Dimitri Watel

A Practical Greedy Approximation for the Directed Steiner Tree Problem

The Directed Steiner Tree (DST) NP-hard problem asks, considering a directed weighted graph with \(n\) nodes and \(m\) arcs, a node \(r\) called root and a set of \(k\) nodes \(X\) called terminals, for a minimum cost directed tree rooted at \(r\) spanning \(X\). The best known polynomial approximation ratio for DST is a \(O(k^\varepsilon )\)-approximation greedy algorithm. However, a much faster \(k\)-approximation, returning the shortest paths from \(r\) to \(X\), is generally used in practice. We give in this paper a new \(O(\sqrt{k})\)-approximation greedy algorithm called Greedy\(_\mathrm{FLAC }\) \(^\triangleright \), derived from a new fast \(k\)-approximation algorithm called Greedy\(_\mathrm{FLAC }\) running in time at most \(O(n m k^2)\).

Directed Steiner Tree with Branching Constraint

Given a directed weighted graph G, a root r and k terminals, the k-Directed Steiner Tree problem is to find a minimum cost tree rooted at r and spanning all terminals. If this problem has several applications in multicast routing in packet switching networks, the modeling is not adapted anymore in networks based upon the circuit switching principle in which some nodes, called non diffusing nodes, are not able to duplicate packets. We define a more general problem, named Directed Steiner Tree with Limited number of Diffusing nodes (DSTLD), able to model the multicast in a network containing at most d diffusing nodes. We show that DSTLD is XP with respect to d, and use this result to build a \(\lceil \frac{k-1}{d} \rceil\)-approximation XP in d for DST. Finally, we prove that, under the assumption that NP \(\not\subseteq\) DTIME[n O(loglogn)], there is no polynomial approximation algorithm for DSTLD with ratio \(1+(\frac{1}{e} - \varepsilon) \cdot \frac{k}{d-1}\) for every constant e > 0.

Directed Steiner trees with diffusion costs

Given a directed arc-weighted graph G with n nodes, a root r and k terminals, the directed steiner tree problem (DST) consists in finding a minimum-weight tree rooted at r and spanning all the terminals. If this problem has several applications in multicast routing in packet switching networks, the modeling is not adapted anymore in networks based upon the circuit switching principle in which some nodes, called non diffusing nodes, are unable to duplicate packets. We define a more general problem, namely the directed steiner tree with a limited number of diffusing nodes (DSTLD), that enables us to model multicast in a network containing at most d diffusing nodes. We show that DSTLD is XP with respect to d, and use this result to build a

Parameterized Complexity and Approximability of Coverability Problems in Weighted Petri Nets

\left\lceil \frac{k-1}{d} \right\rceil

The Maximum Matrix Contraction Problem

k-1d-approximation algorithm for DST that is XP in d. We deduce from that result a strong inapproximability property. In particular, we prove that, under the assumption that NP

Steiner Problems with Limited Number of Branching Nodes

\not \subseteq