We are given an interval graph $$ G = (V,E) $$ G=(V,E) where each interval $$ I \in V$$ I∈V has a weight $$ w_I \in \mathbb {Q}^+ $$ wI∈Q+ . The goal is to color the intervals $$ V$$ V with an arbitrary number of color classes $$ C_1, C_2, \ldots , C_k $$ C1,C2,…,Ck such that $$ \sum _{i=1}^k \max _{I \in C_i} w_I $$ ∑i=1kmaxI∈CiwI is minimized. This problem, called max-coloring interval graphs or weighted coloring interval graphs, contains the classical problem of coloring interval graphs as a special case for uniform weights, and it arises in many practical scenarios such as memory management. Pemmaraju, Raman, and Varadarajan showed that max-coloring interval graphs is NP-hard [21] and presented a 2-approximation algorithm. We settle the approximation complexity of this problem by giving a polynomial-time approximation scheme (PTAS), that is, we show that there is an $$ (1+\epsilon ) $$ (1+ϵ) -approximation algorithm for any $$ \epsilon > 0 $$ ϵ>0 . The PTAS also works for the bounded case where the sizes of the color classes are bounded by some arbitrary $$ k \le n $$ k≤n .