Register Loading via Linear Programming

Gruia Calinescu, Minming Li

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