Bound smoothing under chirality constraints
SIAM Journal on Discrete Mathematics
A robust model for finding optimal evolutionary trees
STOC '93 Proceedings of the twenty-fifth annual ACM symposium on Theory of computing
Efficient algorithms for inverting evolution
STOC '96 Proceedings of the twenty-eighth annual ACM symposium on Theory of computing
An Õ(n3/14)-coloring algorithm for 3-colorable graphs
Information Processing Letters
On constructing radiation hybrid maps (extended abstract)
RECOMB '97 Proceedings of the first annual international conference on Computational molecular biology
Building human genome maps with radiation hybrids
RECOMB '97 Proceedings of the first annual international conference on Computational molecular biology
On the approximability of numerical taxonomy (fitting distances by tree metrics)
Proceedings of the seventh annual ACM-SIAM symposium on Discrete algorithms
The complexity of satisfiability problems
STOC '78 Proceedings of the tenth annual ACM symposium on Theory of computing
Ordinal embeddings of minimum relaxation: general properties, trees, and ultrametrics
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Randomized vs. deterministic distance query strategies for point location on the line
Discrete Applied Mathematics
Ordinal embeddings of minimum relaxation: General properties, trees, and ultrametrics
ACM Transactions on Algorithms (TALG)
Randomized vs. deterministic distance query strategies for point location on the line
Discrete Applied Mathematics
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A natural problem is that of, given an n × n symmetric matrix D, finding an arrangement of n points on the real line such that the so obtained distances agree as well as possible with the by D specified distances. We refer to the variation in which the difference in distance is measured in maximum norm as the MATRIX-TO-LINE problem. The MATRIX-TO-LINE problem has previously been shown to be NP-complete [J.B. Saxe, 17th Allerton Conference in Communication, Control, and Computing, 1979, pp. 480-489]. We show that it can be approximated within 2, but unless P = NP not within 7/5 - δ for any δ 0. We also show a tight lower bound under a stronger assumption. We show that the MATRIX-TO-LINE problem cannot be approximated within 2 - δ unless 3-colorable graphs can be colored with ⌈4/δ⌉ colors in polynomial time. Currently, the best polynomial time algorithm colors a 3-colorable graph with Õ(n3/14) colors [A. Blum, D. Karger, Inform. Process. Lett. 61 (1), (1997), 49-53]. We apply our MATRIX-TO-LINE algorithm to a problem in computational biology, namely, the Radiation Hybrid (RH) problem. That is, the algorithmic part of a physical mapping method called RH mapping. This gives us the first algorithm with a guaranteed convergence for the general RH problem.