Computing Diverse Shortest Paths Efficiently: A Theoretical and Experimental Study

Hanaka, Tesshu; Kobayashi, Yasuaki; Kurita, K.; Lee, See Woo; Otachi, Yota

doi:10.1609/aaai.v36i4.20290

Cited by 7 publications

(9 citation statements)

References 24 publications

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“…These models however do not have fairness aspects in mind. Hanaka et al (2022) are somewhat closer to fairness aspects in path finding. They study shortest paths under diversity aspects, meaning that they search for multiple shortest paths that are maximally different from each other; this fits into the recent trend of finding diverse sets of solutions to optimization problems (Baste et al 2022;Petit and Trapp 2019;Kellerhals, Renken, and Zschoche 2021).…”

Section: Related Workmentioning

confidence: 70%

Fair Short Paths in Vertex-Colored Graphs

Bentert

Kellerhals

Niedermeier

2023

AAAI

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The computation of short paths in graphs with arc lengths is a pillar of graph algorithmics and network science. In a more diverse world, however, not every short path is equally valuable. For the setting where each vertex is assigned to a group (color), we provide a framework to model multiple natural fairness aspects. We seek to find short paths in which the number of occurrences of each color is within some given lower and upper bounds. Among other results, we prove the introduced problems to be computationally intractable (NP-hard and parameterized hard with respect to the number of colors) even in very restricted settings (such as each color should appear with exactly the same frequency), while also presenting an encouraging algorithmic result ("fixed-parameter tractability") related to the length of the sought solution path for the general problem.

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Section: Related Workmentioning

confidence: 70%

Fair Short Paths in Vertex-Colored Graphs

Bentert

Kellerhals

Niedermeier

2023

AAAI

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“…Note, however, that there are no immediate fairness aspects modeled here. Hanaka et al [24] are somewhat closer to fairness aspects in path finding. They study shortest paths under diversity aspects, meaning that they search for multiple shortest paths that are maximally different from each other.…”

Section: Introductionmentioning

confidence: 85%

The structural complexity landscape of finding balance-fair shortest paths

Bentert

Kellerhals

Niedermeier

2022

Theoretical Computer Science

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“…Regarding the diverse tour problems, the oldest known formulation is the Peripatetic Salesman problem (PSP) by Krarup [33], which asks for k-edge disjoint Hamiltonian circuits of minimum total weight in a network. Recently, there has been a surge of interest in the tractability of finding diverse solutions for a number of combinatorial optimization problems [5,8,19], such as hitting sets [4], spanning trees, minimum spanning trees [27], shortest paths [26,28], k-matchings [21], s-t cuts [6], LPs [22], etc. These works are commonly on graph problems, and diversity usually corresponds to the size of the symmetric difference of the edge sets in the solutions.…”

Section: Introductionmentioning

confidence: 99%

Obtaining Approximately Optimal and Diverse Solutions via Dispersion

Gao,

Goswami,

C.S.

et al. 2023

Preprint

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There has been a long-standing interest in computing diverse solutions to optimization problems. In 1995 J. Krarup posed the problem of finding $k$-edge disjoint Hamiltonian Circuits of minimum total weight, called the peripatetic salesman problem (PSP). Since then researchers have investigated the complexity of finding diverse solutions to spanning trees, paths, vertex covers, matchings, and more. Unlike the PSP that has a constraint on the total weight of the solutions, recent work has involved finding diverse solutions that are all optimal. However, sometimes the space of exact solutions may be too small to achieve sufficient diversity. Motivated by this, we initiate the study of obtaining sufficiently-diverse, yet approximately-optimal solutions to optimization problems. Formally, given an integer $k$, an approximation factor $c$, and an instance $I$ of an optimization problem, we aim to obtain a set of $k$ solutions to $I$ that a) are all $c$ approximately-optimal for $I$ and b) maximize the diversity of the $k$ solutions. Finding such solutions, therefore, requires a better understanding of the global landscape of the optimization function. Given a metric on the space of solutions, we first provide a general reduction to an associated budget-constrained optimization (BCO) problem, where one objective function is to optimized subject to a bound on the second objective function. We then prove that bi-approximations to the BCO can be used to give bi-approximations to the diverse approximately optimal solutions problem. As applications of our result, we present polynomial time approximation algorithms for several problems such as diverse $c$-approximate \emph{spanning trees, maximum matchings, $s-t$ shortest paths, global min-cut, and minimum weight bases of a matroid}. In addition, we demonstrate that the \emph{$c$-approximate minimum spanning trees} can be converted into a set of $c$-approximate TSP tours where the tours are empirically diverse, advancing a step towards achieving diverse $c$-approximate TSP tours. We also explore the connection to the field of multiobjective optimization and show that the class of problems to which our result applies includes those for which the associated DUALRESTRICT problem defined by Papadimitriou and Yannakakis, and recently explored by Herzel et al. can be solved in polynomial time.

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Computing Diverse Shortest Paths Efficiently: A Theoretical and Experimental Study

Cited by 7 publications

References 24 publications

Fair Short Paths in Vertex-Colored Graphs

Fair Short Paths in Vertex-Colored Graphs

The structural complexity landscape of finding balance-fair shortest paths

Obtaining Approximately Optimal and Diverse Solutions via Dispersion

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