Resource Allocation in Bounded Degree Trees

We study the

bandwidth allocation problem

(

bap

) in bounded degree trees. In this problem we are given a tree and a set of connection requests. Each request consists of a path in the tree, a bandwidth requirement, and a weight. Our goal is to find a maximum weight subset

S

of requests such that, for every edge

e

, the total bandwidth of requests in

S

whose path contains

e

is at most 1. We also consider the

storage allocation problem

(

sap

), in which it is also required that every request in the solution is given the same contiguous portion of the resource in every edge in its path. We present a deterministic approximation algorithm for

bap

in bounded degree trees with ratio

$(2\sqrt{e}-1)/(\sqrt{e}-1) $

+

ε

< 3.542. Our algorithm is based on a novel application of the

local ratio technique

in which the available bandwidth is divided into narrow strips and requests with very small bandwidths are allocated in these strips. We also present a randomized (2+

ε

)-approximation algorithm for

bap

in bounded degree trees. The best previously known ratio for

bap

in general trees is 5. We present a reduction from

sap

to

bap

that works for instances where the tree is a line and the bandwidths are very small. It follows that there exists a (2+

ε

)-approximation algorithm for

sap

in the line. The best previously known ratio for this problem is 7.