Popular Branchings and Their Dual Certificates
Telikepalli Kavitha, Tam\'as Kir\'aly, Jannik Matuschke, Ildik\'o, Schlotter, Ulrike Schmidt-Kraepelin

TL;DR
This paper characterizes popular branchings in directed graphs with preferences, providing an efficient algorithm to decide their existence, and explores properties under different preference types with applications to liquid democracy.
Contribution
It introduces a dual certificate characterization of popular branchings and develops an efficient combinatorial algorithm for their detection.
Findings
Characterization of popular branchings via dual certificates
Efficient algorithm for the popular branching problem
Existence of approximately popular branchings under strict preferences
Abstract
Let 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 is popular if 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 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…
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