We consider the problem of linearly ordered (LO) coloring of hypergraphs. A hypergraph has an LO coloring if there is a vertex coloring, using a set of ordered colors, so that (i) no edge is monochromatic, and (ii) each edge has a unique maximum color. It is an open question as to whether or not a 2-LO colorable 3-uniform hypergraph can be LO colored with 3 colors in polynomial time. Nakajima and Zivn\'{y} recently gave a polynomial-time algorithm to color such hypergraphs with $\widetilde{O}(n^{1/3})$ colors and asked if SDP methods can be used directly to obtain improved bounds. Our main result is to show how to use SDP-based rounding methods to produce an LO coloring with $\widetilde{O}(n^{1/5})$ colors for such hypergraphs. We first show that we can reduce the problem to cases with highly structured SDP solutions, which we call balanced hypergraphs. Then we show how to apply classic SDP-rounding tools in this case. We believe that the reduction to balanced hypergraphs is novel and could be of independent interest.

翻译：我们考虑超图的线性有序（LO）着色问题。超图具有LO着色，即存在一种使用有序颜色集合的顶点着色方式，满足：(i) 没有边是单色的，(ii) 每条边有唯一的最大颜色。关于2-LO可着色的3一致超图能否在多项式时间内用3种颜色进行LO着色，仍是一个未解决问题。Nakajima和Zivný最近给出了一个多项式时间算法，可用$\widetilde{O}(n^{1/3})$种颜色对此类超图着色，并询问是否可以直接使用SDP方法获得更优的界。我们的主要结果是展示如何利用基于SDP的舍入方法，为此类超图生成具有$\widetilde{O}(n^{1/5})$种颜色的LO着色。我们首先证明可以将问题简化为具有高度结构化SDP解的情形，即平衡超图。然后展示在此情形下如何应用经典SDP舍入工具。我们认为这种简化为平衡超图的方法具有新颖性，可能具有独立的研究价值。