Designing deterministic polynomial-space algorithms by color-coding multivariate polynomials

Gutin, G.Z. and Reidl, Felix and Wahlström, M. and Zehavi, M. (2018) Designing deterministic polynomial-space algorithms by color-coding multivariate polynomials. Journal of Computer and System Sciences 95 , pp. 69-85. ISSN 0022-0000.

Abstract

In recent years, several powerful techniques have been developed to design {\em 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 the following deterministic algorithms (see Introduction for problem definitions). 1. Polynomial-space O∗(3.86k)-time (exponential-space O∗(3.41k)-time) algorithm for {\sc k-Internal Out-Branching}, improving upon the previously fastest {\em exponential-space} O(5.14k)-time algorithm for this problem. 2. Polynomial-space O∗((2e)k+o(k))-time (exponential-space O∗(4.32k)-time) algorithm for {\sc k-Colorful Out-Branching} on arc-colored digraphs and {\sc k-Colorful Perfect Matching} on planar edge-colored graphs. 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.

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