It is difficult to tell if there is a Condorcet spanning tree

Darmann, Andreas

doi:10.1007/s00186-016-0535-3

Cited by 7 publications

(5 citation statements)

References 17 publications

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“…Popular spanning trees were studied in [7][8][9] where the incentive was to find a "socially best" spanning tree. However, in contrast to the popular colorful spanning tree problem, edges have no colors in their model and voters have rankings over the entire edge set.…”

Section: Popular Branchingsmentioning

confidence: 99%

Arborescences, Colorful Forests, and Popularity

Kavitha,

Makino,

Schlotter

et al. 2024

Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)

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Our input is a directed, rooted graph G = (V ∪{r}, E) where each vertex in V has a partial order preference over its incoming edges. The preferences of a vertex extend naturally to preferences over arborescences rooted at r. We seek a popular arborescence in G, i.e., one for which there is no "more popular" arborescence. Popular arborescences have applications in liquid democracy or collective decision making; however, they need not exist in every input instance. The popular arborescence problem is to decide if a given input instance admits a popular arborescence or not. We show a polynomial-time algorithm for this problem, whose computational complexity was not known previously.Our algorithm is combinatorial, and can be regarded as a primal-dual algorithm. It searches for an arborescence along with its dual certificate, a chain of subsets of E, witnessing its popularity. In fact, our algorithm solves the more general popular common base problem in the intersection of two matroids, where one matroid is the partition matroid defined by any partition E = •v∈V δ(v) and the other is an arbitrary matroid M = (E, I) of rank |V |, with each v ∈ V having a partial order over elements in δ(v). We extend our algorithm to the case with forced or forbidden edges.We also study the related popular colorful forest (or more generally, the popular common independent set) problem where edges are partitioned into color classes, and the task is to find a colorful forest that is popular within the set of all colorful forests. For the case with weak rankings, we formulate the popular colorful forest polytope, and thus show that a minimum-cost popular colorful forest can be computed efficiently. By contrast, we prove that it is NP-hard to compute a minimum-cost popular arborescence, even when rankings are strict.

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Section: Popular Branchingsmentioning

confidence: 99%

Arborescences, Colorful Forests, and Popularity

Kavitha,

Makino,

Schlotter

et al. 2024

Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)

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“…The concept of popularity has previously been applied to (undirected) spanning trees [10,11,12]. In contrast to our setting, voters have rankings over the entire edge set.…”

Section: Background and Related Workmentioning

confidence: 99%

“…The top choice for any node a k i on A i is its in-neighbor a k−1 i on A i , its second choice is t i if k = 1 and f i otherwise. 11 Next, for each clause c j we define a clause-gadget as a directed cycle C j on nodes c 1 j , . .…”

Section: For Every Active Node V Domentioning

confidence: 99%

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Popular Branchings and Their Dual Certificates

Kavitha

Király

Matuschke

et al. 2020

Integer Programming and Combinatorial Optimization

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Let G be a digraph where every node has preferences over its incoming edges. The preferences of a node extend naturally to preferences over branchings, i.e., directed forests; a branching B is popular if B does not lose a head-to-head election (where nodes cast votes) against any branching. Such popular branchings have a natural application in liquid democracy. The popular branching problem is to decide if G admits a popular branching or not. We give a characterization of popular branchings in terms of dual certificates and use this characterization to design an efficient combinatorial algorithm for the popular branching problem. When preferences are weak rankings, we use our characterization to formulate the popular branching polytope in the original space and also show that our algorithm can be modified to compute a branching with least unpopularity margin. When preferences are strict rankings, we show that "approximately popular" branchings always exist. 6 See www.democracy.earth and www.interaktive-demokratie.org, respectively. 7 Typically, such a delegation is represented by an edge (y, x); for the sake of consistency with downward edges in a branching, we use (x, y).

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“…Although computing a minimum spanning tree in the single-objective setting is polynomial-time solvable, it appears to be NP-hard for both multi-dimensional optimality concepts. Similar problems are also studied in computation social choice, where the objective is to find one spanning tree such that the satisfaction of the least satisfied agent gets maximized under different "satisfaction" notions (Darmann 2016;Darmann, Klamler, and Pferschy 2009;Escoffier, Gourvès, and Monnot 2013;Gourvès, Monnot, and Tlilane 2015). The key difference between the existing research and our work is that we are not targeting a spanning tree; instead, we are allowed to select a spanning subgraph and aim to please all agents with a minimum number of edges.…”

Section: Introductionmentioning

confidence: 99%

Multiagent MST Cover: Pleasing All Optimally via a Simple Voting Rule

et al. 2023

AAAI

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Given a connected graph on whose edges we can build roads to connect the nodes, a number of agents hold possibly different perspectives on which edges should be selected by assigning different edge weights. Our task is to build a minimum number of roads so that every agent has a spanning tree in the built subgraph whose weight is the same as a minimum spanning tree in the original graph. We first show that this problem is NP-hard and does not admit better than ((1-o(1)) ln k)-approximation polynomial-time algorithms unless P = NP, where k is the number of agents. We then give a simple voting algorithm with an optimal approximation ratio. Moreover, our algorithm only needs to access the agents' rankings on the edges. Finally, we extend our problem to submodular objective functions and Matroid rank constraints.

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It is difficult to tell if there is a Condorcet spanning tree

Cited by 7 publications

References 17 publications

Arborescences, Colorful Forests, and Popularity

Arborescences, Colorful Forests, and Popularity

Popular Branchings and Their Dual Certificates

Multiagent MST Cover: Pleasing All Optimally via a Simple Voting Rule

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