More Like this

1. Random Graph Matching with Improved Noise Robustness

   Mao, Cheng ; Rudelson, Mark ; Tikhomirov, Konstantin ( August 2021 , Proceedings of Machine Learning Research)

   Belkin, Mikhail ; Kpotufe, Samory (Ed.)

   Graph matching, also known as network alignment, refers to finding a bijection between the vertex sets of two given graphs so as to maximally align their edges. This fundamental computational problem arises frequently in multiple fields such as computer vision and biology. Recently, there has been a plethora of work studying efficient algorithms for graph matching under probabilistic models. In this work, we propose a new algorithm for graph matching: Our algorithm associates each vertex with a signature vector using a multistage procedure and then matches a pair of vertices from the two graphs if their signature vectors are close to each other. We show that, for two Erdős–Rényi graphs with edge correlation $1-\alpha$, our algorithm recovers the underlying matching exactly with high probability when $\alpha \le 1 / (\log \log n)^C$, where $n$ is the number of vertices in each graph and $C$ denotes a positive universal constant. This improves the condition $\alpha \le 1 / (\log n)^C$ achieved in previous work. 

   more » « less
2. The Power of D-hops in Matching Power-Law Graphs

   https://doi.org/10.1145/3460094  

   Yu, Liren ; Xu, Jiaming ; Lin, Xiaojun ( June 2021 , Proceedings of the ACM on Measurement and Analysis of Computing Systems)

   null (Ed.)

   This paper studies seeded graph matching for power-law graphs. Assume that two edge-correlated graphs are independently edge-sampled from a common parent graph with a power-law degree distribution. A set of correctly matched vertex-pairs is chosen at random and revealed as initial seeds. Our goal is to use the seeds to recover the remaining latent vertex correspondence between the two graphs. Departing from the existing approaches that focus on the use of high-degree seeds in $1$-hop neighborhoods, we develop an efficient algorithm that exploits the low-degree seeds in suitably-defined D-hop neighborhoods. Specifically, we first match a set of vertex-pairs with appropriate degrees (which we refer to as the first slice) based on the number of low-degree seeds in their D-hop neighborhoods. This approach significantly reduces the number of initial seeds needed to trigger a cascading process to match the rest of graphs. Under the Chung-Lu random graph model with n vertices, max degree Θ(√n), and the power-law exponent 2<β<3, we show that as soon as D> 4-β/3-β, by optimally choosing the first slice, with high probability our algorithm can correctly match a constant fraction of the true pairs without any error, provided with only Ω((log n)4-β) initial seeds. Our result achieves an exponential reduction in the seed size requirement, as the best previously known result requires n1/2+ε seeds (for any small constant ε>0). Performance evaluation with synthetic and real data further corroborates the improved performance of our algorithm. 

   more » « less
3. A Parallel 2/3-Approximation Algorithm for Vertex-Weighted Matching

   https://doi.org/10.1137/1.9781611976229.2  

   Al-Herz, Ahmed ; Pothen, Alex ( January 2020 , SIAM 2020 Workshop on Combinatorial Scientific Computing)

   We consider the maximum vertex-weighted matching problem (MVM), in which non-negative weights are assigned to the vertices of a graph, and the weight of a matching is the sum of the weights of the matched vertices. Although exact algorithms for MVM are faster than exact algorithms for the maximum edge-weighted matching problem, there are graphs on which these exact algorithms could take hundreds of hours. For a natural number k, we design a k/(k + 1)approximation algorithm for MVM on non-bipartite graphs that updates the matching along certain short paths in the graph: either augmenting paths of length at most 2k + 1 or weight-increasing paths of length at most 2k. The choice of k = 2 leads to a 2/3-approximation algorithm that computes nearly optimal weights fast. This algorithm could be initialized with a 2/3-approximate maximum cardinality matching to reduce its runtime in practice. A 1/2-approximation algorithm may be obtained using k = 1, which is faster than the 2/3-approximation algorithm but it computes lower weights. The 2/3-approximation algorithm has time complexity O(Δ2m) while the time complexity of the 1/2-approximation algorithm is O(Δm), where m is the number of edges and Δ is the maximum degree of a vertex. Results from our serial implementations show that on average the 1/2-approximation algorithm runs faster than the Suitor algorithm (currently the fastest 1/2-approximation algorithm) while the 2/3-approximation algorithm runs as fast as the Suitor algorithm but obtains higher weights for the matching. One advantage of the proposed algorithms is that they are well-suited for parallel implementation since they can process vertices to match in any order. The 1/2- and 2/3-approximation algorithms have been implemented on a shared memory parallel computer, and both approximation algorithms obtain good speedups, while the former algorithm runs faster on average than the parallel Suitor algorithm. Care is needed to design the parallel algorithm to avoid cyclic waits, and we prove that it is live-lock free. 

