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Clusters, Orders, and Trees: Methods and Applications pp 13–50Cite as

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

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Part of the book series: Springer Optimization and Its Applications ((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.

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Notes

  1. 1.

    A similar model was developed independently and at the same time in the former USSR; see Mirkin [40] and the references therein.

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Authors and Affiliations

  1. GERAD, HEC Montréal, 3000, chemin de la Côte-Sainte-Catherine, Montréal, Québec, Canada, H3T 2A7

    Pierre Hansen & Christophe Meyer

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  1. Pierre Hansen
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  2. Christophe Meyer
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Correspondence to Pierre Hansen .

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Editors and Affiliations

  1. Department of Higher Mathematics, National Research University Higher School of Economics, Moscow, Russia

    Fuad Aleskerov

  2. Department of Operations, University of Groningen, Groningen, The Netherlands

    Boris Goldengorin

  3. Department of Industrial and Systems Eng, University of Florida, Gainesville, Florida, USA

    Panos M. Pardalos

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Hansen, P., Meyer, C. (2014). A Polynomial Algorithm for a Class of 0–1 Fractional Programming Problems Involving Composite Functions, with an Application to Additive Clustering. In: Aleskerov, F., Goldengorin, B., Pardalos, P. (eds) Clusters, Orders, and Trees: Methods and Applications. Springer Optimization and Its Applications, vol 92. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-0742-7_2

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