RMT 555 - PERSEKITARAN PERUNDANGAN APRIL-MAY 06.

Abstract. We present a linear time algorithm for computing an implicit linear space representation of a minimum cycle basis (MCB) in weighted partial 2-trees, i.e., graphs of treewidth two. The implicit representation can be made explicit in a running time that is proportional to the size of the MCB. For planar graphs, Borradaile, Sankowski, and Wulff-Nilsen [Min st-cut Oracle for Planar Graphs with Near-Linear Preprocessing Time, FOCS 2010] showed how to compute an implicit O(n log n) space representation of an MCB in O(n log 5 n) time. For the special case of partial 2-trees, our algorithm improves this result to linear time and space. Such an improvement was achieved previously only for outerplanar graphs [Liu and Lu: Minimum Cycle Bases of Weighted Outerplanar Graphs, IPL 110:970–974, 2010]

On Minimum Average Stretch Spanning Trees in Polygonal 2-trees

A spanning tree of an unweighted graph is a minimum average stretch spanning tree if it minimizes the ratio of sum of the distances in the tree between the end vertices of the graph edges and the number of graph edges. We consider the problem of computing a minimum average stretch spanning tree in polygonal 2-trees, a super class of 2-connected outerplanar graphs. For a polygonal 2-tree on $n$ vertices, we present an algorithm to compute a minimum average stretch spanning tree in $O(n \log n)$ time. This algorithm also finds a minimum fundamental cycle basis in polygonal 2-trees.Comment: 17 pages, 12 figure

Designing Networks with Good Equilibria under Uncertainty

We consider the problem of designing network cost-sharing protocols with good equilibria under uncertainty. The underlying game is a multicast game in a rooted undirected graph with nonnegative edge costs. A set of k terminal vertices or players need to establish connectivity with the root. The social optimum is the Minimum Steiner Tree. We are interested in situations where the designer has incomplete information about the input. We propose two different models, the adversarial and the stochastic. In both models, the designer has prior knowledge of the underlying metric but the requested subset of the players is not known and is activated either in an adversarial manner (adversarial model) or is drawn from a known probability distribution (stochastic model). In the adversarial model, the designer's goal is to choose a single, universal protocol that has low Price of Anarchy (PoA) for all possible requested subsets of players. The main question we address is: to what extent can prior knowledge of the underlying metric help in the design? We first demonstrate that there exist graphs (outerplanar) where knowledge of the underlying metric can dramatically improve the performance of good network design. Then, in our main technical result, we show that there exist graph metrics, for which knowing the underlying metric does not help and any universal protocol has PoA of $\Omega(\log k)$, which is tight. We attack this problem by developing new techniques that employ powerful tools from extremal combinatorics, and more specifically Ramsey Theory in high dimensional hypercubes. Then we switch to the stochastic model, where each player is independently activated. We show that there exists a randomized ordered protocol that achieves constant PoA. By using standard derandomization techniques, we produce a deterministic ordered protocol with constant PoA.Comment: This version has additional results about stochastic inpu

Approximation Algorithms for Multicoloring Planar Graphs and Powers of Square and Triangular Meshes

International audienceA multicoloring of a weighted graph G is an assignment of sets of colors to the vertices of G so that two adjacent vertices receive two disjoint sets of colors. A multicoloring problem on G is to find a multicoloring of G. In particular, we are interested in a minimum multicoloring that uses the least total number of colors. The main focus of this work is to obtain upper bounds on the weighted chromatic number of some classes of graphs in terms of the weighted clique number. We first propose an 11/6-approximation algorithm for multicoloring any weighted planar graph. We then study the multicoloring problem on powers of square and triangular meshes. Among other results, we show that the infinite triangular mesh is an induced subgraph of the fourth power of the infinite square mesh and we present 2-approximation algorithms for multicoloring a power square mesh and the second power of a triangular mesh, 3-approximation algorithms for multicoloring powers of semi-toroidal meshes and of triangular meshes and 4-approximation algorithm for multicoloring the power of a toroidal mesh. We also give similar algorithms for the Cartesian product of powers of paths and of cycles

Local tree-width, excluded minors, and approximation algorithms

The local tree-width of a graph G=(V,E) is the function ltw^G: N -> N that associates with every natural number r the maximal tree-width of an r-neighborhood in G. Our main graph theoretic result is a decomposition theorem for graphs with excluded minors that essentially says that such graphs can be decomposed into trees of graphs of bounded local tree-width. As an application of this theorem, we show that a number of combinatorial optimization problems, such as Minimum Vertex Cover, Minimum Dominating Set, and Maximum Independent Set have a polynomial time approximation scheme when restricted to a class of graphs with an excluded minor

Random enriched trees with applications to random graphs

We establish limit theorems that describe the asymptotic local and global geometric behaviour of random enriched trees considered up to symmetry. We apply these general results to random unlabelled weighted rooted graphs and uniform random unlabelled $k$-trees that are rooted at a $k$-clique of distinguishable vertices. For both models we establish a Gromov--Hausdorff scaling limit, a Benjamini--Schramm limit, and a local weak limit that describes the asymptotic shape near the fixed root