A Generalized Matching Reconfiguration Problem

The goal in reconfiguration problems is to compute a gradual transformation between two feasible solutions of a problem such that all intermediate solutions are also feasible. In the Matching Reconfiguration Problem (MRP), proposed in a pioneering work by Ito et al. from 2008, we are given a graph G and two matchings M and M\u27, and we are asked whether there is a sequence of matchings in G starting with M and ending at M\u27, each resulting from the previous one by either adding or deleting a single edge in G, without ever going through a matching of size < min{|M|,|M\u27|}-1. Ito et al. gave a polynomial time algorithm for the problem, which uses the Edmonds-Gallai decomposition. In this paper we introduce a natural generalization of the MRP that depends on an integer parameter ? ? 1: here we are allowed to make ? changes to the current solution rather than 1 at each step of the {transformation procedure}. There is always a valid sequence of matchings transforming M to M\u27 if ? is sufficiently large, and naturally we would like to minimize ?. We first devise an optimal transformation procedure for unweighted matching with ? = 3, and then extend it to weighted matchings to achieve asymptotically optimal guarantees. The running time of these procedures is linear. We further demonstrate the applicability of this generalized problem to dynamic graph matchings. In this area, the number of changes to the maintained matching per update step (the recourse bound) is an important quality measure. Nevertheless, the worst-case recourse bounds of almost all known dynamic matching algorithms are prohibitively large, much larger than the corresponding update times. We fill in this gap via a surprisingly simple black-box reduction: Any dynamic algorithm for maintaining a ?-approximate maximum cardinality matching with update time T, for any ? ? 1, T and ? > 0, can be transformed into an algorithm for maintaining a (?(1 +?))-approximate maximum cardinality matching with update time T + O(1/?) and worst-case recourse bound O(1/?). This result generalizes for approximate maximum weight matching, where the update time and worst-case recourse bound grow from T + O(1/?) and O(1/?) to T + O(?/?) and O(?/?), respectively; ? is the graph aspect-ratio. We complement this positive result by showing that, for ? = 1+?, the worst-case recourse bound of any algorithm produced by our reduction is optimal. As a corollary, several key dynamic approximate matching algorithms - with poor worst-case recourse bounds - are strengthened to achieve near-optimal worst-case recourse bounds with no loss in update time

Near-Optimal Dynamic Rounding of Fractional Matchings in Bipartite Graphs

We study dynamic $(1-\epsilon)$-approximate rounding of fractional matchings -- a key ingredient in numerous breakthroughs in the dynamic graph algorithms literature. Our first contribution is a surprisingly simple deterministic rounding algorithm in bipartite graphs with amortized update time $O(\epsilon^{-1} \log^2 (\epsilon^{-1} \cdot n))$, matching an (unconditional) recourse lower bound of $\Omega(\epsilon^{-1})$ up to logarithmic factors. Moreover, this algorithm's update time improves provided the minimum (non-zero) weight in the fractional matching is lower bounded throughout. Combining this algorithm with novel dynamic \emph{partial rounding} algorithms to increase this minimum weight, we obtain several algorithms that improve this dependence on $n$. For example, we give a high-probability randomized algorithm with $\tilde{O}(\epsilon^{-1}\cdot (\log\log n)^2)$-update time against adaptive adversaries. (We use Soft-Oh notation, $\tilde{O}$, to suppress polylogarithmic factors in the argument, i.e., $\tilde{O}(f)=O(f\cdot \mathrm{poly}(\log f))$.) Using our rounding algorithms, we also round known $(1-\epsilon)$-decremental fractional bipartite matching algorithms with no asymptotic overhead, thus improving on state-of-the-art algorithms for the decremental bipartite matching problem. Further, we provide extensions of our results to general graphs and to maintaining almost-maximal matchings.Comment: Full version of STOC 2024 pape

Dynamic Maxflow via Dynamic Interior Point Methods

In this paper we provide an algorithm for maintaining a $(1-\epsilon)$-approximate maximum flow in a dynamic, capacitated graph undergoing edge additions. Over a sequence of $m$-additions to an $n$-node graph where every edge has capacity $O(\mathrm{poly}(m))$ our algorithm runs in time $\widehat{O}(m \sqrt{n} \cdot \epsilon^{-1})$. To obtain this result we design dynamic data structures for the more general problem of detecting when the value of the minimum cost circulation in a dynamic graph undergoing edge additions obtains value at most $F$ (exactly) for a given threshold $F$. Over a sequence $m$-additions to an $n$-node graph where every edge has capacity $O(\mathrm{poly}(m))$ and cost $O(\mathrm{poly}(m))$ we solve this thresholded minimum cost flow problem in $\widehat{O}(m \sqrt{n})$. Both of our algorithms succeed with high probability against an adaptive adversary. We obtain these results by dynamizing the recent interior point method used to obtain an almost linear time algorithm for minimum cost flow (Chen, Kyng, Liu, Peng, Probst Gutenberg, Sachdeva 2022), and introducing a new dynamic data structure for maintaining minimum ratio cycles in an undirected graph that succeeds with high probability against adaptive adversaries.Comment: 30 page

