Given a basic block of instructions, finding a schedule that requires the minimum number of registers for evaluation is a well-known problem. The problem is NP-complete when the dependences among instructions form a directed-acyclic graph instead of a tree. We are striving to find efficient approximation algorithms for this problem not simply because it is an interesting graph optimization problem in theory. A good solution to this problem is also an essential component in solving the more complex instruction scheduling problem on GPU. In this paper, we start with explanations on why this problem is important in GPU instruction scheduling. We then explore two different approaches to tackling this problem. First we model this problem as a constraint-programming problem. Using a state-of-the-art CP-SAT solver, we can find optimal answers for much larger cases than previous works on a modest desktop PC. Second, guided by the optimal answers, we design and evaluate heuristics that can be applied to the polynomial-time list scheduling algorithms. A combination of those heuristics can achieve the register-pressure results that are about 17\% higher than the optimal minimum on average. However, there are still near 6\% cases in which the register pressure by the heuristic approach is 50\% higher than the optimal minimum.

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