# The Planted Matching Problem: Phase Transitions and Exact Results

@article{Moharrami2019ThePM, title={The Planted Matching Problem: Phase Transitions and Exact Results}, author={Mehrdad Moharrami and Cristopher Moore and Jiaming Xu}, journal={ArXiv}, year={2019}, volume={abs/1912.08880} }

We study the problem of recovering a planted matching in randomly weighted complete bipartite graphs $K_{n,n}$. For some unknown perfect matching $M^*$, the weight of an edge is drawn from one distribution $P$ if $e \in M^*$ and another distribution $Q$ if $e \notin M^*$. Our goal is to infer $M^*$, exactly or approximately, from the edge weights. In this paper we take $P=\exp(\lambda)$ and $Q=\exp(1/n)$, in which case the maximum-likelihood estimator of $M^*$ is the minimum-weight matching $M_…

## 7 Citations

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## References

SHOWING 1-10 OF 38 REFERENCES

### Dynamic Programming Optimization over Random Data: The Scaling Exponent for Near-Optimal Solutions

- Computer Science, MathematicsSIAM J. Comput.
- 2009

It is shown that, amongst solutions differing from the optimal solution in a small proportion of places, one can find near-optimal solutions whose objective function value differs from the optimum by a factor of order $\delta^2$ but not of smaller order.

### Proofs of the Parisi and Coppersmith-Sorkin conjectures for the finite random assignment problem

- Business44th Annual IEEE Symposium on Foundations of Computer Science, 2003. Proceedings.
- 2003

Both Parisi's conjecture and Sorkin's conjecture are generalized to the average value of the smallest k-assignment when there are n jobs and m machines and are proved based on a common set of combinatorial and probabilistic arguments.

### Belief Propagation: An Asymptotically Optimal Algorithm for the Random Assignment Problem

- Computer ScienceMath. Oper. Res.
- 2009

The objective method is used to analyze the performance of BP as the size of the underlying graph becomes large and establishes that the dynamic of BP on Knn converges in distribution as n → ∞ to an appropriately defined dynamic on the Poisson weighted infinite tree, and proves correlation decay for this limiting dynamic.

### A proof of Parisi’s conjecture on the random assignment problem

- Mathematics
- 2003

Abstract.An assignment problem is the optimization problem of finding, in an m by n matrix of nonnegative real numbers, k entries, no two in the same row or column, such that their sum is minimal.…

### A Nearly Tight Sum-of-Squares Lower Bound for the Planted Clique Problem

- Mathematics, Computer Science2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)
- 2016

We prove that with high probability over the choice of a random graph G from the Erdös-Rényi distribution G(n,1/2), the nO(d)-time degree d Sum-of-Squares semidefinite programming relaxation for…

### Belief propagation for optimal edge-cover in the random complete graph

- Computer Science, MathematicsArXiv
- 2012

The objective method of Aldous is applied to the problem of finding the minimum-cost edge cover of the complete graph with random independent and identically distributed edge costs and shows that a belief propagation algorithm converges asymptotically to the optimal solution.

### Maximum weight independent sets and matchings in sparse random graphs. Exact results using the local weak convergence method

- MathematicsRandom Struct. Algorithms
- 2006

This work extends the results to maximum weight matchings in G(n, c/n) and G r (n) using the recently developed local weak convergence method further reduced to a certain local optimality property exhibited by the models it considers.

### Sudden Emergence of a Giantk-Core in a Random Graph

- MathematicsJ. Comb. Theory, Ser. B
- 1996

These proofs are based on the probabilistic analysis of an edge deletion algorithm that always find ak-core if the graph has one, and demonstrate that, unlike the 2-core, when ak- core appears for the first time it is very likely to be giant, of size ?pk(?k)n.

### Linear phase transition in random linear constraint satisfaction problems

- MathematicsSODA '04
- 2004

It is proved that the feasibility property experiences a linear phase transition, when n→∞ and m = cn for a constant c, and it is extended to maximum weight b-matchings also in G(n, cn).