Submodular Maximization subject to a Knapsack Constraint: Combinatorial Algorithms with Near-optimal Adaptive Complexity

TL;DR

This work tackles maximizing a (potentially non-monotone) submodular function under a knapsack constraint with a focus on low adaptivity and few value evaluations. It introduces SampleSeq and ThreshSeq, culminating in ParKnapsack, a combinatorial algorithm achieving a $9.465+\varepsilon$-approximation with $O(\log n)$ adaptive rounds and $O(n^2\log^2 n)$ queries, and a near-linear-query variant with $O(\log^2 n)$ adaptivity. The paper extends to monotone objectives and cardinality constraints, providing improved constants (e.g., $3+O(\varepsilon)$ for monotone knapsack) and a unified framework to interpolate adaptivity and query cost via a $\gamma$-ThreshSeq mechanism. By balancing high-value sampling with budget-aware prefixes, the approach yields sublinear adaptivity while maintaining strong approximation guarantees, making it practical for massive instances. The results close the adaptivity gap for knapsack-constrained submodular maximization and offer versatile trade-offs for real-world applications that require parallelizable optimization with limited queries.

Abstract

Submodular maximization is a classic algorithmic problem with multiple applications in data mining and machine learning; there, the growing need to deal with massive instances motivates the design of algorithms balancing the quality of the solution with applicability. For the latter, an important measure is the adaptive complexity, which captures the number of sequential rounds of parallel computation needed by an algorithm to terminate. In this work we obtain the first constant factor approximation algorithm for non-monotone submodular maximization subject to a knapsack constraint with near-optimal $O(\log n)$ adaptive complexity. Low adaptivity by itself, however, is not enough: a crucial feature to account for is represented by the total number of function evaluations (or value queries). Our algorithm asks $\tilde{O}(n^2)$ value queries, but can be modified to run with only $\tilde{O}(n)$ instead, while retaining a low adaptive complexity of $O(\log^2n)$. Besides the above improvement in adaptivity, this is also the first combinatorial approach with sublinear adaptive complexity for the problem and yields algorithms comparable to the state-of-the-art even for the special cases of cardinality constraints or monotone objectives.

Theorems & Definitions (43)

• Definition 1: Adaptive complexity
• Definition 2: Query complexity
• Lemma 1
• Lemma 2: Sampling Lemma
• Lemma 3
• Lemma 4
• proof
• Lemma 5
• proof
• Lemma 6