ResQue Greedy: Rewiring Sequential Greedy for Improved Submodular Maximization

TL;DR

ResQue Greedy addresses submodular maximization under a cardinality constraint by introducing curvature-aware rewiring of the sequential greedy path. The method defines set curvature $\gamma(S|A)$ and expansion curvature $\gamma_e(S)$, and uses a trigger law to detect when rewiring is potentially beneficial, followed by a step-back policy to modify the current solution path. Theoretical guarantees maintain the standard worst-case bound $1-e^{-1}$, while practical performance improves as demonstrated in coverage problems, with only modest additional computational overhead. This curvature-driven rewiring framework provides a practical, scalable approach to elevate greedy-based submodular maximization in resource allocation tasks.

Abstract

This paper introduces Rewired Sequential Greedy (ResQue Greedy), an enhanced approach for submodular maximization under cardinality constraints. By integrating a novel set curvature metric within a lattice-based framework, ResQue Greedy identifies and corrects suboptimal decisions made by the standard sequential greedy algorithm. Specifically, a curvature-aware rewiring strategy is employed to dynamically redirect the solution path, leading to improved approximation performance over the conventional sequential greedy algorithm without significantly increasing computational complexity. Numerical experiments demonstrate that ResQue Greedy achieves tighter near-optimality bounds compared to the traditional sequential greedy method.

Figures (4)

• Figure 1: Hasse diagram for a ground set of five elements $\mathcal{P}=\{1,2,3,4,5\}$. In a), the paths created by the sequential greedy algorithm (arrows) and the observed solutions of the sequential greedy algorithm (dashed). In b), all the possible paths that lead to the optimal solution $\mathcal{S}^\star_4=\{1,2,3,4\}$ and in c), an example of the rewiring operation over the lattice where blue lines stand for the paths created by the sequential greedy algorithm (arrows) and the observed solutions of the sequential greedy algorithm (dashed) and the red lines are the edge added by the rewiring operation at stage $3$.
• Figure 2: A multi-sensor deployment problem: the possible allocation points are shown by bigger black dots, while the information points are shown by blue dots. In this simple case, we can see that ResQue Greedy's rewiring at the last step by dropping the second deployment location chosen by the sequential greedy algorithm and retrying to choose another element leads to an improvement: the sequential greedy solution obtains a coverage of $520$ features collected, whereas ResQue Greedy yields a greater coverage of $657$ features.
• Figure 3: Multi-sensor deployment: $100$ Monte-Carlo simulation.
• Figure 4: Feature points (blue dots) and potential deployment points (black $\star$). Agent deployments: standard sequential greedy (red), $\textsf{ResQue Greedy}$ allocations (orange), and random rewiring greedy allocations (green). All sensors are homogenous with the same circular range. To visualize overlapping deployments using our three different algorithms, sensor ranges are depicted with slightly varying radii for each algorithm. Note the $\textsf{ResQue Greedy}$ resolves some of the overlapped deployments, such as those two in the lower left corner.

Theorems & Definitions (13)

• Definition 1: Total curvature of a submodular function MC-GC:84
• Definition 2: Set curvature
• Lemma III.1
• proof
• Definition 3: Expansion and path curvatures
• Lemma III.2: Properties of expansion and path curvatures
• proof
• Theorem IV.1: Optimality gap of ResQue Greedy
• proof
• Definition 4: Trigger law