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A general framework for finding diverse solutions via network flow and its applications

Yuni Iwamasa, Tomoki Matsuda, Shunya Morihira, Hanna Sumita

TL;DR

The paper addresses the problem of generating multiple diverse solutions for combinatorial optimization by introducing a general framework based on two structural properties: (S) all solutions have equal size and (R) the solution family is the surjective image of ideals of a finite poset. It reduces Sum-$k$-Diverse/Cov-$k$-Diverse to a minimum $k$-potential problem on a poset-derived DAG and solves it via reductions to minimum-cost flow or maximum $s$-$t$ flow, enabling the use of state-of-the-art network-flow algorithms. The authors instantiate the framework for classical problems such as Unweighted Minimum $s$-$t$ Cut and Stable Matching, achieving polynomial-time results with improved running times over prior SFM-based approaches, and show the framework subsumes the De Berg et al. product-of-total-orders approach. Overall, the work provides a scalable, network-flow–based method for producing diverse solution sets and highlights practical gains in running time for generating multiple alternative solutions in key combinatorial domains.

Abstract

In this paper, we present a general framework for efficiently computing diverse solutions to combinatorial optimization problems. Given a problem instance, the goal is to find $k$ solutions that maximize a specified diversity measure; the sum of pairwise Hamming distances or the size of the union of the $k$ solutions. Our framework applies to problems satisfying two structural properties: (i) All solutions are of equal size and (ii) the family of all solutions can be represented by a surjection from the family of ideals of some finite poset. Under these conditions, we show that the problem of computing $k$ diverse solutions can be reduced to the minimum cost flow problem and the maximum $s$-$t$ flow problem. As applications, we demonstrate that both the unweighted minimum $s$-$t$ cut problem and the stable matching problem satisfy the requirements of our framework. By utilizing the recent advances in network flows algorithms, we improve the previously known time complexities of the diverse problems, which were based on submodular function minimization.

A general framework for finding diverse solutions via network flow and its applications

TL;DR

The paper addresses the problem of generating multiple diverse solutions for combinatorial optimization by introducing a general framework based on two structural properties: (S) all solutions have equal size and (R) the solution family is the surjective image of ideals of a finite poset. It reduces Sum--Diverse/Cov--Diverse to a minimum -potential problem on a poset-derived DAG and solves it via reductions to minimum-cost flow or maximum - flow, enabling the use of state-of-the-art network-flow algorithms. The authors instantiate the framework for classical problems such as Unweighted Minimum - Cut and Stable Matching, achieving polynomial-time results with improved running times over prior SFM-based approaches, and show the framework subsumes the De Berg et al. product-of-total-orders approach. Overall, the work provides a scalable, network-flow–based method for producing diverse solution sets and highlights practical gains in running time for generating multiple alternative solutions in key combinatorial domains.

Abstract

In this paper, we present a general framework for efficiently computing diverse solutions to combinatorial optimization problems. Given a problem instance, the goal is to find solutions that maximize a specified diversity measure; the sum of pairwise Hamming distances or the size of the union of the solutions. Our framework applies to problems satisfying two structural properties: (i) All solutions are of equal size and (ii) the family of all solutions can be represented by a surjection from the family of ideals of some finite poset. Under these conditions, we show that the problem of computing diverse solutions can be reduced to the minimum cost flow problem and the maximum - flow problem. As applications, we demonstrate that both the unweighted minimum - cut problem and the stable matching problem satisfy the requirements of our framework. By utilizing the recent advances in network flows algorithms, we improve the previously known time complexities of the diverse problems, which were based on submodular function minimization.

Paper Structure

This paper contains 13 sections, 18 theorems, 29 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Related work.
  3. Organization.
  4. Preliminaries
  5. Reduction to the minimum k-potential problem
  6. Solving the minimum k-potential problem via network flow
  7. Via minimum cost flow
  8. Via maximum s-t flow
  9. Applications
  10. How to construct a poset and a reduction map
  11. The k-diverse unweighted minimum s-t cut problem
  12. The k-diverse stable matching problem
  13. The k-diverse problem on the product of total orders

Key Result

Theorem 1.1

Suppose that a combinatorial problem Prob has the properties cond:S and cond:R. Then, for any instance $\mathbf{I}$ of Prob, the problems Sum-$k$-Diverse and Cov-$k$-Diverse of Prob can be solved in $O(T_P(\mathbf{I}) + T_r(\mathbf{I}) + (|A| + k|E|)^{1+o(1)})$ time and in $O(T_P(\mathbf{I}) +

Figures (1)

  • Figure 1: Illustration of the arcs in $\hat{A}_1$ (solid arrows) and ones in $\hat{A}_2$ (dotted arrows) along with the $0$-$1$ assignment for an arc $(u,v)\in A$. Each circle represents a vertex in $\hat{V}$, with the integer inside the circle indicating the $0$-$1$ assignment to it.

Theorems & Definitions (33)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • Lemma 4.1
  • proof
  • Theorem 4.2
  • proof
  • Lemma 4.3
  • ...and 23 more