Online Submodular Maximization Problem with Vector Packing Constraint

We consider the online vector packing problem in which we have a $d$ dimensional knapsack and items $u$ with weight vectors $\mathbf{w}u \in \mathbb{R}+^d$ arrive online in an arbitrary order. Upon the arrival of an item, the algorithm must decide immediately whether to discard or accept the item into the knapsack. When item $u$ is accepted, $\mathbf{w}u(i)$ units of capacity on dimension $i$ will be taken up, for each $i\in[d]$. To satisfy the knapsack constraint, an accepted item can be later disposed of with no cost, but discarded or disposed of items cannot be recovered. The objective is to maximize the utility of the accepted items $S$ at the end of the algorithm, which is given by $f(S)$ for some non-negative monotone submodular function $f$. For any small constant $\epsilon > 0$, we consider the special case that the weight of an item on every dimension is at most a $(1-\epsilon)$ fraction of the total capacity, and give a polynomial-time deterministic $O(\frac{k}{\epsilon^{2})$-competitive} algorithm for the problem, where $k$ is the (column) sparsity of the weight vectors. We also show several (almost) tight hardness results even when the algorithm is computationally unbounded. We show that under the $\epsilon$-slack assumption, no deterministic algorithm can obtain any $o(k)$ competitive ratio, and no randomized algorithm can obtain any $o(\frac{k}{\log k})$ competitive ratio. For the general case (when $\epsilon = 0$), no randomized algorithm can obtain any $o(k)$ competitive ratio. In contrast to the $(1+\delta)$ competitive ratio achieved in Kesselheim et al. (STOC 2014) for the problem with random arrival order of items and under large capacity assumption, we show that in the arbitrary arrival order case, even when $| \mathbf{w}u |_\infty$ is arbitrarily small for all items $u$, it is impossible to achieve any $o(\frac{\log k}{\log\log k})$ competitive ratio.