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A Polynomial Algorithm for a Class of 0–1 Fractional Programming Problems Involving Composite Functions, with an Application to Additive Clustering

  • Pierre Hansen
  • Christophe Meyer
Chapter
Part of the Springer Optimization and Its Applications book series (SOIA, volume 92)

Abstract

We derive conditions on the functions \(\varphi\), ρ, v and w such that the 0–1 fractional programming problem \(\max \limits _{x\in \{0;1\}^{n}} \frac{\varphi \circ v(x)} {\rho \circ w(x)}\) can be solved in polynomial time by enumerating the breakpoints of the piecewise linear function \(\Phi (\lambda ) =\max \limits _{x\in \{0;1\}^{n}}v(x) -\lambda w(x)\) on [0; +). In particular we show that when \(\varphi\) is convex and increasing, ρ is concave, increasing and strictly positive, v and − w are supermodular and either v or w has a monotonicity property, then the 0–1 fractional programming problem can be solved in polynomial time in essentially the same time complexity than to solve the fractional programming problem \(\max \limits _{x\in \{0;1\}^{n}} \frac{v(x)} {w(x)}\), and this even if \(\varphi\) and ρ are non-rational functions provided that it is possible to compare efficiently the value of the objective function at two given points of {0; 1} n . We apply this result to show that a 0–1 fractional programming problem arising in additive clustering can be solved in polynomial time.

Keywords

0–1 fractional programming Submodular function Polynomial algorithm Composite functions Additive clustering 

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.GERAD, HEC MontréalMontréalCanada

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