Karol Suchan

Pursuing a fast robber on a graph

The Cops and Robbers game as originally defined independently by Quilliot and by Nowakowski and Winkler in the 1980s has been much studied, but very few results pertain to the algorithmic and complexity aspects of it. In this paper we prove that computing the minimum number of cops that are guaranteed to catch a robber on a given graph is NP-hard and that the parameterized version of the problem is W[2]-hard; the proof extends to the case where the robber moves s time faster than the cops. We show that on split graphs, the problem is polynomially solvable if s=1 but is NP-hard if s=2. We further prove that on graphs of bounded cliquewidth the problem is polynomially solvable for [emailxa0protected]?2. Finally, we show that for planar graphs the minimum number of cops is unbounded if the robber is faster than the cops.

k-Chordal Graphs: From Cops and Robber to Compact Routing via Treewidth

Cops and robber games, introduced by Winkler and Nowakowski (in Discrete Math. 43(2–3), 235–239, 1983) and independently defined by Quilliot (in J. Comb. Theory, Ser. B 38(1), 89–92, 1985), concern a team of cops that must capture a robber moving in a graph. We consider the class of k-chordal graphs, i.e., graphs with no induced (chordless) cycle of length greater than k, k≥3. We prove that k−1 cops are always sufficient to capture a robber in k-chordal graphs. This leads us to our main result, a new structural decomposition for a graph class including k-chordal graphs.We present a polynomial-time algorithm that, given a graph G and k≥3, either returns an induced cycle larger than k in G, or computes a tree-decomposition of G, each bag of which contains a dominating path with at most k−1 vertices. This allows us to prove that any k-chordal graph with maximum degree Δ has treewidth at most (k−1)(Δ−1)+2, improving the O(Δ(Δ−1)k−3) bound of Bodlaender and Thilikos (Discrete Appl. Math. 79(1–3), 45–61, 1997. Moreover, any graph admitting such a tree-decomposition has small hyperbolicity).As an application, for any n-vertex graph admitting such a tree-decomposition, we propose a compact routing scheme using routing tables, addresses and headers of size O(klogΔ+logn) bits and achieving an additive stretch of O(klogΔ). As far as we know, this is the first routing scheme with O(klogΔ+logn)-routing tables and small additive stretch for k-chordal graphs.

Computing on Rings by Oblivious Robots: A Unified Approach for Different Tasks

A set of autonomous robots have to collaborate in order to accomplish a common task in a ring-topology where neither nodes nor edges are labeled (that is, the ring is anonymous). We present a unified approach to solve three important problems: the exclusive perpetual exploration, the exclusive perpetual clearing, and the gathering problems. In the first problem, each robot aims at visiting each node infinitely often while avoiding that two robots occupy a same node (exclusivity property); in exclusive perpetual clearing (also known as graph searching), the team of robots aims at clearing the whole ring infinitely often (an edge is cleared if it is traversed by a robot or if both its endpoints are occupied); and in the gathering problem, all robots must eventually occupy the same node. We investigate these tasks in the Look–Compute–Move model where the robots cannot communicate but can perceive the positions of other robots. Each robot is equipped with visibility sensors and motion actuators, and it operates in asynchronous cycles. In each cycle, a robot takes a snapshot of the current global configuration (Look), then, based on the perceived configuration, takes a decision to stay idle or to move to one of its adjacent nodes (Compute), and in the latter case it eventually moves to this neighbor (Move). Moreover, robots are endowed with very weak capabilities. Namely, they are anonymous, asynchronous, oblivious, uniform (execute the same algorithm) and have no common sense of orientation. In this setting, we devise algorithms that, starting from an exclusive and rigid (i.e. aperiodic and asymmetric) configuration, solve the three above problems in anonymous ring-topologies.

k -chordal graphs: from cops and robber to compact routing via treewidth

Cops and robber games concern a team of cops that must capture a robber moving in a graph. We consider the class of k-chordal graphs, i.e., graphs with no induced cycle of length greater than k, k≥3. We prove that k−1 cops are always sufficient to capture a robber in k-chordal graphs. This leads us to our main result, a new structural decomposition for a graph class including k-chordal graphs. n nWe present a quadratic algorithm that, given a graph G and k≥3, either returns an induced cycle larger than k in G, or computes a tree-decomposition of G, each bag of which contains a dominating path with at most k−1 vertices. This allows us to prove that any k-chordal graph with maximum degree Δ has treewidth at most (k−1)(Δ−1)+2, improving the O(Δ(Δ−1)k−3) bound of Bodlaender and Thilikos (1997). Moreover, any graph admitting such a tree-decomposition has small hyperbolicity. n nAs an application, for any n-node graph admitting such a tree-decomposition, we propose a compact routing scheme using routing tables, addresses and headers of size O(logn) bits and achieving an additive stretch of O(klogΔ). As far as we know, this is the first routing scheme with O(k logΔ+logn)-routing tables and small additive stretch for k-chordal graphs.

