A note on maximizing the minimum voter satisfaction on spanning trees

doi:10.1016/j.mathsocsci.2010.04.003

Mathematical Social Sciences

Volume 60, Issue 1, July 2010, Pages 82-85

https://doi.org/10.1016/j.mathsocsci.2010.04.003 Get rights and content

Abstract

A fair spanning tree of a graph maximizes the minimum satisfaction among individuals given their preferences over the edges of the graph. In this note we answer an open question about the computational complexity of determining fair spanning trees raised in Darmann et al. (2009). It is shown that the maximin voter satisfaction problem under choose- $t$ elections is $NP$ -complete for each fixed $t \geq 2$ .

Introduction

The topic of finding a minimum spanning tree is a classical problem in Combinatorial Optimization. Having determined a minimum spanning tree, the natural question may arise how to “fairly” assign the costs of such a tree to individuals. This issue has been widely studied since Claus and Kleitman (1973), e.g., by Bird (1976), Bogomolnaia and Moulin (2009) and Dutta and Kar (2004).

In this paper, however, we focus on another aspect of fairness in connection with spanning trees. In our framework monetary costs are not taken into consideration. Instead, our approach is based on individuals’ preferences over the edges of a graph. Given such preferences, in Darmann et al. (2008) the quality of methods that fairly, i.e., socially acceptably, construct a spanning tree is analyzed. In a companion paper of Darmann et al. (2009), the focus is laid on the computational complexity of methods to fairly (in a maximin sense) construct a spanning tree. In this paper, we study the computational complexity of another method to fairly construct a spanning tree and therewith answer an open question stated in Darmann et al. (2009).

Our framework covers many applications in which a network needs to be installed, e.g., when countries need to agree on transnational traffic systems or oil pipelines, or when homeowners have diverging opinions about the specific links in a sewage or water network that needs to be constructed.

In the Combinatorial Optimization literature maximin fairness can be found in terms of maximizing the minimum of concepts such as utility, costs, time, etc. However, in Social Choice Theory the maximin approach is a well-known concept of formalizing fairness, originally discussed by Rawls (1972); for extensive studies of social choice rules see, e.g., Brams and Fishburn (2002), Nurmi (2007) and Saari (1995). In this paper, we follow the maximin approach based on different social choice rules proposed in Darmann et al. (2009) and show that the maximin voter satisfaction problem (MMVS) under choose- $t$ elections is $NP$ -hard for each fixed $t \geq 2$ .

Section snippets

Problem formulation

In this work $G = (V, E)$ denotes an undirected graph, with a set $V$ of vertices and a set $E$ of edges. Let $n$ be the number of vertices and $m$ be the number of edges of $G$ . We call a subset $T \subseteq E$ a spanning tree of $G$ if the graph $(V, T)$ is connected and acyclic. Let $τ$ be the set of spanning trees of $G$ . A binary relation $≿$ on a set $A$ is called

•
complete if $\forall a, b \in A$ , $a ≿ b$ or $b ≿ a$ , and
•
transitive if $\forall a, b, c \in A$ , $(a ≿ b and b ≿ c) \Rightarrow a ≿ c$ .

A relation is called weak order if it is complete and transitive.

In our framework we are

MMVS under choose- $t$ elections

Theorem 1

MMVS under choose-2 elections is strongly $NP$ -complete.

Proof

Let $Ω$ be an arbitrary instance of 3-SAT with a set $X ≔ {x_{1}, \dots, x_{ℓ}}$ of variables and a set $C ≔ {C_{1}, \dots, C_{q}}$ of clauses over $X$ . Accordingly, the set $Y$ of literals is $Y ≔ {x_{1}, {\bar{x}}_{1}, \dots, x_{ℓ}, {\bar{x}}_{ℓ}}$ . From $Ω$ we create an instance $J$ of MMVS under a choose-2 election by building a graph $G$ (see Fig. 1) and a voter preference profile $π$ (see Table 1) as follows.

In order to create $G = (V, E)$ , for every variable $x_{j}$ we introduce a triangle of edges $(x_{j}, {\bar{x}}_{j}, g_{j})$ and an edge

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There are more references available in the full text version of this article.

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Short communicationA note on maximizing the minimum voter satisfaction on spanning trees

Abstract

Introduction

Section snippets

Problem formulation

MMVS under choose-t elections

Mathematical Social Sciences

Games and Economic Behavior

Approximating min–max (regret) versions of some polynomial problems

On cost allocation for a spanning tree: a game theoretic approach

Networks

Sharing the cost of a minimal cost spanning tree: beyond the folk solution

Games and Economic Behavior

Approval Voting

Short communication
A note on maximizing the minimum voter satisfaction on spanning trees

MMVS under choose- $t$ elections