Finding Diverse Solutions in Combinatorial Problems with a Distributive Lattice Structure

TL;DR

This work addresses the problem of generating a set of $k$ diverse solutions to combinatorial problems by maximizing the sum of pairwise Hamming distances $d_ ext{sum}$. It introduces a general framework that applies when the solution space forms a distributive lattice and three structural properties hold, reducing the search to submodular function minimization on a lattice via a left-right ordered representation. The main contributions are (i) a four-part reduction showing $d_ ext{sum}$-maximization is polynomial-time via SFM on a distributive lattice, (ii) polynomial-time results for diverse minimum $s$-$t$ cuts and diverse stable matchings, and (iii) extensions to two other diversity measures, $d_ ext{cov}$ and $d_ ext{abs}$, with a simpler framework for disjoint solutions. The framework unifies diverse solution generation across classical problems and provides practical implications for robustness and flexibility in optimization under uncertainty or varying constraints. Overall, the paper advances theoretical understanding and algorithmic tooling for producing maximally diverse solution sets in polynomial time for a broad class of problems.

Abstract

We generalize the polynomial-time solvability of $k$-\textsc{Diverse Minimum s-t Cuts} (De Berg et al., ISAAC'23) to a wider class of combinatorial problems whose solution sets have a distributive lattice structure. We identify three structural conditions that, when met by a problem, ensure that a $k$-sized multiset of maximally-diverse solutions -- measured by the sum of pairwise Hamming distances -- can be found in polynomial time. We apply this framework to obtain polynomial time algorithms for finding diverse minimum $s$-$t$ cuts and diverse stable matchings. Moreover, we show that the framework extends to two other natural measures of diversity. Lastly, we present a simpler algorithmic framework for finding a largest set of pairwise disjoint solutions in problems that meet these structural conditions.

Figures (3)

• Figure 1: Example of Birkhoff's representation theorem for distributive lattices. The left is a distributive lattice $L$, the middle is the isomorphic lattice $\mathcal{D}(J(L))$ of ideals of join-irreducibles of $L$, and the right shows the compact representation $G(J(L))$ of $L$. The join irreducible elements of $L$ and $\mathcal{D}(J(L))$ are highlighted in blue.
• Figure 2: Interval representation of the containment of an element $e \in E$ in left-right ordered collections $C_1$ and $C_2$ of feasible solutions, as well as in their join and meet. In the example, there are $k = 8$ solutions in each tuple, and eight corresponding elements in the domain of the intervals. Observe that neither $I_e(C_1 \vee C_2)$ nor $I_e(C_1 \wedge C_2)$ are longer than $I_e(C_1)$ or $I_e(C_2)$. Also, note that the corresponding sums of their lengths are equal.
• Figure 3: Illustration of the order relation $\preceq_i$ over the edges of an $s$-$t$ path $p_i \in \mathcal{P}$.

Theorems & Definitions (50)

• theorem 1.1
• theorem 2.1: Birkhoff's Representation Theorem birkhoff1937rings
• theorem 2.2: murota2003 and markowsky2001overview
• lemma 3.1: Distributivity Lemma
• proof
• proof
• claim 3.3
• proof
• lemma 3.5: Cost-Equivalence Lemma
• proof