正交维数和最小秩的改进NP-近似困难性
Improved NP-Hardness of Approximation for Orthogonality Dimension and Minrank
AI总结：
本文证明了判定图的正交维数（及最小秩）是否在某个阈值之上是NP-困难的，从而改进了近似困难性，无需依赖唯一博弈猜想且适用于任意常数近似因子。
摘要：
图$G$在$\mathbb{R}$上的正交维数是使得可以将非零的$k$维实向量分配给$G$的每个顶点、且每对相邻顶点收到正交向量的最小整数$k$。我们证明，对于每个足够大的整数$k$，判定给定图在$\mathbb{R}$上的正交维数至多为$k$还是至少为$2^{(1-o(1)) \cdot k/2}$是$\mathsf{NP}$-困难的。我们进一步证明了有限域上的正交维数以及与之密切相关的最小秩参数（该参数由信息论中的索引编码问题所激发）也具有这样的困难性结果。这尤其意味着，将这些图量近似到任何常数因子以内都是$\mathsf{NP}$-困难的。此前，近似困难性要么在假设唯一博弈猜想的某些变体成立时已知，要么仅对小于$3/2$的近似因子成立。证明涉及有向线图的概念及其正交维数和其补图的最小秩的界。
英文摘要：
The orthogonality dimension of a graph $G$ over $\mathbb{R}$ is the smallest integer $k$ for which one can assign a nonzero $k$-dimensional real vector to each vertex of $G$, such that every two adjacent vertices receive orthogonal vectors. We prove that for every sufficiently large integer $k$, it is $\mathsf{NP}$-hard to decide whether the orthogonality dimension of a given graph over $\mathbb{R}$ is at most $k$ or at least $2^{(1-o(1)) \cdot k/2}$. We further prove such hardness results for the orthogonality dimension over finite fields as well as for the closely related minrank parameter, which is motivated by the index coding problem in information theory. This in particular implies that it is $\mathsf{NP}$-hard to approximate these graph quantities to within any constant factor. Previously, the hardness of approximation was known to hold either assuming certain variants of the Unique Games Conjecture or for approximation factors smaller than $3/2$. The proofs involve the concept of line digraphs and bounds on their orthogonality dimension and on the minrank of their complement.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
