Register Loading via Linear Programming

Abstract

We study the following optimization problem. The input is a number $k$ and a directed graph with a specified “start” vertex, each of whose vertices may have one “memory bank requirement”, an integer. There are $k$ “registers”, labeled $1, \ldots , k$. A valid solution associates to the vertices with no bank requirement one or more “load instructions” $L[b,j]$, for bank $b$ and register $j$, such that every directed trail from the start vertex to some vertex with bank requirement $c$ contains a vertex $u$ that has been associated $L[c,i]$ (for some register $i \le k$) and no vertex following $u$ in the trail has been associated an $L[b,i]$, for any other bank $b$. The objective is to minimize the total number of associated load instructions. We give a $k(k+1)$-approximation algorithm based on linear programming rounding, with $(k+1)$ being the best possible unless Vertex Cover has approximation $2 - {\epsilon }$ for ${\epsilon }> 0$. We also present a $O(k \log n)$ approximation, with $n$ being the number of vertices in the input directed graph. Based on the same linear program, another rounding method outputs a valid solution with objective at most $2k$ times the optimum for $k$ registers, using $2k-1$ registers.