In the bottleneck multiple knapsack problem, we are given a set of items and a set of knapsacks, where each item has a profit and a weight, and each knapsack has a capacity. Our goal is to assign items to knapsacks so as to maximize the minimum profit received by any knapsack subject to the capacity constraint. When all knapsacks have identical capacity, we give a (2/3 - ε)-approximation algorithm for any constant ε > 0. This result almost matches the (2/3 + ε) inapproximability bound for the bottleneck multiple subset sum problem (Caprara et al., 2000). When the knapsacks can have arbitrary capacities, we propose a (1/2 - ε)-approximation algorithm for any constant ε > 0. We also prove a hardness bound of (1/2 + ε) for any constant ε > 0.

We study the fair allocation of indivisible items to n agents to maximize the utilitarian social welfare, where the fairness criterion is envy-free up to one item and there are only two different utility functions shared by the agents. We present a 2-approximation algorithm when the two utility functions are normalized, improving the previous best ratio of 16 √n shown for general normalized utility functions; thus this constant ratio approximation algorithm confirms the APX-completeness in this special case previously shown APX-hard. When there are only three agents, i.e., n = 3, the previous best ratio is 3 shown for general utility functions, and we present an improved and tight 5/3-approximation algorithm when the two utility functions are normalized, and a best possible and tight 2-approximation algorithm when the two utility functions are unnormalized.

We study a variant of the well-known Path Cover problem where the candidate paths in a solution have orders up to a fixed integer k . In Path Cover, one finds a minimum number of vertex-disjoint paths in an input graph to cover all the vertices; in our variant, not all paths but only those short ones, i.e., containing up to k vertices, can be used as candidates. The problem is NP-hard when k ≥ 3 ; in the literature, there exist quite a number of approximation algorithms, especially for small k 's. We present an improved k 3 -approximation algorithm for k ∈ { 6 , 7 , 8 } , an improved 55 31 -approximation algorithm for k = 5 , and an improved 8 5 -approximation algorithm for k = 4 . The novelty inside these improved algorithms is observing a close connection between an optimal path cover and a certain polynomial-time computed edge set.

Advanced resource reservation is a network protocol to guarantee quality of service (QoS) in data transmission. In a previous paper, Xue (see ICT2000: IEEE International Conference on Telecommunications, p.1071-75) introduced a network model using attribute lists to capture the availability of resources in networks with advanced resource reservations and formally defined various QoS routing problems with advanced resource reservation. In this paper, we present an efficient algorithm for computing an optimal solution to the source-waiting QoS routing problem under certain conditions.

In wavelength routed optical networks, wavelength converters can potentially reduce the requirement on the number of wavelengths. The problem of placing a minimum number of wavelength converters in a WDM network so that any routing can be satisfied using no more wavelengths than if there were wavelength converters at every node was raised by Wilfong and Winkler (1998) as the minimum sufficient set problem. This problem is NP-complete in general WDM networks. Wan et al. (1999), showed that the problem is tractable if every edge in the network is bi-directed and the skeleton of the network is a tree of rings. We show that the minimum sufficient set problem is tractable in any directed graph with a general tree of rings skeleton.