Simple dynamic algorithms for Maximal Independent Set, Maximum Flow and Maximum Matching

Abstract

Most graphs in real life keep changing with time. These changes can be in the form of insertion or deletion of edges. Such rapidly changing graphs motivate us to study dynamic graph algorithms. We address three important dynamic graph problems as follows.Maximal Independent Set (MIS) is one of the most prominently studied problems in the distributed setting. Recently, a lot of work studied the dynamic MIS in both deterministic and randomized settings. Allowing randomization the MIS can be maintained in O(poly log n) time under various models [FOCS19], but the best deterministic algorithm [STOC18] requires O(m 3/4 ) amortized update time, which is fairly complex. We present a simple algorithm to improve this to O(m 2/3 ). Additionally, we present simple algorithms to match the lower bound of amortized Ω(n) update time [ICALP16], for maintaining incremental Maximum Flow and Maximum Matching (for unweighted graphs). These algorithms are simple extensions of the incremental reachability algorithm and blossoms algorithm, respectively. Thus, our results either improve an existing upper bound or match the existing lower bounds. A common trait of our results is that they are surprisingly simple, and use no complex data structures or black-box algorithms for their implementation. Further, they are analysed using basic amortization arguments.

On the Complexity of Weight-Dynamic Network Algorithms

“…For the case of only edge-weight changes, there is an algorithm that has the same worst-case guarantees as recomputation, but better performance on real-world cases [18]. The fully dynamic algorithm we present here also leads to an insertionsonly or deletions-only algorithm with amortized time O(n) per operation [20]. Nonetheless, various approximation algorithms exist: An O(log n log log n)-approximation in time Õ(m 3/4 ) [7] and a n o(1) -approximation in O(n o (1) ) update time [19].…”

Recent Advances in Fully Dynamic Graph Algorithms

“…Note, however, that unlike for matching a maximal independent set does not give an approximate solution for the MIS problem, as shown by a star graph. In a sequence of papers [15,16,26,58,105] the running time for the maximal independent set problem was reduced to O(log 4 ) expected worst-case update time.…”

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

“…Moreover, his approach is based on maintaining fractional matchings, which makes it harder to extend it to the non-bipartite case. Recently, Gupta and Khan [17] gave an algorithm for maintaining an exact matching with an amortized update time of O(n), which is essentially optimal (see below for lower bounds). Other incremental models have been considered in the online algorithms literature, e.g., the bipartite vertex-arrival model of Karp,Vazirani,and Vazirani [24].…”