Following Mettu and Plaxton, we study incremental algorithms for the
$k$-medians problem. Such an algorithm must produce a nested
$F_1 \subseteq F_2 \subseteq \cdots \subseteq F_n$
of sets of
facilities. Mettu and Plaxton show that incremental metric medians has
a (roughly) $40$-competitive deterministic polynomial-time
algorithm. We give improved algorithms, including a
$(24+ε)$-competitive deterministic polynomial-time algorithm
and a $5.44$-competitive, randomized, non-polynomial-time algorithm.
We also consider the competitive ratio with respect to size. An
algorithm is $s$-size-competitive if, for each $k$, the cost of $F_k$
is at most the minimum cost of any set of $k$ facilities, while the
size of $F_k$ is at most $s k$. We present optimally competitive
algorithms for this problem.
Our proofs reduce incremental medians to
the following online bidding problem: faced with some unknown
threshold $T>0$, an algorithm must submit ``bids'' $b>0$ until it
submits a bid as large as $T$. The algorithm pays the sum of its
bids. We describe optimally competitive algorithms for online
Our results on cost-competitive incremental medians extend to
approximately metric distance functions, incremental fractional
medians, and incremental bicriteria approximation.
Journal version of [2006