On-line complexity of monotone set systems

On-line analysis models a player A (randomized or deterministic) who makes immediate responses to incoming elements of an input sequence s = a(1)...a(r). In this paper a(l),...,a(r) are interpreted as elements offered without repetition to the player from a fixed universe and the player's response to each a(i) is a single bit interpreted as pick/not-to-pick, Before seeing the stream s, the player is given a monotone system M of sets indicating all sets of elements that she is allowed to hold at any time during the game and the player's objective is to pick as many elements as possible under these constraints. We assign the performance measure e(M) = max(A) min(s) E(\A(s)\)/OPT(s) to every monotone set system M, where E() is the expectation over the player's random choices and OPT(s) = max(M is an element of M) \M boolean AND s\. Note that for every M, e(M) is an element of [0, 1], and it is easy to show that a system has performance one iff it is a matroid. This model generalizes the apparently unrelated models in [2] and [3]. The video-on-demand problem studied in [1, 2] corresponds to the case when M is the monotone closure of I disjoint sets of size k. Let e(k, l) denote the performance of this set system. Awerbuch et al. [1] show that e(k, l) = Omega(1/(log k log l)). We improve this bound for some values of k and l by showing that e(k, 2) = k/(2k - 1), e(k, 3) = k/(3k - circle minus(root k)), and e(k, l) greater than or equal to k(l-1)/(k(l) - (k - 1)(l)). In particular, for any fixed 1 our bounds tend to 1/l as k grows whether the bound of Awerbuch et al. tends to zero. We also show, that if M = M1U...UMl, and k is an upper bound on the size of the largest set in M, then the performance of M is at least as good as the performance of the worst of the components multiplied by a factor of e(k, l). Our model also generalizes one of Bartal et al. [3] who considered set systems that arise from hereditary graph properties. Bartal et al. use specific graph product to prove that for every n there exists a graph G with O(n) vertices such that the performance of the system of its independent sets is no greater than n(-0.207). Unfortunately their argument contains a flaw. We generalize the graph product to set systems, define the conditional performance of a set system and prove that the conditional performance of the product of set systems is bounded by the product of their conditional performances. As a corollary of this theorem we obtain a correct proof to Bartal et al.'s upper bound and even slightly improve it to n(-0.226) using a better base graph.