Gábor Wiener

On finding spanning trees with few leaves

The problem of finding a spanning tree with few leaves is motivated by the design of communication networks, where the cost of the devices depends on their routing functionality (ending, forwarding, or routing a connection). Besides this application, the problem has its own theoretical importance as a generalization of the Hamiltonian path problem. Lu and Ravi showed that there is no constant factor approximation for minimizing the number of leaves of a spanning tree, unless P=NP. Thus instead of minimizing the number of leaves, we are going to deal with maximizing the number of non-leaves: we give a linear-time 2-approximation for arbitrary graphs, a 32-approximation for claw-free graphs, and a 65-approximation for cubic graphs.

Rounds in Combinatorial Search

A set system

Coloring signed graphs using DFS

\mathcal{H} \subseteq2^{[m]}

Finding a majority ball with majority answers

is said to be separating if for every pair of distinct elements x,y∈[m] there exists a set

Density-Based group testing

H\in\mathcal{H}

Edge Multiplicity and Other Trace Functions

such that H contains exactly one of them. The search complexity of a separating system

Finding a non-minority ball with majority answers

\mathcal{H} \subseteq 2^{[m]}

On constructions of hypotraceable graphs

is the minimum number of questions of type “x∈H?” (where

Strict group testing and the set basis problem

H \in\mathcal{H}

On non-traceable, non-hypotraceable, arachnoid graphs

) needed in the worst case to determine a hidden element x∈[m]. If we receive the answer before asking a new question then we speak of the adaptive complexity, denoted by