Convex quadratic and semidefinite programming relaxations in scheduling
Journal of the ACM (JACM)
Approximation algorithms for MAX-3-CUT and other problems via complex semidefinite programming
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
Approximation algorithms for MAX-3-CUT and other problems via complex semidefinite programming
Journal of Computer and System Sciences - STOC 2001
Ordinal embeddings of minimum relaxation: general properties, trees, and ultrametrics
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
The RPR2 rounding technique for semidefinite programs
Journal of Algorithms
Hardness of fully dense problems
Information and Computation
Ordinal embeddings of minimum relaxation: General properties, trees, and ultrametrics
ACM Transactions on Algorithms (TALG)
Ordinal Embedding: Approximation Algorithms and Dimensionality Reduction
APPROX '08 / RANDOM '08 Proceedings of the 11th international workshop, APPROX 2008, and 12th international workshop, RANDOM 2008 on Approximation, Randomization and Combinatorial Optimization: Algorithms and Techniques
On Random Ordering Constraints
CSR '09 Proceedings of the Fourth International Computer Science Symposium in Russia on Computer Science - Theory and Applications
On random betweenness constraints
FCT'09 Proceedings of the 17th international conference on Fundamentals of computation theory
Betweenness parameterized above tight lower bound
Journal of Computer and System Sciences
ESA'10 Proceedings of the 18th annual European conference on Algorithms: Part I
On random betweenness constraints
Combinatorics, Probability and Computing
Characterization and representation problems for intersection betweennesses
Discrete Applied Mathematics
Maximizing polynomials subject to assignment constraints
ICALP'11 Proceedings of the 38th international colloquim conference on Automata, languages and programming - Volume Part I
Approximation schemes for the betweenness problem in tournaments and related ranking problems
APPROX'11/RANDOM'11 Proceedings of the 14th international workshop and 15th international conference on Approximation, randomization, and combinatorial optimization: algorithms and techniques
On subbetweennesses of trees: Hardness, algorithms, and characterizations
Computers & Mathematics with Applications
Journal of Computer and System Sciences
The Complexity of Finding Multiple Solutions to Betweenness and Quartet Compatibility
IEEE/ACM Transactions on Computational Biology and Bioinformatics (TCBB)
Beating the Random Ordering Is Hard: Every Ordering CSP Is Approximation Resistant
SIAM Journal on Computing
Constraint satisfaction problems parameterized above or below tight bounds: a survey
The Multivariate Algorithmic Revolution and Beyond
An electromagnetism metaheuristic for solving the Maximum Betweenness Problem
Applied Soft Computing
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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 $\alpha 47/48$ finding a total order that satisfies at least $\alpha$ 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 time algorithm that either determines that there is no feasible solution or finds a total order that satisfies at least 1/2 of the m constraints. The algorithm translates the problem into a set of quadratic inequalities and solves a semidefinite relaxation of them in ${\cal R}^n. The n solution points are then projected on a random line through the origin. The claimed performance guarantee is shown using simple geometric properties of the semidefinite programming (SDP) solution.