Thomas Erlebach

• Professor at Durham Univeristy

A temporal graph $\mathcal{G}=(G,\lambda)$ can be represented by an underlying graph $G=(V,E)$ together with a function $\lambda$ that assigns to each edge $e\in E$ the set of time steps during which $e$ is present. The reachability graph of $\mathcal{G}$ is the directed graph $D=(V,A)$ with $(u,v)\in A$ if only if there is a temporal path from $u$...

We study the two-sided stable matching problem with one-sided uncertainty for two sets of agents A and B, with equal cardinality. Initially, the preference lists of the agents in A are given but the preferences of the agents in B are unknown. An algorithm can make queries to reveal information about the preferences of the agents in B. We examine th...

Learning-augmented algorithms have been attracting increasing interest, but have only recently been considered in the setting of explorable uncertainty where precise values of uncertain input elements can be obtained by a query and the goal is to minimize the number of queries needed to solve a problem. We study learning-augmented algorithms for so...

Learning-augmented algorithms have been attracting increasing interest, but have only recently been considered in the setting of explorable uncertainty where precise values of uncertain input elements can be obtained by a query and the goal is to minimize the number of queries needed to solve a problem. We study learning-augmented algorithms for so...

In this paper we study the fixed-parameter tractability of the problem of deciding whether a given temporal graph admits a temporal walk that visits all vertices (temporal exploration) or, in some problem variants, a certain subset of the vertices. Formally, a temporal graph is a sequence <G_1,...,G_L> of graphs with V(G_t) = V(G) and E(G_t) a subs...

A tree t-spanner T of an edge-weighted graph G is a spanning tree with the property that the distance between any pair of nodes in T is at most t times the distance in G. If an edge e in T fails, the tree breaks into two subtrees T A and T B. Let E X denote the set of edges in G that reconnect T A and T B. Every edge f ∈ E X is a potential swap edg...

For the problem of delivering a package from a source node to a destination node in a graph using a set of drones, we study the setting where the movements of each drone are restricted to a certain subgraph of the given graph. We consider the objectives of minimizing the delivery time (problem DDT) and of minimizing the total energy consumption (pr...

In computing with explorable uncertainty, one considers problems where the values of some input elements are uncertain, typically represented as intervals, but can be obtained using queries. Previous work has considered query minimization in the settings where queries are asked sequentially (adaptive model) or all at once (non-adaptive model). We i...

A temporal graph with lifetime L is a sequence of L graphs $$G_1, \ldots ,G_L$$ G 1 , … , G L , called layers, all of which have the same vertex set V but can have different edge sets. The underlying graph is the graph with vertex set V that contains all the edges that appear in at least one layer. The temporal graph is always connected if each lay...

We study how to utilize (possibly erroneous) predictions in a model for computing under uncertainty in which an algorithm can query unknown data. Our aim is to minimize the number of queries needed to solve the minimum spanning tree problem, a fundamental combinatorial optimization problem that has been central also to the research area of explorab...

We study an online scheduling problem that is motivated by applications such as car-sharing. Users submit ride requests, and the scheduler aims to accept requests of maximum total profit using a single server (car). Each ride request specifies the pick-up time and the pick-up location (among two locations, with the other location being the destinat...

An always-connected temporal graph \(\mathcal {G} = \langle G_1, ..., G_L\rangle \) with underlying graph \(G=(V,E)\) is a sequence of graphs \(G_t \subseteq G\) such that \(V(G_t) = V\) and \(G_t\) is connected for all t. This paper considers the property of k-edge-deficiency for temporal graphs; such graphs satisfy \(G_t = (V, E - X_t)\) for all...

Given a hypergraph with uncertain node weights following known probability distributions, we study the problem of querying as few nodes as possible until the identity of a node with minimum weight can be determined for each hyperedge. Querying a node has a cost and reveals the precise weight of the node, drawn from the given probability distributio...

We study two problems where k autonomous mobile agents are initially located on distinct nodes of a weighted graph with n nodes and m edges. Each agent has a predefined velocity and can only move along the edges of the graph. The first problem is to deliver one package from a source node to a destination node. The second is to simultaneously delive...

The temporal graph exploration problem TEXP is the problem of computing a foremost exploration schedule for a temporal graph, i.e., a temporal walk that starts at a given start node, visits all nodes of the graph, and has the smallest arrival time. In the first and second part of the paper, we consider only undirected temporal graphs that are conne...

Mobile sensors are located on a barrier represented by a line segment. Each sensor has a single energy source that can be used for both moving and sensing. A sensor consumes energy in movement in proportion to distance traveled, and it expends energy per time unit for sensing in direct proportion to its radius raised to a constant exponent. We addr...

