In the dynamic minimum set cover problem, a challenge is to minimize the update time while guaranteeing close to the optimal min(O(logn),f) approximation factor. (Throughout, m, n, f, and C
Nearby in the stack
are parameters denoting the maximum number of sets, number of elements, frequency, and the cost range.) In the high-frequency range, when
f=Ω(logn)
, this was achieved by a deterministic
O(logn)
-approximation algorithm with
O(flogn)
amortized update time [Gupta et al. STOC'17]. In the low-frequency range, the line of work by Gupta et al. [STOC'17], Abboud et al. [STOC'19], and Bhattacharya et al. [ICALP'15, IPCO'17, FOCS'19] led to a deterministic
(1+ε)f
-approximation algorithm with
O(flog(Cn)/ε2)
amortized update time. In this paper we improve the latter update time and provide the first bounds that subsume (and sometimes improve) the state-of-the-art dynamic vertex cover algorithms. We obtain: 1.
(1+ε)f
-approximation ratio in
O(flog2(Cn)/ε3)
worst-case update time: No non-trivial worst-case update time was previously known for dynamic set cover. Our bound subsumes and improves by a logarithmic factor the
O(log3n/poly(ε))
worst-case update time for unweighted dynamic vertex cover (i.e., when
f=2
and
C=1
) by Bhattacharya et al. [SODA'17]. 2.
(1+ε)f
-approximation ratio in
O((f2/ε3)+(f/ε2)logC)
amortized update time: This result improves the previous
O(flog(Cn)/ε2)
update time bound for most values of
f
in the low-frequency range, i.e. whenever
f=o(logn)
. It is the first that is independent of
m
and
n
. It subsumes the constant amortized update time of Bhattacharya and Kulkarni [SODA'19] for unweighted dynamic vertex cover (i.e., when