We construct a quasi-polynomial time deterministic approximation algorithm for computing the volume of an independent set polytope with restrictions. Randomized polynomial time approximation algorithms for computing the volume of a convex body have been known now for several decades, but the corresponding deterministic counterparts are not available, and our algorithm is the first of this kind. The class of polytopes for which our algorithm applies arises as linear programming relaxation of the independent set problem with the additional restriction that each variable takes value in the interval $[0,1-\alpha]$ for some $\alpha<1/2$. (We note that the $\alpha\ge 1/2$ case is trivial). We use the correlation decay method for this problem applied to its appropriate and natural discretization. The method works provided $\alpha> 1/2-O(1/\Delta^2)$, where $\Delta$ is the maximum degree of the graph. When $\Delta=3$ (the sparsest non-trivial case), our method works provided $0.488<\alpha<0.5$. Interestingly, the interpolation method, which is based on analyzing complex roots of the associated partition functions, fails even in the trivial case when the underlying graph is a singleton.

翻译：我们提出一种拟多项式时间的确定性近似算法，用于计算带约束的独立集多面体体积。凸体体积的随机多项式时间近似算法已存在数十年，但相应的确定性算法尚属空白，本文提出的算法为首个此类成果。该算法适用的多面体类型源于独立集问题的线性规划松弛，附加约束为每个变量取值区间为$[0,1-\alpha]$（其中$\alpha<1/2$；需注意$\alpha\ge 1/2$的情形是平凡的）。我们对该问题采用相关性衰减方法，并施加适当的自然离散化处理。该方法的有效条件为$\alpha> 1/2-O(1/\Delta^2)$，其中$\Delta$为图的最大度。当$\Delta=3$（最稀疏的非平凡情形）时，方法在$0.488<\alpha<0.5$条件下有效。值得注意的是，基于关联配分函数复根分析的插值方法，即使在底图为单位元图的平凡情形下也会失效。