# 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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