Optimal Cuts and Bisections on the Real Line in Polynomial Time
  The exact complexity of geometric cuts and bisections is the longstanding open problem including even the dimension one. In this paper, we resolve this problem for dimension one (the real line) by designing an exact polynomial time algorithm. Our results depend on a new technique of dealing with metric equalities and their connection to dynamic programming. The method of our solution could be also of independent interest. 
Marek Karpinski, Andrzej Lingas, Dzmitry Sledneu
cs.DS cs.CC cs.DM