Testing popularity in linear time via maximum matching

TL;DR

A new algorithm for testing popularity is given by reducing the problem to maximum matching testing, thus attaining a linear running time O(|E|) and derive a more structured dual witness than previous ones.

Abstract

Popularity is an approach in mechanism design to find fair structures in a graph, based on the votes of the nodes. Popular matchings are the relaxation of stable matchings: given a graph G=(V,E) with strict preferences on the neighbors of the nodes, a matching M is popular if there is no other matching M' such that the number of nodes preferring M' is more than those preferring M. This paper considers the popularity testing problem, when the task is to decide whether a given matching is popular or not. Previous algorithms applied reductions to maximum weight matchings. We give a new algorithm for testing popularity by reducing the problem to maximum matching testing, thus attaining a linear running time O(|E|). Linear programming-based characterization of popularity is often applied for proving the popularity of a certain matching. As a consequence of our algorithm we derive a more structured dual witness than previous ones. Based on this result we give a combinatorial characterization of fractional popular matchings, which are a special class of popular matchings.

Figures (3)

• Figure 1: An example for $G_M^*$. Matching $M=\{ab, cd, ef\}$, blocking edges are $\{ac,de,df\}$. There is one star $S=\{de,df\}$ with middle node $d$. There are several options to see that $M$ is not popular. First, there is an augmenting path $b_c-c-d-b_d$ in $G_M^*$ connecting $b_c$ and $b_d$, thus there is an alternating path $a-c-d-f$ connecting blocking edges $ac$ and $df$, so matching $M'=\{ac,df\}$ is more popular than $M$. There is another alternating path $b_c-c-d-u$ in $G_M^*$ giving that matching $M"=\{ac,dg,ef\}$ is also more popular than $M$. Finally alternating path $b_S-e-f-u$ shows that matching $M"'=\{ab,de,fh\}$ is also more popular than $M$.
• Figure 2: An example for the existence of structure a) in Theorem \ref{['thm:char-trula-popular']}. For matching $M=\{ab,cd\}$, nodes $\{a-b-b_S\}$ form an odd component in $X\cap D$ and an odd alternating cycle as well. Thus there is an odd alternating cycle $C=c-a-b-c$ through a star in $G$ and a fractional matching $p$ more popular than $M$ exists.
• Figure 3: An example for the existence of structure b) in Theorem \ref{['thm:char-trula-popular']}. For matching $M=\{ab,cd, ef\}$, alternating odd cycle $\{d-e-f-d\}$ can be reached on an alternating path from both $b_c$ and $b_a$, but the path from $b_c$ is shorter. Thus there is an odd alternating path $P=a-c-d$ to cycle $Q=d-e-f-d$ in $G$, and a fractional matching $p$ more poplar than $M$ exists.

Theorems & Definitions (30)

• Theorem 1: Kavitha, Kav19
• Theorem 2: Kavitha Kav19, Lemma 7
• Theorem 3: Huang and Kavitha HK13
• Theorem 4
• proof
• Claim 3.1
• proof
• Claim 3.2
• proof
• Claim 3.3