en Discrete Optimization Discrete Optimization پژوهشی Original Cai et al.(2013) and  Cai and Han (2014) presented the polynomial time algorithms for two-pair and three-pair networks with common bottleneck links, respectively. Also, Chen and HaiBin(2012) proposed a non-polynomial time algorithms for $n$-pair networks with common bottleneck links, where $n$ is an arbitrary integer. This paper presents a new sufficient and necessary condition to determine the solvability of single rate $n$-pair networks with common bottleneck links, which concludes a polynomial time algorithm for $n$-pair networks with common bottleneck links, where $n$ is an arbitrary integer. Our algorithm runs in $O(|V||E|^{2})$ time, where $|V|$ and $|E|$ are the number of nodes and links, respectively. Network coding, Single rate n-pair networks, Bottleneck links, Solvability. 122 132 http://iors.ir/journal/browse.php?a_code=A-10-310-6&slc_lang=en&sid=1 Sepideh Ghazvineh s.ghazvineh@sci.basu.ac.ir 00031947532846002944 00031947532846002944 No Mehdi Ghiyasvand meghiyasvand@yahoo.com 00031947532846002945 00031947532846002945 Yes Bu-Ali Sina University