Protein NMR peak assignment refers to the process of assigning a group of "spin systems" obtained experimentally to a protein sequence of amino acids. The automation of this process is still an unsolved and challenging problem in NMR protein structure determination. Recently, protein NMR peak assignment has been formulated as an interval scheduling problem (ISP), where a protein sequence P of amino acids is viewed as a discrete time interval I (the amino acids on P one-to-one correspond to the time units of I), each subset S of spin systems that are known to originate from consecutive amino acids from P is viewed as a "job" j(s), the preference of assigning S to a subsequence P of consecutive amino acids on P is viewed as the profit of executing job j(s) in the subinterval of I corresponding to P, and the goal is to maximize the total profit of executing the jobs (on a single machine) during I. The interval scheduling problem is max SNP-hard in general; but in the real practice of protein NMR peak assignment, each job j(s) usually requires at most 10 consecutive time units, and typically the jobs that require one or two consecutive time units are the most difficult to assign/schedule. In order to solve these most difficult assignments, we present an efficient 13/7-approximation algorithm for the special case of the interval scheduling problem where each job takes one or two consecutive time units. Combining this algorithm with a greedy filtering strategy for handling long jobs (i.e., jobs that need more than two consecutive time units), we obtain a new efficient heuristic for protein NMR peak assignment. Our experimental study shows that the new heuristic produces the best peak assignment in most of the cases, compared with the NMR peak assignment algorithms in the recent literature. The above algorithm is also the first approximation algorithm for a nontrivial case of the well-known interval scheduling problem that breaks the ratio 2 barrier.