On approximating the rank of graph divisors

Krist'of B'erczi, H. P. Hoang, Lilla T'othm'er'esz
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Abstract

Baker and Norine initiated the study of graph divisors as a graph-theoretic analogue of the Riemann-Roch theory for Riemann surfaces. One of the key concepts of graph divisor theory is the {\it rank} of a divisor on a graph. The importance of the rank is well illustrated by Baker's {\it Specialization lemma}, stating that the dimension of a linear system can only go up under specialization from curves to graphs, leading to a fruitful interaction between divisors on graphs and curves. Due to its decisive role, determining the rank is a central problem in graph divisor theory. Kiss and T\'othm\'eresz reformulated the problem using chip-firing games, and showed that computing the rank of a divisor on a graph is NP-hard via reduction from the Minimum Feedback Arc Set problem. In this paper, we strengthen their result by establishing a connection between chip-firing games and the Minimum Target Set Selection problem. As a corollary, we show that the rank is difficult to approximate to within a factor of $O(2^{\log^{1-\varepsilon}n})$ for any $\varepsilon>0$ unless $P=NP$. Furthermore, assuming the Planted Dense Subgraph Conjecture, the rank is difficult to approximate to within a factor of $O(n^{1/4-\varepsilon})$ for any $\varepsilon>0$.
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关于图因子秩的逼近
Baker和Norine开创了图因子的研究,作为黎曼曲面的黎曼-洛克理论的图论类比。图除数理论的一个关键概念是图上的除数的{\it秩}。贝克的{\it专门化引理}很好地说明了秩的重要性,指出线性系统的维数只能在从曲线到图的专门化下上升,从而导致图和曲线上的除数之间富有成效的相互作用。由于秩的决定作用,确定秩是图除数理论中的一个中心问题。Kiss和Tóthméresz使用芯片发射游戏重新表述了这个问题,并通过最小化反馈弧集问题的简化表明,计算图上一个除数的秩是np困难的。在本文中,我们通过建立掷片对策与最小目标集选择问题之间的联系来加强他们的结果。作为推论,我们表明,对于任何$\varepsilon>0$,除非$P=NP$,秩很难近似到$O(2^{\log^{1-\varepsilon}n})$的一个因子内。此外,假设种植密集子图猜想,对于任何$\varepsilon>0$,秩很难近似到$O(n^{1/4-\varepsilon})$的一个因子内。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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