In this paper, we obtain an (1 − e −1)-approximation algorithm for maximizing a nondecreasing submodular set function subject to a knapsack constraint. This algorithm requires O(n 5) function value computations.
A directed multigraph is said to be <i>d</i>-regular if the indegree and outdegree of every vertex is exactly <i>d</i>. By Hall's theorem, one can represent such a multigraph as a combination of at most <i>n</i><sup>2</sup> cycle covers, each taken with an appropriate multiplicity. We prove that if the <i>d</i>-regular multigraph does not contain more than… (More)
We consider two types of buffering policies that are used in network switches supporting QoS (Quality of Service). In the <italic>FIFO</italic> type, packets must be released in the order they arrive; the difficulty in this case is the limited buffer space. In the <italic>bounded-delay</italic> type, each packet has a maximum delay time by which it must be… (More)
The uncapacitated facility location problem in the following formulation is considered: max SI Z(S) = X j2J max i2S b ij ? X i2S c i ; where I and J are nite sets, and b ij , c i 0 are real numbers. Let Z denote the optimal value of the problem and Z R = P j2J min i2I b ij ? P i2I c i. Cornuejols, Fisher and Nemhauser (1977) prove that for the problem with… (More)
We study the following packing problem: Given a collection of d-dimensional rectangles of specified sizes, pack them into the minimum number of unit cubes. We show that unlike the one-dimensional case, the two-dimensional packing problem cannot have an asymptotic polynomial time approximation scheme (APTAS), unless P = NP. On the positive side, we give an… (More)
A <i>separable assignment problem</i> (SAP) is defined by a set of bins and a set of items to pack in each bin; a value, <i>f</i><inf><i>ij</i></inf>, for assigning item <i>j</i> to bin <i>i;</i> and a separate packing constraint for each bin - i.e. for bin <i>i</i>, a family <i>L</i><inf><i>i</i></inf> of subsets of items that fit in bin <i>i</i>. The goal… (More)
We consider the following problem: The Santa Claus has n presents that he wants to distribute among m kids. Each kid has an arbitrary value for each present. Let p<sub>ij</sub> be the value that kid i has for present j. The Santa's goal is to distribute presents in such a way that the least lucky kid is as happy as possible, i.e he tries to maximize… (More)