Deterministic near-optimal approximation algorithms for dynamic set cover

In the dynamic minimum set cover problem, the challenge is to minimize the update time while guaranteeing a close-to-optimal min{O(log n), f\} approximation factor. (Throughout, n, m, f, and C are parameters denoting the maximum number of elements, the number of sets, the frequency, and the cost range.) In the high-frequency range, when f = \Omega (log n), this was achieved by a deterministic O(log n)-approximation algorithm with O(f log n) amortized update time by Gupta et al. [Online and dynamic algorithms for set cover, in Proceedings STOC 2017, ACM, pp. 537550]. In this paper we consider the low-frequency range, when f = O(log n), and obtain deterministic algorithms with a (1 + \epsilon )f-approximation ratio and the following guarantees on the update time. (1) O ((f /\epsilon ) \cdot log(Cn)) amortized update time: Prior to our work, the best approximation ratio guaranteed by deterministic algorithms was O(f2) of Bhattacharya, Henzinger, and Italiano [Design of dynamic algorithms via primal-dual method, in Proceedings ICALP 2015, Springer, pp. 206--218]. In contrast, the only result with O(f)-approximation was that of Abboud et al. [Dynamic set cover: Improved algorithms and lower bounds, in Proceedings STOC 2019, ACM, pp. 114--125], who designed a randomized (1 + \epsilon )f-approximation algorithm with O((f2/\epsilon ) \cdot log n) amortized update time. (2) O \bigl(f2/\epsilon3 + (f/\epsilon2) \cdot log C\bigr) amortized update time: This result improves the above update time bound for most values off in the low-frequency range, i.e., f = o(log n). It is also the first result that is independent of m and n. It subsumes the constant amortized update time of Bhattacharya and Kulkarni [Deterministically maintaining a (2 + \epsilon )-approximate minimum vertex cover in O(1/\epsilon2) amortized update time, in Proceedings SODA 2019, SIAM, pp. 1872--1885] for unweighted dynamic vertex cover (i.e., when f = 2 and C = 1). (3) O((f/\epsilon3) \cdot log2(Cn)) worst-case update time: No nontrivial worst-case update time was previously known for the dynamic set cover problem. Our bound subsumes and improves by a logarithmic factor the O(log3 n/poly(\epsilon )) worst-case update time for the unweighted dynamic vertex cover problem (i.e., when f = 2 and C = 1) of Bhattacharya, Henzinger, and Nanongkai [Fully dynamic approximate maximum matching and minimum vertex cover in O(log3)n worst case update time, in Proceedings SODA 2017, SIAM, pp. 470--489]. We achieve our results via the primal-dual approach, by maintaining a fractional packing solution as a dual certificate. Prior work in dynamic algorithms that employs the primal-dual approach uses a local update scheme that maintains relaxed complementary slackness conditions for every set. For our first result we use instead a global update scheme that does not always maintain complementary slackness conditions. For our second result we combine the global and the local update schema. To achieve our third result we use a hierarchy of background schedulers. It is an interesting open question whether this background scheduler technique can also be used to transform algorithms with amortized running time bounds into algorithms with worst-case running time bounds.