A Geometric Approach to Betweenness

An input to the betweenness problem contains m constraints over n real variables (points). Each constraint consists of three points, where one of the points is specified to lie inside the interval defined by the other two. The order of the other two points (i.e., which one is the largest and which one is the smallest) is not specified. This problem comes up in questions related to physical mapping in molecular biology. In 1979, Opatrny showed that the problem of deciding whether the n points can be totally ordered while satisfying the m betweenness constraints is NP-complete [SIAM J. Comput., 8 (1979), pp. 111–114]. Furthermore, the problem is MAX SNP complete, and for every α > 47/48 finding a total order that satisfies at least α of the m constraints is NP-hard (even if all the constraints are satisfiable). It is easy to find an ordering of the points that satisfies 1/3 of the m constraints (e.g., by choosing the ordering at random). This paper presents a polynomial ti...

• Type: Conference
• Year: 1995
• Where: ESA
• Authors: Benny Chor, Madhu Sudan
• Keywords: Algorithms, Betweenness Constraints, Betweenness Problem, ESA 1995, Total Order