Paper accepted at ICALP

We present a simple randomized reduction from fully-dynamic integral matching algorithms to fully-dynamic "approximately-maximal" fractional matching algorithms. Applying this reduction to the recent fractional matching algorithm of Bhattacharya, Henzinger, and Nanongkai (SODA 2017), we obtain a novel result for the integral problem. Specifically, our main result is a randomized fully-dynamic \((2+\varepsilon)\)-approximate integral matching algorithm with small polylog worst-case update time. For the \((2+\varepsilon)\)-approximation regime only a fractional fully-dynamic \((2+\varepsilon)\)-matching algorithm with worst-case polylog update time was previously known, due to Bhattacharya et al. (SODA 2017). Our algorithm is the first algorithm that maintains approximate matchings with worst-case update time better than polynomial, for any constant approximation ratio. As a consequence, we also obtain the first constant-approximate worst-case polylogarithmic update time maximum weight matching algorithm.

We study a natural problem in graph sparsification, the Spanning Tree Congestion (STC) problem. Informally, it seeks a spanning tree with no tree-edge routing too many of the original edges. For any general connected graph with n vertices and m edges, we show that its STC is at most \( \mathcal{O(\sqrt{mn}) \), which is asymptotically optimal since we also demonstrate graphs with STC at least \( \Omega(\sqrt{mn}) \). We present a polynomial-time algorithm which computes a spanning tree with congestion \( \mathcal{O(\sqrt{mn}) \log n \) \( \mathcal{O(\sqrt{mn} \log n ) \). We also present another algorithm for computing a spanning tree with congestion \( \mathcal{O(\sqrt{mn}) \); this algorithm runs in sub-exponential time when \(m = \omega(n \log^2 n ) \). For achieving the above results, an important intermediate theorem is generalized Györi-Lovász theorem. Chen et al. [Jiangzhuo Chen et al., 2007] gave a non-constructive proof. We give the first elementary and constructive proof with a local search algorithm of running time \( \mathcal{O^*( 4^n ) \). We discuss some consequences of the theorem concerning graph partitioning, which might be of independent interest. We also show that for any graph which satisfies certain expanding properties, its STC is at most \( \mathcal{O^*(n) \), and a corresponding spanning tree can be computed in polynomial time. We then use this to show that a random graph has STC \( \Theta(n) \) with high probability.

A common way to accelerate shortest path algorithms on graphs is the use of a bidirectional search, which simultaneously explores the graph from the start and the destination. It has been observed recently that this strategy performs particularly well on scale-free real-world networks. Such networks typically have a heterogeneous degree distribution (e.g., a power-law distribution) and high clustering (i.e., vertices with a common neighbor are likely to be connected themselves). These two properties can be obtained by assuming an underlying hyperbolic geometry. To explain the observed behavior of the bidirectional search, we analyze its running time on hyperbolic random graphs and prove that it is \(\tilde{O}(n\)\(^{2 - 1/ \alpha}+\)\(n^{1/(2\alpha)}\)\(+ \delta_{\max})\) with high probability, where \(\alpha\)\(\in\)\((0.5, 1)\) controls the power-law exponent of the degree distribution, and \(\delta_{\max}\) is the maximum degree. This bound is sublinear, improving the obvious worst-case linear bound. Although our analysis depends on the underlying geometry, the algorithm itself is oblivious to it.