Computer Science/Discrete Mathematics Seminar I

It is known that any opened necklace with beads of n types can be partitioned by at most (k-1)n cuts into intervals that can be distributed into k collections, each containing the same number of beads of each type (up to 1). The proof is topological and provides no efficient procedure for finding such cuts. In fact, for k=2 the problem of finding such cuts, and even an easier approximate version of it, are known to be PPA hard.

I will discuss efficient online and offline algorithms for solving the problem and its approximate version by increasing the number of cuts, and show that the number of cuts for the online case is essentially optimal. A representative result is an efficient (offline) algorithm that solves the problem for a necklace with n types of beads, m beads of each type and k=2 by making at most n(log m+2) cuts. The method provides an exponential improvement of a result of Bhatt and Leighton from the 80s.

Joint work with Andrei Graur.