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

Abstract

We derive conditions on the functions ', , v and w such that the 0-1 fractional programming problem max w.x/ on OE0I C1/. 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, 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 1g n . We apply this result to show that a 0-1 fractional programming problem arising in additive clustering can be solved in polynomial time.

On the complexity and approximation of the maximum expected value all-or-nothing subset

“…The complexity of the algorithm is determined by at most [n] \ N 1/2 ≤ n invocations of (8). Hence, it is…”

“…Also [11] develops an FPTAS for a general quasiconcave minimization problem. Note that in contrast, the maximization variant of the ratio problem (2) with S = 2 [n] can be solved in polynomial time via the Dinkelbach algorithm; see [8] and references therein. We first establish the NP-hardness of the maximum expected value all-ornothing subset.…”

“…. , n} κ • ẑ(h, j) = ẑ(n, n + 1) ≥ (1 − ǫ) • C * • P (n, C * ),and the algorithm must determine j since it enumerates all elements of [n]\N 1/2 in the main loop (in lines 2-7).The complexity of the algorithm is determined by at most [n] \ N 1/2 ≤ n invocations of(8). Hence, it isO(n 2 C) ⊆ O…”