The area of computing with uncertainty considers problems where some information about the input elements is uncertain, but can be obtained using queries. For example, instead of the weight of an element, we may be given an interval that is guaranteed to contain the weight, and a query can be performed to reveal the weight. While previous work has...

We introduce a novel adversarial model for scheduling with explorable uncertainty. In this model, the processing time of a job can potentially be reduced (by an a priori unknown amount) by testing the job. Testing a job j takes one unit of time and may reduce its processing time from the given upper limit \(\bar{p}_j\) (which is the time taken to e...

We study how to utilize (possibly machine-learned) predictions in a model for optimization under uncertainty that allows an algorithm to query unknown data. The goal is to minimize the number of queries needed to solve the problem. Considering fundamental problems such as finding the minima of intersecting sets of elements or sorting them, as well...

A temporal graph \(\mathcal {G} = \langle G_1, ..., G_L\rangle \) is a sequence of graphs \(G_i \subseteq G\), for some given underlying graph G of order n. We consider the non-strict variant of the Temporal Exploration problem, in which we are asked to decide if \(\mathcal {G}\) admits a sequence W of consecutively crossed edges \(e \in G\), such...

The original version of the article was inadvertently published with misinterpretations on author’s proof corrections on mathematical expressions.

This paper considers a game in which a single cop and a single robber take turns moving along the edges of a given graph G. If there exists a strategy for the cop which enables it to be positioned at the same vertex as the robber eventually, then G is called cop-win, and robber-win otherwise. In contrast to previous work, we study this classical co...

This paper considers a game in which a single cop and a single robber take turns moving along the edges of a given graph $G$. If there exists a strategy for the cop which enables it to be positioned at the same vertex as the robber eventually, then $G$ is called cop-win, and robber-win otherwise. We study this classical combinatorial game in a nove...

We study a problem where k autonomous mobile agents are initially located on distinct nodes of a weighted graph (with n nodes and m edges). Each autonomous mobile agent has a predefined velocity and is only allowed to move along the edges of the graph. We are interested in delivering a package, initially positioned in a source node s, to a destinat...

Motivated by applications in optical networks and job scheduling, we consider the interval coloring problem in a setting where an increasing cost is associated with using a higher color index. The cost of a coloring at any point of the line is the cost of the maximum color index used at that point, and the cost of the overall coloring is the integr...

We study a problem where k autonomous mobile agents are initially located on distinct nodes of a weighted graph (with n nodes and m edges). Each autonomous mobile agent has a predefined velocity and is only allowed to move along the edges of the graph. We are interested in delivering a package, initially positioned in a source node s, to a destinat...

Consider a problem where 4k given vectors need to be partitioned into k clusters of four vectors each. A cluster of four vectors is called a quad, and the cost of a quad is the sum of the component-wise maxima of the four vectors in the quad. The problem is to partition the given 4k vectors into k quads with minimum total cost. We analyze a straigh...

Explorable uncertainty refers to settings where parts of the input data are initially unknown, but can be obtained at a certain cost using queries. In a typical setting, initially only intervals that contain the exact input values are known, and queries can be made to obtain exact values. An algorithm must make queries one by one until it has obtai...

We study an on-line scheduling problem that is motivated by applications such as car-sharing. Users submit ride requests, and the scheduler aims to accept requests of maximum total profit using a single server (car). Each ride request specifies the pick-up time and the pick-up location (among two locations, with the other location being the destina...

We introduce a novel model for scheduling with explorable uncertainty. In this model, the processing time of a job can potentially be reduced (by an a priori unknown amount) by testing the job. Testing a job $j$ takes one unit of time and may reduce its processing time from the given upper limit $\bar{p}_j$ (which is the time taken to execute the j...

Data collection is one of the predominant operations in wireless sensor networks (WSNs). This paper focuses on the problem of efficient data collection in a setting where some nodes may not possess data each time data is collected. In that case, idle listening slots may occur, which lead to a waste of energy and an increase in latency. To alleviate...

Queryable uncertainty refers to settings where the input of a problem is initially not known precisely, but exact information about the input can be obtained at a cost using queries. A natural goal is then to minimize the number of the queries that are required until the precise information that has been obtained about the input is sufficient for s...

In activation network problems we are given a directed or undirected graph \(G=(V,E)\) with a family \(\{f_{uv} : (u,v)\in E\}\) of monotone non-decreasing activation functions from \(D^2\) to \(\{0,1\}\), where D is a constant-size subset of the non-negative real numbers, and the goal is to find activation values \(x_v \in D\) for all \(v\in V\) o...