Online Duet between Metric Embeddings and Minimum-Weight Perfect Matchings

Low-distortional metric embeddings are a crucial component in the modern algorithmic toolkit. In an online metric embedding, points arrive sequentially and the goal is to embed them into a simple space irrevocably, while minimizing the distortion. Our first result is a deterministic online embedding of a general metric into Euclidean space with distortion $O(\log n)\cdot\min\{\sqrt{\log\Phi},\sqrt{n}\}$ (or, $O(d)\cdot\min\{\sqrt{\log\Phi},\sqrt{n}\}$ if the metric has doubling dimension $d$), solving a conjecture by Newman and Rabinovich (2020), and quadratically improving the dependence on the aspect ratio $\Phi$ from Indyk et al.\ (2010). Our second result is a stochastic embedding of a metric space into trees with expected distortion $O(d\cdot \log\Phi)$, generalizing previous results (Indyk et al.\ (2010), Bartal et al.\ (2020)). Next, we study the \emph{online minimum-weight perfect matching} problem, where a sequence of $2n$ metric points arrive in pairs, and one has to maintain a perfect matching at all times. We allow recourse (as otherwise the order of arrival determines the matching). The goal is to return a perfect matching that approximates the \emph{minimum-weight} perfect matching at all times, while minimizing the recourse. Our third result is a randomized algorithm with competitive ratio $O(d\cdot \log \Phi)$ and recourse $O(\log \Phi)$ against an oblivious adversary, this result is obtained via our new stochastic online embedding. Our fourth result is a deterministic algorithm against an adaptive adversary, using $O(\log^2 n)$ recourse, that maintains a matching of weight at most $O(\log n)$ times the weight of the MST, i.e., a matching of lightness $O(\log n)$. We complement our upper bounds with a strategy for an oblivious adversary that, with recourse $r$, establishes a lower bound of $\Omega(\frac{\log n}{r \log r})$ for both competitive ratio and lightness.Comment: 53 pages, 8 figures, to be presented at the ACM-SIAM Symposium on Discrete Algorithms (SODA24

Online Minimum Cost Matching with Recourse on the Line

In online minimum cost matching on the line, n requests appear one by one and have to be matched immediately and irrevocably to a given set of servers, all on the real line. The goal is to minimize the sum of distances from the requests to their respective servers. Despite all research efforts, it remains an intriguing open question whether there exists an O(1)-competitive algorithm. The best known online algorithm by Raghvendra [S. Raghvendra, 2018] achieves a competitive factor of ?(log n). This result matches a lower bound of ?(log n) [A. Antoniadis et al., 2018] that holds for a quite large class of online algorithms, including all deterministic algorithms in the literature. In this work, we approach the problem in a recourse model where we allow to revoke online decisions to some extent, i.e., we allow to reassign previously matched edges. We show an O(1)-competitive algorithm for online matching on the line with amortized recourse of O(log n). This is the first non-trivial result for min-cost bipartite matching with recourse. For so-called alternating instances, with no more than one request between two servers, we obtain a near-optimal result. We give a (1+?)-competitive algorithm that reassigns any request at most O(?^{-1.001}) times. This special case is interesting as the aforementioned quite general lower bound ?(log n) holds for such instances

Adaptive Out-Orientations with Applications

We give simple algorithms for maintaining edge-orientations of a fully-dynamic graph, such that the out-degree of each vertex is bounded. On one hand, we show how to orient the edges such that the out-degree of each vertex is proportional to the arboricity $\alpha$ of the graph, in a worst-case update time of $O(\log^2 n \log \alpha)$. On the other hand, motivated by applications in dynamic maximal matching, we obtain a different trade-off, namely the improved worst case update time of $O(\log n \log \alpha)$ for the problem of maintaining an edge-orientation with at most $O(\alpha + \log n)$ out-edges per vertex. Since our algorithms have update times with worst-case guarantees, the number of changes to the solution (i.e. the recourse) is naturally limited. Our algorithms make choices based entirely on local information, which makes them automatically adaptive to the current arboricity of the graph. In other words, they are arboricity-oblivious, while they are arboricity-sensitive. This both simplifies and improves upon previous work, by having fewer assumptions or better asymptotic guarantees. As a consequence, one obtains an algorithm with improved efficiency for maintaining a $(1+\varepsilon)$ approximation of the maximum subgraph density, and an algorithm for dynamic maximal matching whose worst-case update time is guaranteed to be upper bounded by $O(\alpha + \log n\log \alpha)$, where $\alpha$ is the arboricity at the time of the update