TY - GEN

T1 - Maximizing non-linear concave functions in fixed dimension

AU - Toledo, Sivan

N1 - Publisher Copyright:
© 1992 IEEE.

PY - 1992

Y1 - 1992

N2 - Consider a convex set P in Rd and a piece wise polynomial concave function F: P to R. Let A be an algorithm that given a point x in IRd computes F(x) if x in P, or returns a concave polynomial p such that p(x) or= 0. The author assumes that d is fixed and that all comparisons in A depend on the sign of polynomial functions of the input point. He shows that under these conditions, one can find maxP F in time which is polynomial in the number of arithmetic operations of A. Using this method he gives the first strongly polynomial algorithms for many nonlinear parametric problems in fixed dimension, such as the parametric max flow problem, the parametric minimum s-t distance, the parametric spanning tree problem and other problems. In addition he shows that in one dimension, the same result holds even if one only knows how to approximate the value of F. Specifically, if one can obtain an alpha -approximation for F(x) then one can alpha -approximate the value of maxF. He thus obtains the first polynomial approximation algorithms for many NP-hard problems such as the parametric Euclidean traveling salesman problem.

AB - Consider a convex set P in Rd and a piece wise polynomial concave function F: P to R. Let A be an algorithm that given a point x in IRd computes F(x) if x in P, or returns a concave polynomial p such that p(x) or= 0. The author assumes that d is fixed and that all comparisons in A depend on the sign of polynomial functions of the input point. He shows that under these conditions, one can find maxP F in time which is polynomial in the number of arithmetic operations of A. Using this method he gives the first strongly polynomial algorithms for many nonlinear parametric problems in fixed dimension, such as the parametric max flow problem, the parametric minimum s-t distance, the parametric spanning tree problem and other problems. In addition he shows that in one dimension, the same result holds even if one only knows how to approximate the value of F. Specifically, if one can obtain an alpha -approximation for F(x) then one can alpha -approximate the value of maxF. He thus obtains the first polynomial approximation algorithms for many NP-hard problems such as the parametric Euclidean traveling salesman problem.

UR - http://www.scopus.com/inward/record.url?scp=85041014728&partnerID=8YFLogxK

U2 - 10.1109/SFCS.1992.267783

DO - 10.1109/SFCS.1992.267783

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AN - SCOPUS:85041014728

T3 - Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS

SP - 676

EP - 685

BT - Proceedings - 33rd Annual Symposium on Foundations of Computer Science, FOCS 1992

PB - IEEE Computer Society

T2 - 33rd Annual Symposium on Foundations of Computer Science, FOCS 1992

Y2 - 24 October 1992 through 27 October 1992

ER -