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An Exact Solver for Submodular Knapsack Problems

Sabine Münch, Stephen Raach

TL;DR

This work tackles Submodular Knapsack-Constrained Maximization, formalizing the problem as maximizing a monotone submodular function $f$ under a knapsack constraint $B$ with element weights $w_i$. It develops an exact branch-and-bound solver that uses a fractional knapsack based bound and greedy node ordering, augmented by three acceleration techniques: Lazy Evaluations, Early Pruning, and Candidate Reduction. The approach is evaluated on artificial and real-world instances, comparing against Sakaue–Ishihata (2018) solvers and a Gurobi-based branch-and-cut, with the LE^A+EP+CR combination delivering the best performance across most benchmarks. The results demonstrate practical feasibility of exact solutions for general-weight submodular knapsack problems and point to parallelization and instance-specific pruning as fruitful directions for further improvements.

Abstract

We study the problem of maximizing a monotone increasing submodular function over a set of weighted elements subject to a knapsack constraint. Although this problem is NP-hard, many applications require exact solutions, as approximate solutions are often insufficient in practice. To address this need, we propose an exact branch-and-bound algorithm tailored for the submodular knapsack problem and introduce several acceleration techniques to enhance its efficiency. We evaluate these techniques on artificial instances of three benchmark problems as well as on instances derived from real-world data. We compare the proposed solver with two solvers by Sakaue and Ishihata (2018), which currently achieve the strongest performance reported in the literature, as well as with a branch-and-cut algorithm implemented using Gurobi that solves a binary linear reformulation of the submodular knapsack problem, demonstrating that our methods are highly successful.

An Exact Solver for Submodular Knapsack Problems

TL;DR

This work tackles Submodular Knapsack-Constrained Maximization, formalizing the problem as maximizing a monotone submodular function under a knapsack constraint with element weights . It develops an exact branch-and-bound solver that uses a fractional knapsack based bound and greedy node ordering, augmented by three acceleration techniques: Lazy Evaluations, Early Pruning, and Candidate Reduction. The approach is evaluated on artificial and real-world instances, comparing against Sakaue–Ishihata (2018) solvers and a Gurobi-based branch-and-cut, with the LE^A+EP+CR combination delivering the best performance across most benchmarks. The results demonstrate practical feasibility of exact solutions for general-weight submodular knapsack problems and point to parallelization and instance-specific pruning as fruitful directions for further improvements.

Abstract

We study the problem of maximizing a monotone increasing submodular function over a set of weighted elements subject to a knapsack constraint. Although this problem is NP-hard, many applications require exact solutions, as approximate solutions are often insufficient in practice. To address this need, we propose an exact branch-and-bound algorithm tailored for the submodular knapsack problem and introduce several acceleration techniques to enhance its efficiency. We evaluate these techniques on artificial instances of three benchmark problems as well as on instances derived from real-world data. We compare the proposed solver with two solvers by Sakaue and Ishihata (2018), which currently achieve the strongest performance reported in the literature, as well as with a branch-and-cut algorithm implemented using Gurobi that solves a binary linear reformulation of the submodular knapsack problem, demonstrating that our methods are highly successful.

Paper Structure

This paper contains 18 sections, 3 theorems, 30 equations, 2 figures, 8 tables, 7 algorithms.

Table of Contents

  1. Introduction.
  2. Results.
  3. Related work.
  4. Outline.
  5. A basic branch-and-bound algorithm.
  6. A basic pruning rule.
  7. Improving the pruning rule in Algorithm \ref{['e_alg1']}.
  8. Acceleration techniques.
  9. Lazy Evaluations.
  10. Early Pruning.
  11. Candidate Reduction.
  12. Experiments.
  13. A binary linear reformulation of Submodular Knapsack-Constrained Maximization.
  14. Experimental setup and data.
  15. Artificial instances.
  16. ...and 3 more sections

Key Result

Theorem 1

Given an instance $(I,f,w,B)\in\mathop{\mathrm{\mathcal{I}}}\nolimits$, Algorithm e_alg1 terminates and returns the objective value $s^*$ of an optimal solution to $(I,f,w,B)$.

Figures (2)

  • Figure 1: Number of solved instances for all solvers on artificial $\mathop{\mathrm{\textsc{COV}}}\nolimits$-, $\mathop{\mathrm{\textsc{LOC}}}\nolimits$-, and $\mathop{\mathrm{\textsc{INF}}}\nolimits$- instances generated following the method by sakaue2018.
  • Figure 2: Number of solved instances for all solvers on artificial $\mathop{\mathrm{\textsc{COV}}}\nolimits$-, $\mathop{\mathrm{\textsc{LOC}}}\nolimits$-, and $\mathop{\mathrm{\textsc{INF}}}\nolimits$- instances generated following our method.

Theorems & Definitions (22)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • Theorem 1
  • proof
  • Definition 8
  • ...and 12 more