Parameterized Algorithms for Computing MAD Trees

Abstract

We consider the well-studied problem of finding a spanning tree with minimum average distance between vertex pairs (called a MAD tree). This is a classic network design problem which is known to be NP-hard. While approximation algorithms and polynomial-time algorithms for some graph classes are known, the parameterized complexity of the problem has not been investigated so far. We start a parameterized complexity analysis with the goal of determining the border of algorithmic tractability for the MAD tree problem. To this end, we provide a linear-time algorithm for graphs of constant modular width and a polynomial-time algorithm for graphs of bounded treewidth; the degree of the polynomial depends on the treewidth. That is, the problem is in FPT with respect to modular width and in XP with respect to treewidth. Moreover, we show it is in FPT when parameterized by vertex integrity or by an above-guarantee parameter. We complement these algorithms with NP-hardness on split graphs.

Figures (4)

• Figure 1: The four subtrees of forgotten vertices are highlighted. The set $\{a,b,c\}$ of leaves of the leftmost one is the only below connection in $T$. There is no vertices above or below $e$, there are 13 above and 3 below $f$, and 1 above and none below $g$.
• Figure 2: The vertex $u$ is introduced in $t$ and thus only has neighbors in $B_{t'}$ or above. The connected component $T[B_t]^{u}_{uv}$ of the forest $T[B_t]-uv$ containing $u$ is highlighted in grey. The number of vertices above $v$ with respect to $t$ is 0 while it is 14 with respect to $t'$.
• Figure 3: A solution $T$ with respect to a join node (at the top) and its children (below). The below connection $\{a,c\}$ is also a below connection in the left child but not in the right child. There are three vertices below $f$ in the parent node, two of which are below $f$ in the left node and the last one is below $f$ in the right node.
• Figure 7: The three top figures represent the case with two modules: on the left is the input graph, in the middle is the quotient graph (in grey dashed edges) and the internal edges of each module, and on the right is a double-star with centers highlighted in blue. The bottom figure represents a poly-star in solid edges. The root and the root module are highlighted in orange. One maximum-degree vertex per module is highlighted in blue. Internal edges unused in the poly-star are dotted.

Theorems & Definitions (7)

• Lemma 1: DDR04
• Theorem 2: thm:treewidth-XP
• Theorem 2: thm:treewidth-XP
• Definition 3
• Definition 4
• Definition 5
• Definition 6