Designing Deterministic Polynomial-Space Algorithms by Color-Coding Multivariate Polynomials∗

In recent years, several powerful techniques have been developed to design randomized polynomial-space parameterized algorithms. In this paper, we introduce an enhancement of color coding to design deterministic polynomial-space parameterized algorithms. Our approach aims at reducing the number of random choices by exploiting the special structure of a solution. Using our approach, we derive polynomial-space O∗(3.86k)time (exponential-space O∗(3.41k)-time) deterministic algorithm for kInternal Out-Branching, improving upon the previously fastest exponential-space O∗(5.14k)-time algorithm for this problem. (The notation O∗ hides factors polynomial in the input size.) We also design polynomialspace O∗((2e)k+o(k))-time (exponential-space O∗(4.32k)-time) deterministic algorithm for k-Colorful Out-Branching on arc-colored digraphs and k-Colorful Perfect Matching on planar edge-colored graphs. In k-Colorful Out-Branching, given an arc-colored digraph D, decide whether D has an out-branching with arcs of at least k colors. In kColorful Perfect Matching, given an undirected graph G, decide whether G has a perfect matching with edges of at least k colors. To obtain our polynomial space algorithms, we show that (n, k, αk)-splitters (α > 1) and in particular (n, k)-perfect hash families can be enumerated one by one with polynomial delay using polynomial space.