   more » « less
4. Generalized Stochastic Matching

   https://doi.org/10.1609/aaai.v36i9.21239  

   Farhadi, Alireza ; Gilbert, Jacob ; Hajiaghayi, MohammadTaghi ( June 2022 , Proceedings of the AAAI Conference on Artificial Intelligence)

   In this paper, we generalize the recently studied stochastic matching problem to more accurately model a significant medical process, kidney exchange, and several other applications. Up until now the stochastic matching problem that has been studied was as follows: given a graph G= (V,E), each edge is included in the realized sub-graph of G independently with probability pe, and the goal is to find a degree-bounded sub-graph Q of G that has an expected maximum matching that approximates the expected maximum matching of G. This model does not account for possibilities of vertex dropouts, which can be found in several applications, e.g. in kidney exchange when donors or patients opt out of the exchange process as well as in online freelancing and online dating when online profiles are found to be faked. Thus, we will study a more generalized model of stochastic matching in which vertices and edges are both realized independently with some probabilities pv, pe, respectively, which more accurately fits important applications than the previously studied model. We will discuss the first algorithms and analysis for this generalization of the stochastic matching model and prove that they achieve good approximation ratios. In particular, we show that the approximation factor of a natural algorithm for this problem is at least 0.6568 in unweighted graphs, and 1/2+ε in weighted graphs for some constant ε >0. We further improve our result for unweighted graphs to 2/3 using edge degree constrained sub-graphs (EDCS). 

   more » « less
5. Local Access to Random Walks , January 2022

   Biswas, A.S. ; Pyne, E. ; Rubinfeld, R. ( January 2022 , Innovations in Theoretical Computer Science (ITCS 2022))

   For a graph G on n vertices, naively sampling the position of a random walk of at time t requires work Ω(t). We desire local access algorithms supporting positionG(t) queries, which return the position of a random walk from some fixed start vertex s at time t, where the joint distribution of returned positions is 1/ poly(n) close to those of a uniformly random walk in ℓ1 distance. We first give an algorithm for local access to random walks on a given undirected d-regular graph with eO( 1 1−λ √ n) runtime per query, where λ is the second-largest eigenvalue of the random walk matrix of the graph in absolute value. Since random d-regular graphs G(n, d) are expanders with high probability, this gives an eO(√ n) algorithm for a graph drawn from G(n, d) whp, which improves on the naive method for small numbers of queries. We then prove that no algorithm with subconstant error given probe access to an input d-regular graph can have runtime better than Ω(√ n/ log(n)) per query in expectation when the input graph is drawn from G(n, d), obtaining a nearly matching lower bound. We further show an Ω(n1/4) runtime per query lower bound even with an oblivious adversary (i.e. when the query sequence is fixed in advance). We then show that for families of graphs with additional group theoretic structure, dramatically better results can be achieved. We give local access to walks on small-degree abelian Cayley graphs, including cycles and hypercubes, with runtime polylog(n) per query. This also allows for efficient local access to walks on polylog degree expanders. We show that our techniques apply to graphs with high degree by extending or results to graphs constructed using the tensor product (giving fast local access to walks on degree nϵ graphs for any ϵ ∈ (0, 1]) and Cartesian product. 

   more » « less