On-line Complexity of Monotone Set Systems

On-line models assume a player, A (randomized or deterministic), who makes immediate responses to incomming elements of an input sequence s = a 1 : : : a r . In this paper we study the case, when the response for every a i is a single bit, which is interpreted as pick/not-topick. The player's objective is to pick as many elements as possible under the condition, that the set of picked elements, A(s), satisfies a certain prescribed property. Bartal et al. in [3] give bounds for the randomized player's worst case performance, when the incomming elements are vertices of a fixed graph G, and the property in question is some hereditary graph property. In their model G is known to the player, but she does not know when the input sequence ends. In this paper we allow A(s) to belong to an arbitrary, but fixed monotone set system M, where M is known to the player. We assign the performance measure e(M) = maxA min s E(jA(s)j)=OPT (s), to every monotone set system M, where E() is the expectation..