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

We derive conditions on the functions ', , v and w such that the 0–1 fractional programming problem max x2f0I1gn 'ıv.x/ ıw.x/ can be solved in polynomial time by enumerating the breakpoints of the piecewise linear functionˆ. / D max x2f0I1gn v.x/ w.x/ on Œ0IC1/. In particular we show that when ' is convex and increasing, 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 x2f0I1gn v.x/ w.x/ , and this even if ' and are nonrational functions provided that it is possible to compare efficiently the value of the objective function at two given points of f0I 1gn. We apply this result to show that a 0–1 fractional programming problem arising in additive clustering can be solved in polynomial time.