Better Approximation Algorithms for NMR Spectral Peak Assignment

We study a constrained bipartite matching problem where the input is a weighted bipartite graph G = (U, V,E), U is a set of vertices following a sequential order, V is another set of vertices partitioned into a collection of disjoint subsets, each following a sequential order, and E is a set of edges between U and V with non-negative weights. The objective is to find a matching in G with the maximum weight that satisfies the given sequential orders on both U and V , i.e. if ui+1 follows ui in U and if vj+1 follows vj in V , then ui is matched with vj if and only if ui+1 is matched with vj+1. The problem has recently been formulated as a crucial step in an algorithmic approach for interpreting NMR spectral data [13]. The interpretation of NMR spectral data is known as a key problem in protein structure determination via NMR spectroscopy. Unfortunately, the constrained bipartite matching problem is NP-hard [13]. We first propose a 2-approximation algorithm for the problem, which follows directly from the recent result of Bar-Noy et al. [2] on interval scheduling. However, our extensive experimental results on real NMR spectral data illustrate that the algorithm performs poorly in terms of recovering the target-matching (i.e. correct) edges. We then propose another approximation algorithm that tries to take advantage of the “density” of the sequential order information in V . Although we are only able to prove an approximation ratio of 3 log2 D for this algorithm, where D is the length of a longest string in V , the experimental results demonstrate that this new algorithm performs much better on real data, i.e. it is able to recover a large fraction of the target-matching ? Supported in part by the Grant-in-Aid for Scientific Research of the Ministry of Education, Science, Sports and Culture of Japan, under Grant No. 12780241. Email: [email protected]. Part of the work done while visiting at UC Riverside. ?? Supported in part by a UCR startup grant and NSF Grants CCR-9988353 and ITR-0085910. Email: [email protected]. ? ? ? Supported in part by NSERC grants RGPIN249633 and A008599, and Startup Grant REE-P5-01-02-Sci from the University of Alberta. Email: [email protected]. † Supported by NSF Grant CCR-9988353. Email: [email protected]. ‡ Supported by the Office of Biological and Environmental Research, U.S. Department of Energy, under Contract DE-AC05-00OR22725, managed by UT-Battelle, LLC. Email: xud,[email protected].