New Results on a General Class of Minimum Norm Optimization Problems

TL;DR

This work presents a unifying framework for minimizing symmetric monotone norms over feasible combinatorial structures by reducing MinNorm to a Logarithmic Budgeted Optimization (LogBgt) problem. The authors obtain constant-factor approximations for several covering variants (e.g., interval cover, multi-dimensional knapsack cover, and set cover) via LP-rounding and a multi-budgeted perspective, and they study min-norm versions of PM, $s$-$t$ path, and $s$-$t$ cut, demonstrating large LP integrality gaps for these problems. To complement the negative results, they provide a bi-criterion approximation for perfect matching and a nontrivial $O(\log\log n)$-approximation for min-norm $s$-$t$ path, supported by an approximate dynamic programming approach. The paper also develops a general reduction framework and explores several problem-specific reductions (e.g., interval to tree cover) with accompanying algorithmic techniques, paving the way for future extensions to broader norm objectives and network-design problems.

Abstract

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.

Figures (4)

• Figure 1: An example for $c = 2, k = 3$. The red solid edges are in $S^{\operatorname{path}}_{1}$. The black solid edges are in $S^{\operatorname{path}}_{2}$. The dashed edges are in $S^{\operatorname{path}}_{3}$.
• Figure 2: Outline of \ref{['sec:proofof6.1']}.
• Figure 3: An example of adding a new interval. The blue interval represents $I \in S^{\operatorname{ld}}_{k+1}$. We remove some intervals, extend some intervals, and split $I$ into at most two parts.
• Figure 4: An example of converting $\operatorname{\eta^{lam}}$ to a LogBgt-TreeCov problem. The red intervals or nodes represent a solution.

Theorems & Definitions (93)

• Definition 1.1
• Claim 3.1
• Lemma 3.2
• Lemma 3.3
• Lemma 3.4
• Lemma 3.5
• Lemma 3.6
• proof
• Definition 4.1: Logarithmic Budgeted Optimization (LogBgt)
• Theorem 4.1