Optimal Aggregation of Blocks into Subproblems in Linear-Programs with Block-Diagonal-Structure

TL;DR

This paper investigates how to optimally combine blocks into subproblems for linear programs with block-diagonal structure to minimize total solution time, considering parallel processing constraints and block sizes.

Contribution

It provides a polynomial-time algorithm for optimal block aggregation when blocks are of equal size and proves NP-hardness for unequal-sized blocks.

Findings

Optimal aggregation reduces total solution time under limited parallel resources.

Polynomial-time solution exists for equal-sized block aggregation.

NP-hardness established for unequal-sized block aggregation.

Abstract

Wall-clock-time is minimized for a solution to a linear-program with block-diagonal-structure, by decomposing the linear-program into as many small-sized subproblems as possible, each block resulting in a separate subproblem, when the number of available parallel-processing-units is at least equal to the number of blocks. This is not necessarily the case when the parallel processing capability is limited, causing multiple subproblems to be serially solved on the same processing-unit. In such a situation, it might be better to aggregate blocks into larger sized subproblems. The optimal aggregation strategy depends on the computing-platform used, and minimizes the average-case running time for the set of subproblems. We show that optimal aggregation is NP-hard when blocks are of unequal size, and that optimal aggregation can be achieved within polynomial-time when blocks are of equal size.