Fast Polynomial-Space Algorithms Using Möbius Inversion: Improving on Steiner Tree and Related Problems

We give an O∗(2k)-time algorithm for STEINER TREE using polynomial space, where k is the number of terminals. To obtain this result, we apply Möbius inversion, also known as Inclusion-Exclusion, to the famous Dreyfus-Wagner recurrence. Among our results are also polynomial-space O∗(2n) algorithms for several NP-complete spanning tree and partition problems. With these results, we solve open problems from three papers, and improve resuls from a STOC 07 paper. The previous best known algorithms for these problems use the technique of dynamic programming among subsets, and they require exponential space. We use Möbius inversion, also known as Inclusion-Exclusion, as an algebraic tool to get improved algorithms. More specifically, we apply the zeta-transform to the recurrence associated with the dynamic programming approach and show that the number of states becomes polynomially bounded.