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.