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). 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) instances for which the greedy algorithm is optimal. In the present paper, we focus on 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), 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.