Distributed computing of efficient routing schemes in generalized chordal graphs

Efficient algorithms for computing routing tables should take advantage of particular properties arising in large scale networks. Two of them are of special interest: low (logarithmic) diameter and high clustering coefficient. High clustering coefficient implies the existence of few large induced cycles. Considering this fact, we propose here a routing scheme that computes short routes in the class of k-chordal graphs, i.e., graphs with no induced cycles of length more than k. In the class of k-chordal graphs, our routing scheme achieves an additive stretch of at most k-1, i.e., for all pairs of nodes, the length of the route never exceeds their distance plus k-1. In order to compute the routing tables of any n-node graph with diameter D we propose a distributed algorithm which uses O(logn)-bit messages and takes O(D) time. The corresponding routing scheme achieves the stretch of k-1 on k-chordal graphs. We then propose a routing scheme that achieves a better additive stretch of 1 in chordal graphs (notice that chordal graphs are 3-chordal graphs). In this case, distributed computation of routing tables takes O(min{@DD,n}) time, where @D is the maximum degree of the graph. Our routing schemes use addresses of size logn bits and local memory of size 2(d-1)logn bits per node of degree d.

On Dissemination Thresholds in Regular and Irregular Graph Classes

AbstractWe investigate the natural situation of the dissemination of information on various graph classes starting with a random set of informed vertices called active. Initially active vertices are chosen independently with probability p, and at any stage in the process, a vertex becomes active if the majority of its neighbours are active, and thereafter never changes its state. This process is a particular case of bootstrap percolation. We show that in any cubic graph, with high probability, the information will not spread to all vertices in the graph if n

Allowing each node to communicate only once in a distributed system: shared whiteboard models

p<frac{1}{2}

Complexity of splits reconstruction for low-degree trees

n. We give families of graphs in which information spreads to all vertices with high probability for relatively small values ofxa0p.

Minimum Size Tree-Decompositions

In this paper we study distributed algorithms on massive graphs where links represent a particular relationship between nodes (for instance, nodes may represent phone numbers and links may indicate telephone calls). Since such graphs are massive they need to be processed in a distributed way. When computing graph-theoretic properties, nodes become natural units for distributed computation. Links do not necessarily represent communication channels between the computing units and therefore do not restrict the communication flow. Our goal is to model and analyze the computational power of such distributed systems where one computing unit is assigned to each node. Communication takes place on a whiteboard where each node is allowed to write at most one message. Every node can read the contents of the whiteboard and, when activated, can write one small message based on its local knowledge. When the protocol terminates its output is computed from the final contents of the whiteboard. We describe four synchronization models for accessing the whiteboard. We show that message size and synchronization power constitute two orthogonal hierarchies for these systems. We exhibit problems that separate these models, i.e., that can be solved in one model but not in a weaker one, even with increased message size. These problems are related to maximal independent set and connectivity. We also exhibit problems that require a given message size independently of the synchronization model.

On dissemination thresholds in regular and irregular graph classes

Given a vertex-weighted tree T, the split of an edge xy in T is min{sx, sy} where sx (respectively, sy) is the sum of all weights of vertices that are closer to x than to y (respectively, closer to y than to x) in T. Given a set of weighted vertices V and a multiset of splits S, we consider the problem of constructing a tree on V whose splits correspond to S. The problem is known to be NP-complete, even when all vertices have unit weight and the maximum vertex degree of T is required to be no more than 4. We show that n n the problem is strongly NP-complete when T is required to be a path. For this variant we exhibit an algorithm that runs in polynomial time when the number of distinct vertex weights is constant.We also show that n nthe problem is NP-complete when all vertices have unit weight and the maximum degree of T is required to be no more than 3, and n nit remains NP-complete when all vertices have unit weight and T is required to be a caterpillar with unbounded hair length and maximum degree at most 3. n nFinally, we shortly discuss the problem when the vertex weights are not given but can be freely chosen by an algorithm. n nThe considered problem is related to building libraries of chemical compounds used for drug design and discovery. In these inverse problems, the goal is to generate chemical compounds having desired structural properties, as there is a strong correlation between structural properties, such as the Wiener index, which is closely connected to the considered problem, and biological activity.