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

Part of the Springer Optimization and Its Applications book series (SOIA, volume 92)


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.


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


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© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.GERAD, HEC MontréalMontréalCanada

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