(1 + ε)-Approximate Incremental Matching in Constant Deterministic Amortized Time ∗ Fabrizio Grandoni † , Stefano Leonardi ‡ , Piotr Sankowski § Chris Schwiegelshohn ¶ , Shay Solomon ‖ Abstract We study the matching problem in the incremental setting, where we are given a sequence of edge insertions and aim at maintaining a near-maximum cardinality matching of the graph with small update time. We present a deterministic algorithm that, for any constant ε> 0, maintains a (1 + ε)- approximate matching with constant amortized update time per insertion. 1 Introduction Let G = (V,E) be an n-node m-edge undirected graph. Finding a large cardinality matching in G is a fundamental optimization problem. For bipartite graphs, the currently best available time bounds are O(m √ n) due to Hopcroft and Karp [21], O(n ω ) due to Mucha and Sankowski [29] and ˜ O(m 10/7 ) due to Madry [27]. The former two algorithms have been extend to finding matchings in general (non-bipartite) graphs as well [28, 29]. In contrast to this static case (where the graph is given up-front), there has been recently a lot of interest in the dynamic matching problem. In dynamic setting we must maintain a (near-)optimal matching as the graph changes over time. Most of the results have been given in the fully-dynamic model where edges are added or deleted over time. It is known how to maintain the size of the maximum matching with O(n 1.495 ) worst- ∗ This work was done in part while a subset of the authors was visiting the Algorithms and Uncertainty and Bridging Discrete and Continuous Optimization programs at the Simons Institute for the Theory of Computing. F. Grandoni is partially supported by the SNSF Grant 200021 159697/1 and the SNSF Excellence Grant 200020B 182865/1. S. Leonardi and C. Schwiegelshohn are partially supported by the ERC Advanced Grant 788893 AM- DROMA. P. Sankowski is partially supported by ERC Consolida- tor Grant 772346 TUgbOAT and Polish National Science Centre grant 2014/13/B/ST6/00770. † IDSIA, USI-SUPSI ‡ Sapienza University of Rome § University of Warsaw ¶ Sapienza University of Rome ‖ Tel Aviv University case update time [33]. And we known that maintaining the exact value of the maximum matching requires polynomial update time under reasonable complexity conjectures [2, 20, 26]. Hence, we turn our attention to approximate matchings. In this case we know how to maintain 2-approximate matchings with constant amortized update time [34], but algorithms achieving better-than-2 approximations all require polynomial update time. In particular, we can maintain a (1 + ε)- approximate matching in the fully-dynamic setting with update time O( √ m/ε 2 )[18], a (3/2+ ε)-approximate matching with update time O(m 1/4 /ε 2.5 )[8], and for every sufficiently large integer K, an α K -approximation to the matching size with update time O(n 2/K ) where α K ∈ (1, 2). (We survey the existing results in detail in Section 1.2.) This suggests the question: can we achieve better approximation in some natural dynamic settings? In this paper, we consider the incremental model for dynamic algorithms, where the edges of the graph can only be inserted but not deleted. We show that in this case we can give much stronger results than in the fully-dynamic model: Theorem 1.1. Given a sequence of edge insertions to a graph G and a constant ε > 0, there exists a deterministic algorithm that maintains a (1 + ε)- approximate matching with O ε (1) amortized update time per insertion. We remark that by [26], maintaining a maximum matching requires polynomial amortized update time even in the incremental case assuming the 3-SUM conjecture. Hence, our result is asymptotically optimal, up to deamortization. The only previous result for approximate matchings in the incremental model is due to Gupta [16], who gave an amortized O(log 2 n) update-time algorithm to maintain (1 + ε)-approximate matchings in bipartite graphs. Hence, we improve the update time from polylogarithmic to constant. Moreover, we also extend the result from bipartite graphs to general graphs. Copyright © 2019 by SIAM Unauthorized reproduction of this article is prohibited 1886 Downloaded 10/26/20 to 130.225.0.251. Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/page/terms