Abstract
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.
Highlights
Let G be a directed graph where every node has preferences over its incoming edges
Theorem 1.4 A branching with minimum unpopularity margin in a digraph where every node has a weak ranking over its incoming edges can be efficiently computed
The functions unpopularity factor/margin were introduced in [32] to measure the unpopularity of a matching; it was shown in [32] that it is NP-hard to compute a matching that minimizes either of these quantities
Summary
Let G be a directed graph where every node has preferences (in partial order) over its incoming edges. As suggested in [6], we assume voters to accept themselves as their least preferred approved representative. This reveals the connection to branchings in simple graphs where nodes correspond to voters and the edge (x, y) indicates that voter x is an approved delegate of voter y.2. We assume that voters rate branchings only based on their predecessors This is justified when approved delegates are considered to be more competent both in deciding on the issue as well as in assessing the competence of others. It is easy to check that this instance has no popular branching
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