
Recognizing and Eliciting Weakly Single Crossing Profiles on Trees
The domain of single crossing preference profiles is a widely studied do...
read it

Deleting to Structured Trees
We consider a natural variant of the wellknown Feedback Vertex Set prob...
read it

Polynomial tuning of multiparametric combinatorial samplers
Boltzmann samplers and the recursive method are prominent algorithmic fr...
read it

New Algorithms for Unordered Tree Inclusion
The tree inclusion problem is, given two nodelabeled trees P and T (the...
read it

Incomplete Preferences in SinglePeaked Electorates
Incomplete preferences are likely to arise in realworld preference aggr...
read it

Approximating the Minimum kSection Width in BoundedDegree Trees with Linear Diameter
Minimum kSection denotes the NPhard problem to partition the vertex se...
read it

Determining a Slater Winner is Complete for Parallel Access to NP
We consider the complexity of deciding the winner of an election under t...
read it
Preferences SinglePeaked on a Tree: Multiwinner Elections and Structural Results
A preference profile is singlepeaked on a tree if the candidate set can be equipped with a tree structure so that the preferences of each voter are decreasing from their top candidate along all paths in the tree. This notion was introduced by Demange (1982), and subsequently Trick (1989) described an efficient algorithm for deciding if a given profile is singlepeaked on a tree. We study the complexity of multiwinner elections under several variants of the ChamberlinCourant rule for preferences singlepeaked on trees. We show that the egalitarian version of this problem admits a polynomialtime algorithm. For the utilitarian version, we prove that winner determination remains NPhard, even for the Borda scoring function; however, a winning committee can be found in polynomial time if either the number of leaves or the number of internal vertices of the underlying tree is bounded by a constant. To benefit from these positive results, we need a procedure that can determine whether a given profile is singlepeaked on a tree that has additional desirable properties (such as, e.g., a small number of leaves). To address this challenge, we develop a structural approach that enables us to compactly represent all trees with respect to which a given profile is singlepeaked. We show how to use this representation to efficiently find the best tree for a given profile for use with our winner determination algorithms: Given a profile, we can efficiently find a tree with the minimum number of leaves, or a tree with the minimum number of internal vertices among trees on which the profile is singlepeaked. We also consider several other optimization criteria for trees: for some we obtain polynomialtime algorithms, while for others we show NPhardness results.
READ FULL TEXT
Comments
There are no comments yet.