k-distinct in- and out-branchings in digraphs

An out-branching and an in-branching of a digraph D are called k-distinct if each of them has k arcs absent in the other. Bang-Jensen, Saurabh and Simonsen (2016) proved that the problem of deciding whether a strongly connected digraph D has k-distinct out- branching and in-branching is fixed-parameter tractable (FPT) when parameterized by k. They asked whether the problem remains FPT when extended to arbitrary digraphs. Bang-Jensen and Yeo (2008) asked whether the same problem is FPT when the out-branching and in-branching have the same root. By linking the two problems with the problem of whether a digraph has an out-branching with at least k leaves (a leaf is a vertex of out-degree zero), we first solve the problem of Bang-Jensen and Yeo (2008). We then develop a new digraph decomposition and using it prove that the problem of Bang-Jensen et al. (2016) is FPT for all digraphs

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

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

Approximating solution structure

Approximations can aim at having close to optimal value or, alternatively, they can aim at structurally resembling an optimal solution. Whereas value-approximation has been extensively studied by complexity theorists over the last three decades, structural-approximation has not yet been defined, let alone studied. However, structuralapproximation is theoretically no less interesting, and has important applications in cognitive science. Building on analogies with existing valueapproximation algorithms and classes, we develop a general framework for analyzing structural (in)approximability. We identify dissociations between solution value and solution structure, and generate a list of open problems that may stimulate future research