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Added by
Suthee Ruangwises
Publication date
2014
fields
Informatics Engineering
and research's language is
English
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For a set $A$ of $n$ people and a set $B$ of $m$ items, with each person having a preference list that ranks all items from most wanted to least wanted, we consider the problem of matching every person with a unique item. A matching $M$ is called $epsilon$-popular if for any other matching $M$, the number of people who prefer $M$ to $M$ is at most $epsilon n$ plus the number of those who prefer $M$ to $M$. In 2006, Mahdian showed that when randomly generating peoples preference lists, if $m/n > 1.42$, then a 0-popular matching exists with $1-o(1)$ probability; and if $m/n < 1.42$, then a 0-popular matching exists with $o(1)$ probability. The ratio 1.42 can be viewed as a transition point, at which the probability rises from asymptotically zero to asymptotically one, for the case $epsilon=0$. In this paper, we introduce an upper bound and a lower bound of the transition point in more general cases. In particular, we show that when randomly generating each persons preference list, if $alpha(1-e^{-1/alpha}) > 1-epsilon$, then an $epsilon$-popular matching exists with $1-o(1)$ probability (upper bound); and if $alpha(1-e^{-(1+e^{1/alpha})/alpha}) < 1-2epsilon$, then an $epsilon$-popular matching exists with $o(1)$ probability (lower bound).
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For a set A of n applicants and a set I of m items, we consider a problem of computing a matching of applicants to items, i.e., a function M mapping A to I; here we assume that each applicant $x in A$ provides a preference list on items in I. We say that an applicant $x in A$ prefers an item p than an item q if p is located at a higher position than q in its preference list, and we say that x prefers a matching M over a matching M if x prefers M(x) over M(x). For a given matching problem A, I, and preference lists, we say that M is more popular than M if the number of applicants preferring M over M is larger than that of applicants preferring M over M, and M is called a popular matching if there is no other matching that is more popular than M. Here we consider the situation that A is partitioned into $A_{1},A_{2},...,A_{k}$, and that each $A_{i}$ is assigned a weight $w_{i}>0$ such that w_{1}>w_{2}>...>w_{k}>0$. For such a matching problem, we say that M is more popular than M if the total weight of applicants preferring M over M is larger than that of applicants preferring M over M, and we call M an k-weighted popular matching if there is no other matching that is more popular than M. In this paper, we analyze the 2-weighted matching problem, and we show that (lower bound) if $m/n^{4/3}=o(1)$, then a random instance of the 2-weighted matching problem with $w_{1} geq 2w_{2}$ has a 2-weighted popular matching with probability o(1); and (upper bound) if $n^{4/3}/m = o(1)$, then a random instance of the 2-weighted matching problem with $w_{1} geq 2w_{2}$ has a 2-weighted popular matching with probability 1-o(1).
We show that the ratio of matched individuals to blocking pairs grows linearly with the number of propose--accept rounds executed by the Gale--Shapley algorithm for the stable marriage problem. Consequently, the participants can arrive at an almost stable matching even without full information about the problem instance; for each participant, knowing only its local neighbourhood is enough. In distributed-systems parlance, this means that if each person has only a constant number of acceptable partners, an almost stable matching emerges after a constant number of synchronous communication rounds. This holds even if ties are present in the preference lists. We apply our results to give a distributed $(2+epsilon)$-approximation algorithm for maximum-weight matching in bicoloured graphs and a centralised randomised constant-time approximation scheme for estimating the size of a stable matching.
The min-cost matching problem suffers from being very sensitive to small changes of the input. Even in a simple setting, e.g., when the costs come from the metric on the line, adding two nodes to the input might change the optimal solution completely. On the other hand, one expects that small changes in the input should incur only small changes on the constructed solutions, measured as the number of modified edges. We introduce a two-stage model where we study the trade-off between quality and robustness of solutions. In the first stage we are given a set of nodes in a metric space and we must compute a perfect matching. In the second stage $2k$ new nodes appear and we must adapt the solution to a perfect matching for the new instance. We say that an algorithm is $(alpha,beta)$-robust if the solutions constructed in both stages are $alpha$-approximate with respect to min-cost perfect matchings, and if the number of edges deleted from the first stage matching is at most $beta k$. Hence, $alpha$ measures the quality of the algorithm and $beta$ its robustness. In this setting we aim to balance both measures by deriving algorithms for constant $alpha$ and $beta$. We show that there exists an algorithm that is $(3,1)$-robust for any metric if one knows the number $2k$ of arriving nodes in advance. For the case that $k$ is unknown the situation is significantly more involved. We study this setting under the metric on the line and devise a $(10,2)$-robust algorithm that constructs a solution with a recursive structure that carefully balances cost and redundancy.
A matching $M$ in a graph $G$ is said to be uniquely restricted if there is no other matching in $G$ that matches the same set of vertices as $M$. We describe a polynomial-time algorithm to compute a maximum cardinality uniquely restricted matching in an interval graph, thereby answering a question of Golumbic et al. (Uniquely restricted matchings, M. C. Golumbic, T. Hirst and M. Lewenstein, Algorithmica, 31:139--154, 2001). Our algorithm actually solves the more general problem of computing a maximum cardinality strong independent set in an interval nest digraph, which may be of independent interest. Further, we give linear-time algorithms for computing maximum cardinality uniquely restricted matchings in proper interval graphs and bipartite permutation graphs.
We study the computational complexity of several problems connected with finding a maximal distance-$k$ matching of minimum cardinality or minimum weight in a given graph. We introduce the class of $k$-equimatchable graphs which is an edge analogue of $k$-equipackable graphs. We prove that the recognition of $k$-equimatchable graphs is co-NP-complete for any fixed $k ge 2$. We provide a simple characterization for the class of strongly chordal graphs with equal $k$-packing and $k$-domination numbers. We also prove that for any fixed integer $ell ge 1$ the problem of finding a minimum weight maximal distance-$2ell$ matching and the problem of finding a minimum weight $(2 ell - 1)$-independent dominating set cannot be approximated in polynomial time in chordal graphs within a factor of $delta ln |V(G)|$ unless $mathrm{P} = mathrm{NP}$, where $delta$ is a fixed constant (thereby improving the NP-hardness result of Chang for the independent domination case). Finally, we show the NP-hardness of the minimum maximal induced matching and independent dominating set problems in large-girth planar graphs.
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