to approximately maximize non-monotone submodular functions when the constraints are given by the intersection of p matroid constraints; their algorithm is based on local-search procedures that consider p-swaps, and hence the running time may be n^Omega(p), implying their algorithm is polynomial-time only for constantly many matroids. In this paper, we give algorithms that work for p-independence systems (which generalize constraints given by the intersection of p matroids), where the running time is poly(n,p). Our algorithm essentially reduces the non-monotone maximization problem to multiple runs of the greedy algorithm previously used in the monotone case. Our idea of using existing algorithms for monotone functions to solve the non-monotone case also works for maximizing a submodular function with respect to a knapsack constraint: we get a simple greedy-based constant-factor approximation for this problem. With these simpler algorithms, we are able to adapt our approach to constrained non-monotone submodular maximization to the (online) secretary setting, where elements arrive one at a time in random order, and the algorithm must make irrevocable decisions about whether or not to select each element as it arrives. We give constant approximations in this secretary setting when the algorithm is constrained subject to a uniform matroid or a partition matroid, and give an O(log k) approximation when it is constrained by a general matroid of rank k.Comment: In the Proceedings of WINE 201