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Register a new userAlmost stable matchings in constant time
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Added by
Jukka Suomela
Publication date
2008
fields
Informatics Engineering
and research's language is
English
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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.
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We study the problem of finding solutions to the stable matching problem that are robust to errors in the input and we obtain a polynomial time algorithm for a special class of errors. In the process, we also initiate work on a new structural question concerning the stable matching problem, namely finding relationships between the lattices of solutions of two nearby instances. Our main algorithmic result is the following: We identify a polynomially large class of errors, $D$, that can be introduced in a stable matching instance. Given an instance $A$ of stable matching, let $B$ be the random variable that represents the instance that results after introducing {em one} error from $D$, chosen via a given discrete probability distribution. The problem is to find a stable matching for $A$ that maximizes the probability of being stable for $B$ as well. Via new structural properties of the type described in the question stated above, we give a combinatorial polynomial time algorithm for this problem. We also show that the set of robust stable matchings for instance $A$, under probability distribution $p$, forms a sublattice of the lattice of stable matchings for $A$. We give an efficient algorithm for finding a succinct representation for this set; this representation has the property that any member of the set can be efficiently retrieved from it.
Correlation clustering is a central topic in unsupervised learning, with many applications in ML and data mining. In correlation clustering, one receives as input a signed graph and the goal is to partition it to minimize the number of disagreements. In this work we propose a massively parallel computation (MPC) algorithm for this problem that is considerably faster than prior work. In particular, our algorithm uses machines with memory sublinear in the number of nodes in the graph and returns a constant approximation while running only for a constant number of rounds. To the best of our knowledge, our algorithm is the first that can provably approximate a clustering problem on graphs using only a constant number of MPC rounds in the sublinear memory regime. We complement our analysis with an experimental analysis of our techniques.
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).
We show fully polynomial time randomized approximation schemes (FPRAS) for counting matchings of a given size, or more generally sampling/counting monomer-dimer systems in planar, not-necessarily-bipartite, graphs. While perfect matchings on planar graphs can be counted exactly in polynomial time, counting non-perfect matchings was shown by [Jer87] to be #P-hard, who also raised the question of whether efficient approximate counting is possible. We answer this affirmatively by showing that the multi-site Glauber dynamics on the set of monomers in a monomer-dimer system always mixes rapidly, and that this dynamics can be implemented efficiently on downward-closed families of graphs where counting perfect matchings is tractable. As further applications of our results, we show how to sample efficiently using multi-site Glauber dynamics from partition-constrained strongly Rayleigh distributions, and nonsymmetric determinantal point processes. In order to analyze mixing properties of the multi-site Glauber dynamics, we establish two notions for generating polynomials of discrete set-valued distributions: sector-stability and fractional log-concavity. These notions generalize well-studied properties like real-stability and log-concavity, but unlike them robustly degrade under useful transformations applied to the distribution. We relate these notions to pairwise correlations in the underlying distribution and the notion of spectral independence introduced by [ALO20], providing a new tool for establishing spectral independence based on geometry of polynomials. As a byproduct of our techniques, we show that polynomials avoiding roots in a sector of the complex plane must satisfy what we call fractional log-concavity; this extends a classic result established by [Gar59] who showed homogeneous polynomials that have no roots in a half-plane must be log-concave over the positive orthant.
We consider the well-studied problem of finding a perfect matching in $d$-regular bipartite graphs with $2n$ vertices and $m = nd$ edges. While the best-known algorithm for general bipartite graphs (due to Hopcroft and Karp) takes $O(m sqrt{n})$ time, in regular bipartite graphs, a perfect matching is known to be computable in $O(m)$ time. Very recently, the $O(m)$ bound was improved to $O(min{m, frac{n^{2.5}ln n}{d}})$ expected time, an expression that is bounded by $tilde{O}(n^{1.75})$. In this paper, we further improve this result by giving an $O(min{m, frac{n^2ln^3 n}{d}})$ expected time algorithm for finding a perfect matching in regular bipartite graphs; as a function of $n$ alone, the algorithm takes expected time $O((nln n)^{1.5})$. To obtain this result, we design and analyze a two-stage sampling scheme that reduces the problem of finding a perfect matching in a regular bipartite graph to the same problem on a subsampled bipartite graph with $O(nln n)$ edges that has a perfect matching with high probability. The matching is then recovered using the Hopcroft-Karp algorithm. While the standard analysis of Hopcroft-Karp gives us an $tilde{O}(n^{1.5})$ running time, we present a tighter analysis for our special case that results in the stronger $tilde{O}(min{m, frac{n^2}{d} })$ time mentioned earlier. Our proof of correctness of this sampling scheme uses a new correspondence theorem between cuts and Halls theorem ``witnesses for a perfect matching in a bipartite graph that we prove. We believe this theorem may be of independent interest; as another example application, we show that a perfect matching in the support of an $n times n$ doubly stochastic matrix with $m$ non-zero entries can be found in expected time $tilde{O}(m + n^{1.5})$.
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