Streaming algorithm for balance gain and cost with cardinality constraint on the integer lattice
Jingjing Tan
TL;DR
The paper tackles maximizing $g(\boldsymbol{x}) - c(\boldsymbol{x})$ under a cardinality constraint on the integer lattice, where $g$ is a monotone nonnegative submodular function on $\mathbb{N}^{E}$ and $c$ is a nonnegative linear cost. It proposes bicriteria streaming algorithms that combine thresholding with lattice binary search to produce scalable online solutions for both submodular and $\alpha$-weakly submodular objectives. For the submodular case, the authors derive a tunable approximation ratio dependent on parameters $t$, $\mu$, and $\nu$, with memory $O(k \log k / \varepsilon)$ and per-element queries $O(\log K)$; for the $\alpha$-weakly submodular case, they obtain a distinct bicriteria ratio $( \frac{\alpha}{1 + \alpha + \mu - \nu}, \frac{1 + \alpha}{1 + \alpha + \mu - \nu} )$ and memory $O(k \log (k/\alpha)/\varepsilon)$. The results enable efficient, near-optimal decision-making for lattice-based team formation and related online allocation problems under knapsack-like constraints. The work also provides detailed analyses and extends prior lattice-submodular streaming results to non-submodular settings.
Abstract
Team formation problem is a very important problem in the labor market, and it is proved to be NP-hard. In this paper, we design an efficient bicriteria streaming algorithms to construct a balance between gain and cost in a team formation problem with cardinality constraint on the integer lattice. To solve this problem, we establish a model for maximizing the difference between a nonnegative normalized monotone submodule function and a nonnegative linear function. Further, we discuss the case where the first function of the object function is $α$--weakly submodular. Combining the lattice binary search with the threshold method, we present an online algorithm called bicriteria streaming algorithms. Meanwhile, we give detailed analysis for both of these models.