Evangelos Bampas

Almost optimal asynchronous rendezvous in infinite multidimensional grids

Two anonymous mobile agents (robots) moving in an asynchronous manner have to meet in an infinite grid of dimension δ > 0, starting from two arbitrary positions at distance at most d. Since the problem is clearly infeasible in such general setting, we assume that the grid is embedded in a δ-dimensional Euclidean space and that each agent knows the Cartesian coordinates of its own initial position (but not the one of the other agent). We design an algorithm permitting the agents to meet after traversing a trajectory of length O(dδ polylog d). This bound for the case of 2D-grids subsumes the main result of [12]. The algorithm is almost optimal, since the Ω(dδ) lower bound is straightforward. Further, we apply our rendezvous method to the following network design problem. The ports of the δ-dimensional grid have to be set such that two anonymous agents starting at distance at most d from each other will always meet, moving in an asynchronous manner, after traversing a O(dδ polylog d) length trajectory. We can also apply our method to a version of the geometric rendezvous problem. Two anonymous agents move asynchronously in the δ-dimensional Euclidean space. The agents have the radii of visibility of r1 and r2, respectively. Each agent knows only its own initial position and its own radius of visibility. The agents meet when one agent is visible to the other one. We propose an algorithm designing the trajectory of each agent, so that they always meet after traveling a total distance of O((d/r)δ polylog(d/r)), where r = min(r1, r2) and for r ≥ 1.

Euler Tour Lock-In Problem in the Rotor-Router Model

The rotor-router model, also called the Propp machine, was first considered as a deterministic alternative to the random walk. It is known that the route in an undirected graph G = (V, E), where |V| = n and |E| = m, adopted by an agent controlled by the rotor-router mechanism forms eventually an Euler tour based on arcs obtained via replacing each edge in G by two arcs with opposite direction. The process of ushering the agent to an Euler tour is referred to as the lock-in problem. In recent work [11] Yanovski et al. proved that independently of the initial configuration of the rotor-router mechanism in G the agent locks-in in time bounded by 2mD, where D is the diameter of G. In this paper we examine the dependence of the lock-in time on the initial configuration of the rotor-router mechanism. The case study is performed in the form of a game between a player P intending to lock-in the agent in an Euler tour as quickly as possible and its adversary A with the counter objective. First, we observe that in certain (easy) cases the lock-in can be achieved in time O(m). On the other hand we show that if adversary A is solely responsible for the assignment of ports and pointers, the lock-in time Ω(mċD) can be enforced in any graph with m edges and diameter D. Furthermore, we show that if A provides its own port numbering after the initial setup of pointers by P, the complexity of the lock-in problem is bounded by O(mċmin{log m, D}). We also propose a class of graphs in which the lock-in requires time Ω(m ċ log m). In the remaining two cases we show that the lock-in requires time Ω(m ċ D) in graphs with the worst-case topology. In addition, however, we present non-trivial classes of graphs with a large diameter in which the lock-in time is O(m).

On a noncooperative model for wavelength assignment in multifiber optical networks

We propose and investigate Selfish Path MultiColoring games as a natural model for noncooperative wavelength assignment in multifiber optical networks. In this setting, we view the wavelength assignment process as a strategic game in which each communication request selfishly chooses a wavelength in an effort to minimize the maximum congestion that it encounters on the chosen wavelength. We measure the cost of a certain wavelength assignment as the maximum, among all physical links, number of parallel fibers employed by this assignment. We start by settling questions related to the existence and computation of and convergence to pure Nash equilibria in these games. Our main contribution is a thorough analysis of the price of anarchy of such games, that is, the worst-case ratio between the cost of a Nash equilibrium and the optimal cost. We first provide upper bounds on the price of anarchy for games defined on general network topologies. Along the way, we obtain an upper bound of 2 for games defined on star networks. We next show that our bounds are tight even in the case of tree networks of maximum degree 3, leading to nonconstant price of anarchy for such topologies. In contrast, for network topologies of maximum degree 2, the quality of the solutions obtained by selfish wavelength assignment is much more satisfactory: We prove that the price of anarchy is bounded by 4 for a large class of practically interesting games defined on ring networks.

Linear Search by a Pair of Distinct-Speed Robots

Two mobile robots are initially placed at the same point on an infinite line. Each robot may move on the line in either direction not exceeding its maximal speed. The robots need to find a stationary target placed at an unknown location on the line. The search is completed when both robots arrive at the target point. The target is discovered at the moment when either robot arrives at its position. The robot knowing the placement of the target may communicate it to the other robot. We look for the algorithm with the shortest possible search time (i.e. the worst-case time at which both robots meet at the target) measured as a function of the target distance from the origin (i.e. the time required to travel directly from the starting point to the target at unit velocity).

Network verification via routing table queries

The network verification problem is that of establishing the accuracy of a high-level description of its physical topology, by making as few measurements as possible on its nodes. This task can be formalized as an optimization problem that, given a graph and a query model specifying the information returned by a query at a node, asks for finding a minimum-size subset of nodes to be queried so as to univocally identify the graph. This problem has been studied with respect to different query models, assuming that a node had some global knowledge about the network. Here, we propose a new query model based on the local knowledge a node instead usually has. Quite naturally, we assume that a query at a given node returns the associated routing table, i.e., a set of entries which provides, for each destination node, a corresponding (set of) first-hop node(s) along an underlying shortest path. Any network of n nodes needs ? ( log ? log ? n ) queries to be verified.Constant diameter networks need ? ( log ? n ) queries.There is no o ( log ? n ) -approximation algorithm for diameter 2 networks, unless P = NP .We give an O ( log ? n ) -approximation algorithm for diameter 2 networks.We give exact linear-time algorithms for paths, trees, and cycles of even length.

