Balancing minimum spanning trees and shortest-path trees

We give a simple algorithm to find a spanning tree that simultaneously approximates a shortest-path tree and a minimum spanning tree. The algorithm provides a continuous tradeoff: given the two trees and a>0, the algorithm returns a spanning tree in which the distance between any vertex and the root of the shortest-path tree is at most 1+2 times the shortest-path distance, and yet the total weight of the tree is at most 1+2/ times the weight of a minimum spanning tree. Our algorithm runs in linear time and obtains the best-possible tradeoff. It can be implemented on a CREW PRAM to run a logarithmic time using one processor per vertex.Current research supported by NSF Research Initiation Award CCR-9307462. This work was done while this author was supported by NSF Grants CCR-8906949, CCR-9103135, and CCR-9111348.Part of this work was done while this, author was at the University of Maryland Institute for Advanced Computer Studies (UMIACS) and supported by NSF Grants CCR-8906949 and CCR-9111348.     

Variations of the maximum leaf spanning tree problem for bipartite graphs

The maximum leaf spanning tree problem is known to be NP-complete. In [M.S. Rahman, M. Kaykobad, Complexities of some interesting problems on spanning trees, Inform. Process. Lett. 94 (2005) 93-97], a variation on this problem was posed. This variation restricts the problem to bipartite graphs and asks, for a fixed integer K, whether or not the graph contains a spanning tree with at least K leaves in one of the partite sets. We show not only that this problem is NP-complete, but that it remains NP-complete for planar bipartite graphs of maximum degree 4. We also consider a generalization of a related decision problem, which is known to be polynomial-time solvable. We show the problem is still polynomial-time solvable when generalized to weighted graphs.     

Vertex rankings of chordal graphs and weighted trees

In this paper we consider the vertex ranking problem of weighted trees. We show that this problem is strongly NP-hard. We also give a polynomial-time reduction from the problem of vertex ranking of weighted trees to the vertex ranking of (simple) chordal graphs, which proves that the latter problem is NP-hard. In this way we solve an open problem of Aspvall and Heggernes. We use this reduction and the algorithm of Bodlaender et al.'s for vertex ranking of partial k-trees to give an exact polynomial-time algorithm for vertex ranking of a tree with bounded and integer valued weight functions. This algorithm serves as a procedure in designing a PTAS for weighted vertex ranking problem of trees with bounded weight functions.     

Finding a dominating set on bipartite graphs

Finding a dominating set of minimum cardinality is an NP-hard graph problem, even when the graph is bipartite. In this paper we are interested in solving the problem on graphs having a large independent set. Given a graph G with an independent set of size z, we show that the problem can be solved in time O^∗(²n−z), where n is the number of vertices of G. As a consequence, our algorithm is able to solve the dominating set problem on bipartite graphs in time O^∗(²n/2). Another implication is an algorithm for general graphs whose running time is O(n^1.7088).     

Distributed verification of minimum spanning trees

The problem of verifying a Minimum Spanning Tree (MST) was introduced by Tarjan in a sequential setting. Given a graph and a tree that spans it, the algorithm is required to check whether this tree is an MST. This paper investigates the problem in the distributed setting, where the input is given in a distributed manner, i.e., every node “knows” which of its own emanating edges belong to the tree. Informally, the distributed MST verification problem is the following. Label the vertices of the graph in such a way that for every node, given (its own state and label and) the labels of its neighbors only, the node can detect whether these edges are indeed its MST edges. In this paper, we present such a verification scheme with a maximum label size of O(log n log W), where n is the number of nodes and W is the largest weight of an edge. We also give a matching lower bound of Ω(log n log W) (as long as W > (log n)^1+ε for some fixed ε > 0). Both our bounds improve previously known bounds for the problem. For the related problem of tree sensitivity also presented by Tarjan, our method yields rather efficient schemes for both the distributed and the sequential settings. A preliminary version of this work was presented in ACM PODC 2006. A. Korman was supported in part at the Technion by an Aly Kaufman fellowship. S. Kutten was supported in part by a grant from the Israeli Ministry for Science and Technology.     

A simple linear time certifying LBFS-based algorithm for recognizing trivially perfect graphs and their complements

We introduce a simple, linear time algorithm for recognizing trivially perfect (TP) graphs. It improves upon the algorithm of Yan et al. [J.-H. Yan, J.-J. Chen, G.J. Chang, Quasi-threshold graphs, Discrete Appl. Math. 69 (3) (1996) 247–255] in that it is certifying, producing a P₄ or a C₄ when the graph is not TP. In addition, our algorithm can be easily modified to recognize the complement of TP graphs (co-TP) in linear time as well. It is based on lexicographic BFS, and in particular the technique of partition refinement, which has been used in the recognition of many other graph classes [D.G. Corneil, Lexicographic breadth first search—a survey, in: WG 2004, in: Lecture Notes in Comput. Sci., vol. 3353, Springer, 2004, pp. 1–19].     

Planar packing of trees and spider trees

In [A. García, C. Hernando, F. Hurtado, M. Noy, J. Tejel, Packing trees into planar graphs, J. Graph Theory (2002) 172-181] García et al. conjectured that for every two non-star trees there exists a planar graph containing them as edge-disjoint subgraphs. In this paper we prove the conjecture in the case in which one of the trees is a spider tree.