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arXiv · 2504.02369

Finding Diverse Solutions in Combinatorial Problems with a Distributive Lattice Structure

Mark de Berg·Andrés López Martínez·Frits Spieksma
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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.

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Mark de Berg, Andrés López Martínez, Frits Spieksma. 2025-04-03. Finding Diverse Solutions in Combinatorial Problems with a Distributive Lattice Structure. https://arxiv.org/abs/2504.02369

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