### Abstract:

Given a metric space X, a set of clients C X, and a set of facilities F X, such that X = F S C. the fault-tolerant k-median problem (FTM) requires to open k facilities in F so as to minimize the sum of distances from each client to r nearest open facilities. Fault-tolerant facility location (FTFL) allows the opening of a variable number of facilities, with an opening cost FTFL and FTM clustering are known NP-Hard problems, in any general metric space. Various approximation algorithms have been proposed to solve the problem in polynomial time. In this work, we consider the input space to be a line metric. M. Hajiaghayi et al. have proposed an LP-based technique to solve FTM in polynomial time. We propose the rst combinatorial algorithm to solve FTFL in polynomial time. This paper talks about the structure theorem that clearly de nes the structure of the underlying solution. The structure theorem is exploited to build a min-cost max ow model to solve FTFL on a line metric. The eventual goal is to nd a combinatorial polynomial-time solution to FTM on a line metric and extend it to tree metric space. Keywords :