Coordination Complexity of Parallel Price-Directive Decomposition

Abstract

The general block-angular convex resource sharing problem in K blocks and M nonnegative block-separable coupling constraints is considered. Applications of this model are in combinatorial optimization, network flows, scheduling, communication networks, engineering design, and finance. This paper studies the coordination complexity of approximate price-directive decomposition (PDD) for this problem, i.e., the number of iterations required to solve the problem to a fixed relative accuracy as a function of K and M. First a simple PDD method based on the classical logarithmic potential is shown to be optimal up to a logarithmic factor in M in the class of all PDD methods that work with the original (unrestricted) blocks. It is then shown that logarithmic and exponential potentials generate a polylogarithmically-optimal algorithm for a wider class of PDD methods which can restrict the blocks by the coupling constraints. As an application, the fastest currently-known deterministic approximation algorithm for minimum-cost multicommodity flows is obtained.

Approximation schemes for job shop scheduling problems with controllable processing times

“…To find an approximate solution for the linear program we first rewrite it as a convex block-angular resource-sharing problem, and then use the algorithm of [9] to solve it with a given accuracy. A convex block-angular resource sharing problem has the form: …”

“…Since the number of fractional assignments is O(n), using the rounding technique described in [11], we can obtain in linear time a new feasible solution with only a constant number of fractional assignments. The algorithm in [9] works by choosing a starting solution x 0 ∈ B j and then it repeats the following three steps for at most O(mg log(mg)) times:…”

“…To achieve the promised running time we additionally need that λ(x 0 ) ≤ cλ * [9], where c is a constant and λ(x 0 ) is the value of λ corresponding to x 0 . This is accomplished as follows.…”

Grouping Techniques for Scheduling Problems: Simpler and Faster

“…Their algorithm has to solve at least (m 3 /ε 2 ) m many linear programs. In order to obtain a linear running time for the case when m is fixed, they use the price-directive decomposition method proposed by Grigoriadis and Khachiyan [5] for computing approximate solutions of block structured convex programs. The final ingredient is an intricate rounding technique based on the solution of a linear program and a partition of the job set.…”

“…Williamson et al [24] proved that when the number of machines, jobs, and operations per job are part of the input there does not exist a polynomial time approximation algorithm with worst case bound smaller than 5 4 unless P = N P . When m and µ are part of the input the best known result [4] is an approximation algorithm with worst case bound O((log(mµ) log(min(mµ, p max ))/ log log(mµ)) 2 ), where p max is the largest processing time among all operations.…”