Selfish Resource Allocation in Optical Networks

We introduce Colored Resource Allocation Games as a new model for selfish routing and wavelength assignment in multifiber all-optical networks. Colored Resource Allocation Games are a generalization of congestion and bottleneck games where players have their strategies in multiple copies (colors). We focus on two main subclasses of these games depending on the player cost: in Colored Congestion Games the player cost is the sum of latencies of the resources allocated to the player, while in Colored Bottleneck Games the player cost is the maximum of these latencies. We investigate the pure price of anarchy for three different social cost functions and prove tight bounds for each separate case. We first consider a social cost function which is particularly meaningful in the setting of multifiber all-optical networks, where it captures the objective of fiber cost minimization. Additionally, we consider the two usual social cost functions (maximum and average player cost) and obtain improved bounds that could not have been derived using earlier results for the standard models for congestion and bottleneck games.

An experimental study of maximum profit wavelength assignment in WDM rings

We are interested in the problem of satisfying a maximum-profit subset of undirected communication requests in an optical ring that uses the Wavelength Division Multiplexing technology. We present four deterministic and purely combinatorial algorithms for this problem, and give theoretical guarantees for their worst-case approximation ratios. Two of these algorithms are novel, whereas the rest are adaptation of earlier approaches. An experimental evaluation of the algorithms in terms of attained profit and execution time reveals that the theoretically best algorithm performs only marginally better than one of the new algorithms, while at the same time being several orders of magnitude slower. Furthermore, an extremely fast greedy heuristic with nonconstant approximation ratio performs reasonably well and may be favored over the other algorithms whenever it is crucial to minimize execution time.

On Mobile Agent Verifiable Problems

We consider decision problems that are solved in a distributed fashion by synchronous mobile agents operating in an unknown, anonymous network. Each agent has a unique identifier and an input string and they have to decide collectively a property which may involve their input strings, the graph on which they are operating, and their particular starting positions. Building on recent work by Fraigniaud and Pelc [LATIN 2012, LNCS 7256, pp. 362–374], we introduce several natural new computability classes allowing for a finer classification of problems below \(\mathsf {co\text {-}MAV}\) or \(\mathsf {MAV}\), the latter being the class of problems that are verifiable when the agents are provided with an appropriate certificate. We provide inclusion and separation results among all these classes. We also determine their closure properties with respect to set-theoretic operations. Our main technical tool, which is of independent interest, is a new meta-protocol that enables the execution of a possibly infinite number of mobile agent protocols essentially in parallel, similarly to the well-known dovetailing technique from classical computability theory.

Improved periodic data retrieval in asynchronous rings with a faulty host

The exploration problem has been extensively studied in unsafe networks containing malicious hosts of a highly harmful nature, called black holes, which completely destroy mobile agents that visit them. In a recent work, Kralovic and Miklik (SIROCCO 2010, LNCS 6058, pp. 157-167) 20 considered various types of malicious host behavior in the context of the Periodic Data Retrieval problem in asynchronous ring networks with exactly one malicious host. In this problem, a team of initially co-located agents must report data from all safe nodes of the network to the homebase, infinitely often. The malicious host can choose whether to kill visiting agents or allow them to pass through (gray hole). In another variation of the model, the malicious host can, in addition, alter its whiteboard contents in order to deceive visiting agents. The goal is to design a protocol for Periodic Data Retrieval using as few agents as possible.In this paper, we present the first nontrivial lower bounds on the number of agents for Periodic Data Retrieval in asynchronous ring networks. Specifically, we show that at least 4 agents are needed when the malicious host is a gray hole, and at least 5 agents are needed when the malicious host whiteboard is unreliable. This improves the previous lower bound of 3 in both cases and answers an open question posed in the aforementioned paper.On the positive side, we propose an optimal protocol for Periodic Data Retrieval in asynchronous rings with a gray hole, which solves the problem with only 4 agents. This improves the previous upper bound of 9 agents and settles the question of the optimal number of agents in the gray-hole case. Finally, we propose a protocol with 7 agents when the whiteboard of the malicious host is unreliable, significantly improving the previously known upper bound of 27 agents. Along the way, we set forth a detailed framework for studying networks with malicious hosts of varying capabilities.

On the Connection between Interval Size Functions and Path Counting

We investigate the complexity of hard counting problems that belong to the class #P but have easy decision version; several well-known problems such as # Perfect Matchings , # DNFSat share this property. We focus on classes of such problems which emerged through two disparate approaches: one taken by Hemaspaandra et al. [1] who defined classes of functions that count the size of intervals of ordered strings, and one followed by Kiayias et al. [2] who defined the class TotP, consisting of functions that count the total number of paths of NP computations. We provide inclusion and separation relations between TotP and interval size counting classes, by means of new classes that we define in this work. Our results imply that many known #P-complete problems with easy decision are contained in the classes defined in [1]--but are unlikely to be complete for these classes under certain types of reductions. We also define a new class of interval size functions which strictly contains FP and is strictly contained in TotP under reasonable complexity-theoretic assumptions. We show that this new class contains some hard counting problems.