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
\((24+\epsilon)\)-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