Nonlinear Optimization over a Weighted Independence System

We consider the problem of optimizing a nonlinear objective function over a weighted independence system presented by a linear-optimization oracle. We provide a polynomial-time algorithm that determines an r-best solution for nonlinear functions of the total weight of an independent set, where r is a constant that depends on certain Frobenius numbers of the individual weights and is independent of the size of the ground set. In contrast, we show that finding an optimal (0-best) solution requires exponential time even in a very special case of the problem.

💡 Research Summary

The paper investigates the problem of minimizing a nonlinear objective f(w·x) over a weighted independence system S ⊆ {0,1}ⁿ when the system is accessed only through a linear‑optimization oracle and the objective is accessed only through a comparison oracle. The weight vector w has entries drawn from a fixed primitive p‑tuple a = (a₁,…,a_p) of distinct positive integers with gcd = 1. The authors introduce the notion of an r‑best solution: a feasible solution x* such that at most r other feasible solutions achieve a strictly better objective value. An optimal solution corresponds to r = 0.

The main contributions are twofold. First, they prove that for any fixed primitive a there exists a constant r(a) independent of the ground set size n such that a polynomial‑time algorithm can compute an r(a)‑best solution. The algorithm works for arbitrary independence systems presented by a linear‑optimization oracle, arbitrary weight vectors w ∈ {a₁,…,a_p}ⁿ, and arbitrary univariate objectives f given only by a comparison oracle. Moreover, two special cases are highlighted: (i) if the entries of a form a divisibility chain (a₁ | a₂ | … | a_p), then r(a) = 0, i.e., the algorithm finds an optimal solution; (ii) when p = 2, the algorithm guarantees r(a) = F(a), where F(a) is the classical Frobenius number of the pair (a₁,a₂). Since F(2,3) = 1, a 1‑best solution can be obtained efficiently for weights {2,3}.

The algorithmic framework proceeds in four stages. A naïve approach first uses the linear‑optimization oracle to obtain a maximizer \bar{x} of w·x and then searches all subsets of \bar{x} to minimize f; this alone is insufficient because many feasible weight values may be missed. The authors then partition S according to the vector λ of maximal counts of each weight type, defining blocks S_{λ}^{μ} where the number of selected elements of weight a_i is fixed to μ_i (≤ λ_i). Within each block the feasible total weights form a restricted monoid M(a,λ) = {μ·a | 0 ≤ μ ≤ λ}. By exploiting classical properties of monoids—symmetry of M(a,λ) and the structure of its gap set G(a) (the set of integers not representable as a non‑negative combination of a)—the authors show that the naïve strategy applied to a single block can miss at most r(a) weight values, where r(a) depends only on the Frobenius numbers of certain sub‑tuples of a. Since the number of blocks is polynomial (bounded by the product of the λ_i), the overall procedure runs in polynomial time.

The second major contribution is a hardness result. For the simple weight set {2,3}, the authors construct an independence system that corresponds to the set of matchings in a bipartite graph. By defining f to be the identity on the total weight (which equals the matching size), the problem of finding an optimal solution becomes exactly the “Exact Matching” problem, known to require exponential time under standard complexity assumptions. Consequently, while an r‑best solution is efficiently obtainable, achieving r = 0 is provably intractable even in this highly restricted setting.

In summary, the paper establishes a clear dichotomy: for fixed weight families the nonlinear optimization problem admits a constant‑approximation (in the sense of r‑best) computable in polynomial time, with the approximation quality governed by Frobenius numbers; however, computing a truly optimal solution is generally exponential‑time hard. The results bridge combinatorial optimization, number theory (via Frobenius numbers), and oracle‑based algorithm design, and they open avenues for applying similar techniques to multi‑criteria optimization, matching problems, and broader classes of integer programs with nonlinear objectives.