Deterministic Near-Optimal Approximation Algorithms for Dynamic Set Cover
Sayan Bhattacharya, Monika Rauch Henzinger, Danupon Nanongkai, Xiaowei Wu · SIAM Journal on Computing · 2023
Abstract. In the dynamic minimum set cover problem, the challenge is to minimize the update time while guaranteeing a close-to-optimal [Formula: see text] approximation factor. (Throughout, [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text] are parameters denoting the maximum number of elements, the number of sets, the frequency, and the cost range.) In the high-frequency range, when [Formula: see text], this was achieved by a deterministic [Formula: see text]-approximation algorithm with [Formula: see text] amortized update time by Gupta et al. [ Online and dynamic algorithms for set cover, in Proceedings STOC 2017, ACM, pp. 537–550]. In this paper we consider the low-frequency range, when [Formula: see text], and obtain deterministic algorithms with a [Formula: see text]-approximation ratio and the following guarantees on the update time. (1) [Formula: see text] amortized update time: Prior to our work, the best approximation ratio guaranteed by deterministic algorithms was [Formula: see text] 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 [Formula: see text]-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 [Formula: see text]-approximation algorithm with [Formula: see text] amortized update time. (2) [Formula: see text] amortized update time: This result improves the above update time bound for most values of [Formula: see text] in the low-frequency range, i.e., [Formula: see text]. It is also the first result that is independent of [Formula: see text] and [Formula: see text]. It subsumes the constant amortized update time of Bhattacharya and Kulkarni [ Deterministically maintaining a [Formula: see text]-approximate minimum vertex cover in [Formula: see text] amortized update time, in Proceedings SODA 2019, SIAM, pp. 1872–1885] for unweighted dynamic vertex cover (i.e., when [Formula: see text] and [Formula: see text]). (3) [Formula: see text] 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 [Formula: see text] worst-case update time for the unweighted dynamic vertex cover problem (i.e., when [Formula: see text] and [Formula: see text]) of Bhattacharya, Henzinger, and Nanongkai [ Fully dynamic approximate maximum matching and minimum vertex cover in [Formula: see text] 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.