Anders Dessmark

Deterministic Rendezvous in Graphs

Two mobile agents having distinct identifiers and located in nodes of an unknown anonymous connected graph, have to meet at some node of the graph. We seek fast deterministic algorithms for this rendezvous problem, under two scenarios: simultaneous startup, when both agents start executing the algorithm at the same time, and arbitrary startup, when starting times of the agents are arbitrarily decided by an adversary. The measure of performance of a rendezvous algorithm is its cost: for a given initial location of agents in a graph, this is the number of steps since the startup of the later agent until rendezvous is achieved. We first show that rendezvous can be completed at cost O(n + log l) on any n-node tree, where l is the smaller of the two identifiers, even with arbitrary startup. This complexity of the cost cannot be improved for some trees, even with simultaneous startup. Efficient rendezvous in trees relies on fast network exploration and cannot be used when the graph contains cycles. We further study the simplest such network, i.e., the ring. We prove that, with simultaneous startup, optimal cost of rendezvous on any ring is Θ(D log l), where D is the initial distance between agents. We also establish bounds on rendezvous cost in rings with arbitrary startup. For arbitrary connected graphs, our main contribution is a deterministic rendezvous algorithm with cost polynomial in n, τ and log l, where τ is the difference between startup times of the agents. We also show a lower bound Ω (n2) on the cost of rendezvous in some family of graphs. If simultaneous startup is assumed, we construct a generic rendezvous algorithm, working for all connected graphs, which is optimal for the class of graphs of bounded degree, if the initial distance between agents is bounded.

Broadcasting in geometric radio networks

We consider deterministic broadcasting in geometric radio networks (GRN) whose nodes know only a limited part of the network. Nodes of a GRN are situated in the plane and each of them is equipped with a transmitter of some range r. A signal from this node can reach all nodes at distance at most r from it but if a node u is situated within the range of two nodes transmitting simultaneously, then a collision occurs at u and u cannot get any message. Each node knows the part of the network within knowledge radius s from it, i.e., it knows the positions, labels and ranges of all nodes at distance at most s. The aim of this paper is to study the impact of knowledge radius s on the time of deterministic broadcasting in a GRN with n nodes and eccentricity D of the source. Our results show sharp contrasts between the efficiency of broadcasting in geometric radio networks as compared to broadcasting in arbitrary graphs. They also show quantitatively the impact of various types of knowledge available to nodes on broadcasting time in GRN. Efficiency of broadcasting is influenced by knowledge radius, knowledge of individual positions when knowledge radius is zero, and awareness of collisions.

Faster Algorithms for Subgraph Isomorphism of k-Connected Partial k-Trees

The problem of determining whether a k -connected partial k -tree is isomorphic to a subgraph of another partial k -tree is shown to be solvable in time O(n k+2 ) . The presented time-bounds considerably improve the corresponding bounds known in the literature. They rely in part on a new characterization of width-k tree-decomposition of k -connected partial k -trees.

Optimal Graph Exploration without Good Maps

A robot has to visit all nodes and traverse all edges of an unknown undirected connected graph, using as few edge traversals as possible. The quality of an exploration algorithm A is measured by comparing its cost (number of edge traversals) to that of the optimal algorithm having full knowledge of the graph. The ratio between these costs, maximized over all starting nodes in the graph and over all graphs in a given class U, is called the overhead of algorithm A for the class U of graphs. We construct natural exploration algorithms, for various classes of graphs, that have smallest, or - in one case - close to smallest, overhead. An important contribution of this paper is establishing lower bounds that prove optimality of these exploration algorithms.

Tradeoffs between knowledge and time of communication in geometric radio networks

We consider deterministic broadcasting in geometric radio networks (GRN) whose nodes know only a limited part of the network Nodes of a GRN are situated in the plane and each of them is equipped with a transmitter of some range r. A signal from this node can reach all nodes at distance at most r from it but if a node is situated within range of two nodes transmitting simultaneously it cannot get any message. Each node knows the part of the network within knowledge radius s from it, i.e., it knows the positions, labels and ranges of all nodes at distance at most s. The aim of this paper is to investigate tradeoffs between knowledge radius s and time of deterministic broadcasting in a GRN with n nodes and eccentricity D of the source. For s exceeding the largest range, or s exceeding the largest distance between any two nodes, we design an (optimal) broadcasting algorithm working in time O(D), while for any positive s we show how to broadcast in time &Ogr;(D(1+log(n/D))). For s = 0, i.e., when knowledge of each node is limited to itself, broadcasting can always be performed in time &Ogr;(n) and cannot be improved even for some symmetric GRN of constant diameter. In contrast to the upper bound&Ogr;(n) which assumes that each node knows its own position, we show a surprising result that broadcasting requires time &OHgr;(n log n) for some GRN whose nodes do not have this knowledge. If the collision detection capability is additionally assumed, we show that optimal broadcasting time in symmetric GRN is &THgr;(D + log n). These results show sharp contrasts between the efficiency of broadcasting in geometric radio networks as compared to broadcasting in arbitrary graphs. They also show quantitatively the impact of various types of knowledge available to nodes, on broadcasting time in GRN. The type of knowledge influencing efficiency of broadcasting includes knowledge radius, knowledge of individual positions when knowledge radius is zero, and awareness of collisions.

