A Tight Deterministic Algorithm for the Submodular Multiple Knapsack Problem

Submodular function maximization has been a central topic in the theoretical computer science community over the last decade. Plenty of well-performing approximation algorithms have been designed for the maximization of (monotone or non-monotone) submodular functions over a variety of constraints. In this paper, we consider the submodular multiple knapsack problem (SMKP), which is the submodular version of the well-studied multiple knapsack problem (MKP). Roughly speaking, the problem asks to maximize a monotone submodular function over multiple bins (knapsacks). Recently, Fairstein et al. (ESA20) presented a tight $(1-1/e-\epsilon)$-approximation randomized algorithm for SMKP. Their algorithm is based on the continuous greedy technique which inherently involves randomness. However, the deterministic algorithm of this problem has not been understood very well previously. In this paper, we present a tight $(1-1/e-\epsilon)$ deterministic algorithm for SMKP. Our algorithm is based on reducing SMKP to an exponential-size submodular maximizaion problem over a special partition matroid which enjoys a tight deterministic algorithm. We develop several techniques to mimic the algorithm, leading to a tight deterministic approximation for SMKP

Approximability of Monotone Submodular Function Maximization under Cardinality and Matroid Constraints in the Streaming Model

Maximizing a monotone submodular function under various constraints is a classical and intensively studied problem. However, in the single-pass streaming model, where the elements arrive one by one and an algorithm can store only a small fraction of input elements, there is much gap in our knowledge, even though several approximation algorithms have been proposed in the literature. In this work, we present the first lower bound on the approximation ratios for cardinality and matroid constraints that beat $1-\frac{1}{e}$ in the single-pass streaming model. Let $n$ be the number of elements in the stream. Then, we prove that any (randomized) streaming algorithm for a cardinality constraint with approximation ratio $\frac{2}{2+\sqrt{2}}+\varepsilon$ requires $\Omega\left(\frac{n}{K^2}\right)$ space for any $\varepsilon>0$, where $K$ is the size limit of the output set. We also prove that any (randomized) streaming algorithm for a (partition) matroid constraint with approximation ratio $\frac{K}{2K-1}+\varepsilon$ requires $\Omega\left(\frac{n}{K}\right)$ space for any $\varepsilon>0$, where $K$ is the rank of the given matroid. In addition, we give streaming algorithms when we only have a weak oracle with which we can only evaluate function values on feasible sets. Specifically, we show weak-oracle streaming algorithms for cardinality and matroid constraints with approximation ratios $\frac{K}{2K-1}$ and $\frac{1}{2}$, respectively, whose space complexity is exponential in $K$ but is independent of $n$. The former one exactly matches the known inapproximability result for a cardinality constraint in the weak oracle model. The latter one almost matches our lower bound of $\frac{K}{2K-1}$ for a matroid constraint, which almost settles the approximation ratio for a matroid constraint that can be obtained by a streaming algorithm whose space complexity is independent of $n$