Greedy colorings for the binary paintshop problem

Cars have to be painted in two colors in a sequence where each car occurs twice; assign the two colors to the two occurrences of each car so as to minimize the number of color changes. This problem is denoted by PPW(2,1)PPW(2,1). This version and a more general version—with an arbitrary multiset of colors for each car—were proposed and studied for the first time in 2004 by Epping, Hochstättler and Oertel. Since then, other results have been obtained: for instance, Meunier and Sebő have found a class of PPW(2,1)PPW(2,1) instances for which the greedy algorithm is optimal. In the present paper, we focus on PPW(2,1)PPW(2,1) and find a larger class of instances for which the greedy algorithm is still optimal. Moreover, we show that when one draws uniformly at random an instance w   of PPW(2,1)PPW(2,1), the greedy algorithm needs at most 1/3 of the length of w color changes. We conjecture that asymptotically the true factor is not 1/3 but 1/4. Other open questions are emphasized.