On the approximability of maximum and minimum edge clique partition problems

We consider the following clustering problems: given a general undirected graph, partition its vertices into disjoint clusters such that each cluster forms a clique and the number of edges within the clusters is maximized (Max-ECP), or the number of edges between clusters is minimized (Min-ECP). These problems arise naturally in the DNA clone classification. We investigate the hardness of finding such partitions and provide approximation algorithms. Further, we show that greedy strategies yield constant factor approximations for graph classes for which maximum cliques can be found efficiently.

Faster Algorithms for Subgraph Isomorphism of k-Connected Partial k-Trees.

Abstract. The problem of determining whether a k -connected partial k -tree is isomorphic to a subgraph of another partial k -tree is shown to be solvable in time O(nk+2) . The presented time-bounds considerably improve the corresponding bounds known in the literature. They rely in part on a new characterization of width-k tree-decomposition of k -connected partial k -trees.

The Maximum k-Dependent and f-Dependent Set Problem

Let k be a positive integer. A k-dependent set in an undirected graph G = (V,E) is a subset of the set V of vertices such that no vertex in the subset is adjacent to more than k vertices of the subset. This subset induces a subgraph of G of maximum degree bounded by k. A 0-dependent set in G is simply an independent set of vertices in G. Furthermore, an 1-dependent set is in general a set of independent vertices and edges and a 2-dependent set is a set of independent paths and cycles. The problem of constructing a maximum k-dependent set and its decision version have been studied in [6]. The NP-completeness of the decision version has been shown for arbitrary graphs and each k _> 0. On the other hand a linear-time algorithm for the construction problem restricted to trees has been presented in [6]. Furthermore, for each constant k the problem of finding a maximum k-dependent set for a graph with constant treewidth has been observed to be solvable in linear time. The problem of finding a maximmn k-dependent set for k = 2 has several applications, for example in information dissemination in hypercubes with a large number of faulty processors [:~]. A generalization of this problem called the maximum f-dependeut set problem has been given in [7]. Given weights f(v) E ]No for v E V, an f-dependent set is a subset A of V such that each vertex v E A is adjacent to at most f(v) vertices in A. In [7] parallel algorithms for finding maximal k and f-dependent sets have been given. In this paper we analyze both problems for bipartite graphs, cographs, trees, split graphs and graphs with bounded treewidth. Among others, we show that the decision version of the maximum k-dependent set problem restricted to planar, bipartite graphs is NP-complete for any given k >_ 1. This contrast with the well known fact that the maxinmm 0-dependent (i.e. independent) set in a bipartite graph can be found in polynomial time by reduction to maximum matching (see [10]) via KSnig-Egervary theorem (see [15] and [8]). Next, we give polynomial algorithms for both problems restricted to cographs, trees and graphs with bounded treewidth. On the other hand, we show that the complexity differs for split graphs; we give a polynomial time algorithm for the maximum k-dependent set problem and show the NP-completeness for

On Parallel Complexity of Maximum f-matching and the Degree Sequence Problem

We present a randomized NC solution to the problem of constructing a maximum (cardinality) f-matching. As a corollary, we obtain a randomized NC algorithm for the problem of constructing a graph satisfying a sequence d1, d2,..., dn of equality degree constraints. We provide an optimal NC algorithm for the decision version of the degree sequence problem and an approximation NC algorithm for the construction version of this problem. Our main result is an NC algorithm for constructing if possible a graph satisfying the degree constraints d1, d2,..., dn in case di ≤ \(\sqrt {\Sigma _{j = 1}^n d_j /5 }\)for i=1, ..., n.

Maximum packing for k -connected partial k -trees in polynomial time

Abstract The problem of determining the maximum number of node-disjoint subgraphs of a partial k -tree H on n H nodes that are isomorphic to a k -connected partial k -tree G on n G nodes is shown to be solvable in time O(n G k+1 n H +n H k ) .