Compact Forbidden-Set Routing

Abstract

We study labelling schemes for X-constrained path problems. Given a graph (V, E) and X ⊆ V , a path is X-constrained if all intermediate vertices avoid X. We study the problem of assigning labels J(x) to vertices so that given {J(x) : x ∈ X} for any X ⊆ V , we can route on the shortest X-constrained path between x, y ∈ X. This problem is motivated by Internet routing, where the presence of routing policies means that shortest-path routing is not appropriate. For graphs of tree width k, we give a routing scheme using routing tables of size O(k 2 log 2 n). We introduce m-clique width, generalizing clique width, to show that graphs of m-clique width k also have a routing scheme using size O(k 2 log 2 n) tables.

Graph Operations Characterizing Rank-Width and Balanced Graph Expressions

“…The exponent 2 k+1 is part of the large size of constants in FPT algorithms. [16,7,6,13] For every undirected graph G,…”

“…Here twd, cwd and mcwd denote respectively tree-width [17], clique-width [6] and m-clique-width (we recall below the definition of m-clique-width [7]). The following constants will be used: for A ⊆ L we let A be a constant denoting the graph G with single vertex x and δ G (x) = A.…”

“…. , k}, like in the variant of clique-width called m-clique-width (see definitions of Section 2 and [6,7]), but vertex colors are manipulated by linear transformations on the GF(2) vector space {0, 1} k rather than with set union over subsets of {1, . .…”

“…This is the case for instance, of the labeling schemes considered in [8,7]. Another practical use of balanced terms is the design of parallel algorithms.…”

Connectivity check in 3-connected planar graphs with obstacles

“…If nodes or links fail in a network, then recomputation of the labels is generally required. Courcelle and Twigg studied in [4] the forbidden-set labelling problem, where in addition to source and destination, the query algorithm is given a set of failed nodes and edges and it must construct a path that avoids this set. These labels can be used to quickly recover from failures in a network.…”