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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2504.13489 |
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| _version_ | 1866915266512814080 |
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| author | Chen, Kuowen Li, Jian Rabani, Yuval Zhang, Yiran |
| author_facet | Chen, Kuowen Li, Jian Rabani, Yuval Zhang, Yiran |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_13489 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | New Results on a General Class of Minimum Norm Optimization Problems Chen, Kuowen Li, Jian Rabani, Yuval Zhang, Yiran Data Structures and Algorithms 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. |
| title | New Results on a General Class of Minimum Norm Optimization Problems |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2504.13489 |