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.
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)
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)
Submodular function maximization is a central problem in combinatorial optimization, generalizing many important problems including Max Cut in directed/undirected graphs and in hypergraphs, certain constraint satisfaction problems, maximum entropy sampling, and maximum facility location problems. Unlike submodular minimization, submodular maximization is… (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)
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)
Submodular-function maximization is a central problem in combinatorial optimization , generalizing many important NP-hard problems including Max Cut in digraphs, graphs and hypergraphs, certain constraint satisfaction problems, maximum-entropy sampling, and maximum facility-location problems. Our main result is that for any k ≥ 2 and any ε > 0, there is a… (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)