(1 + ∊)-Approximate Incremental Matching in Constant Deterministic Amortized Time

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.

Recent Advances in Fully Dynamic Graph Algorithms

“…For worst-case bounds, the best results are a (1 + )-approximation in O( √ / ) update time by Gupta and Peng [107] (see [183] for a 3/2approximation in the same time), a (3/2 + )-approximation in O( 1/4 / 2.5 ) time by Bernstein and Stein [34], and a (2 + )-approximation in O(polylog ) time by Charikar and Solomon [57] and Arar et al [13]. Recently, Grandoni et al [100] gave an incremental matching algorithm that achieves a (1 + )-approximate matching in constant deterministic amortized time. Finally, Bernstein et al [35] improved the maximal matching algorithm of Baswana et al [22] to O(log 5 ) worst-case time with high probability.…”

Fully-dynamic Weighted Matching Approximation in Practice

“…For worstcase bounds, the best results are by (i) Gupta and Peng [29] [17] as well as Arar et al [4] (using [13]) presented independently the first algorithms requiring O(poly log n) worst-case update time while maintaining a (2 + )-approximation. Recently, Grandoni et al [28] gave an incremental matching algorithm that achieves a (1 + )-approximate matching in constant deterministic amortized time. Barenboim and Maimon [6] present an algorithm that hasÕ( √ n) update time for graphs with constant neighborhood independence.…”

Fully-dynamic Weighted Matching Approximation in Practice

“…However, the first deterministic data structure improving [32] was given by Bhattacharya et al [10]; it maintains a (3 + ) approximate matching in Õ(min( √ n, m [14] as well as Arar et al [3] (using [12]) presented independently the first algorithms requiring O(poly log n) worst-case update time while maintaining a (2 + )-approximation. Recently, Grandoni et al [25] gave an incremental matching algorithm that achieves a (1 + )-approximate matching in constant deterministic amortized time. Barenboim and Maimon [5] present an algorithm that has Õ( √ n) update time for graphs with constant neighborhood independence.…”