New Results on a General Class of Minimum Norm Optimization Problems

We study the general norm optimization for combinatorial problems, initiated by Chakrabarty and Swamy (STOC 2019). We propose a general formulation that captures a large class of combinatorial structures: we are given a set U of n weighted elements and a family of feasible subsets F. Each subset S ∈ F is called a feasible solution/set of the problem. We denote the value vector by v = {vi}_i∈[n], where vi ≥ 0 is the value of element i. For any subset S ⊆ U, we use v[S] to denote the n-dimensional vector {ve · 1[e ∈ S]}_e∈U (i.e., we zero out all entries that are not in S). Let f : ℝⁿ → ℝ+ be a symmetric monotone norm function. Our goal is to minimize the norm objective f(v[S]) over feasible subset S ∈ F. The problem significantly generalizes the corresponding min-sum and min-max problems. We present a general equivalent reduction of the norm minimization problem to a multi-criteria optimization problem with logarithmic budget constraints, up to a constant approximation factor. Leveraging this reduction, we obtain constant factor approximation algorithms for the norm minimization versions of several covering problems, such as interval cover, multi-dimensional knapsack cover, and logarithmic factor approximation for set cover. We also study the norm minimization versions for perfect matching, s-t path and s-t cut. We show the natural linear programming relaxations for these problems have a large integrality gap. To complement the negative result, we show that, for perfect matching, it is possible to obtain a bi-criteria result: for any constant ϵ, δ > 0, we can find in polynomial time a nearly perfect matching (i.e., a matching that matches at least 1 − ϵ proportion of vertices) and its cost is at most (8 + δ) times of the optimum for perfect matching. Moreover, we establish the existence of a polynomial-time O(log log n)-approximation algorithm for the norm minimization variant of the s-t path problem. Specifically, our algorithm achieves an α-approximation with a time complexity of n^{O(log log n/α)}, where 9 ≤ α ≤ log log n.