en Discrete Optimization Discrete Optimization پژوهشی Original A three-pair network consists of a directed acyclic graph with two sets $S={s_1,s_2,s_3}$ and $T={t_1,t_2,t_3}$  such that each $s_{i}in S$ generates a message $X_{i}$ and each $t_{i}in T$ wants to get the message $X_{i}$.  This paper presents an $O(|E|^2)$ time algorithm to diagnose the solvability or unsolvability of three-pair networks with common bottleneck links, where $|E|$ is the number of links. The algorithm is a region decomposition method, which improves upon the previous polynomial time of $O(|E|^3)$ due to cite{CH}. Network coding, Region decomposition method, Three-pair networks. 165 198 http://iors.ir/journal/browse.php?a_code=A-10-310-8&slc_lang=en&sid=1 Sepideh Ghazvineh s.ghazvineh@sci.basu.ac.ir 00031947532846003166 00031947532846003166 No Bu-Ali Sina University Mehdi Ghiyasvand mghiyasvand@basu.ac.ir 00031947532846003167 00031947532846003167 Yes Bu-Ali Sina University