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 $S\in F$ is called a feasible solution/set of the problem. We denote the value vector by $v=\{v_i\}_{i\in [n]}$, where $v_i\geq 0$ is the value of element $i$. For any subset $S\subseteq U$, we use $v[S]$ to denote the $n$-dimensional vector $\{v_e\cdot \mathbf{1}[e\in S]\}_{e\in U}$. Let $f: \mathbb{R}^n\rightarrow\mathbb{R}_+$ be a symmetric monotone norm function. Our goal is to minimize the norm objective $f(v[S])$ over feasible subset $S\in F$. 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, there is 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.