Considering the model of computing under uncertainty where element weights are uncertain but can be obtained at a cost by query operations, we study the problem of identifying a cheapest (minimum-weight) set among a given collection of feasible sets using a minimum number of queries of element weights. For the general case we present an algorithm t...

In a capacitated network design game, each of n players selects a path from her source to her sink. The cost of each edge is shared equally among the players using the edge. Every edge has a finite capacity that limits the number of players using the edge. We study the price of stability for such games with respect to the max-cost objective, i.e.,...

A temporal graph is a graph in which the edge set can change from step to step. The temporal graph exploration problem TEXP is the problem of computing a foremost exploration schedule for a temporal graph, i.e., a temporal walk that starts at a given start node, visits all nodes of the graph, and has the smallest arrival time. We consider only temp...

Data collection is one of the predominant operations in wireless sensor networks. This paper focuses on the problem of efficient data collection in a setting where some nodes may not possess data each time data is collected. In that case, idle listening slots may occur, which lead to a waste of energy and an increase in latency. To alleviate these...

In the verification under uncertainty setting, an algorithm is given, for each input item, an uncertainty area that is guaranteed to contain the exact input value, as well as an assumed input value. An update of an input item reveals its exact value. If the exact value is equal to the assumed value, we say that the update verifies the assumed value...

In activation network problems we are given a directed or undirected graph G = (V,E) with a family {f uv : (u,v) ∈ E} of monotone non-decreasing activation functions from D 2 to {0,1}, where D is a constant-size subset of the non-negative real numbers, and the goal is to find activation values x v for all v ∈ V of minimum total cost ∑ v ∈ V x v suc...

Wireless mesh networks have emerged as a viable means of communicating between points that are not within wireless range of each other. There are still, however, a number of challenges involved in designing and implementing such wireless multi-hop networks, particularly with respect to optimising the user throughput. This paper presents some prelim...

The use of multi-channel MAC protocols improves the capacity of wireless networks. Efficient multi-channel MAC protocols aim to utilize multiple channels effectively. Our proposed multi-channel MAC protocol called LCV-MMAC effectively utilizes the multiple channels by handling the control channel saturation. LCV-MMAC demonstrates significantly bett...

Due to limited battery life and fault tolerance issues posed by Wireless Sensor Networks (WSNs), efficient methods which ensure reliable coverage are highly desirable. One solution is to use disjoint set covers to cover the targets. We formulate a problem called MDC which addresses the maximum coverage by using disjoint set covers S 1 and S 2. We p...

In network activation problems we are given a directed or undirected graph G = (V,E) with a family {f uv (x u ,x v ): (u,v) ∈ E} of monotone non-decreasing activation functions from D 2 to {0,1}, where D is a constant-size domain. The goal is to find activation values x v for all v ∈ V of minimum total cost ∑ v ∈ V x v such that the activated set o...

This paper proposes a simple scheme to optimise the end-to-end user throughput of small, multi-hop wireless networks. The scheme which involves the introduction of small delays between successive packet transmissions at the source node has been tested on an indoor wireless mesh testbed for uni-directional flows of data traffic with nodes that use C...

Harmful Internet hijacking incidents put in evidence how fragile the Border Gateway Protocol (BGP) is, which is used to exchange routing information between Autonomous Systems (ASes). As proved by recent research contributions, even S-BGP, the secure variant of BGP that is being deployed, is not fully able to blunt traffic attraction attacks. Given...

In recent years, wireless mesh networks have emerged as a viable means of communications, not least because of their ease of deployment and cost-effectiveness. There are still, however, a number of challenges involved in designing and implementing such wireless multi-hop networks. This paper describes some of the preliminary work carried out for th...

This paper presents preliminary results from a simulation-based study of static, wireless multi-hop networks with linear chain topologies supporting the uni-directional flow of data. Chain lengths of up to 10 nodes have been examined. The effects of varying parameters such as the number of nodes, the MAC data rate, the offered load and the packet s...

We study the problem of maximising the lifetime of a sensor network for fault-tolerant target coverage in a setting with composite events. Here, a composite event is the simultaneous occurrence of a combination of atomic events, such as the detection of smoke and high temperature. We are given sensor nodes that have an initial battery level and can...

Scheduling workflows in Grids with the goal of minimizing the completion time is a challenging optimisation problem. Communication times between dependent tasks affect the schedule length only if the tasks are scheduled on different resources, and this makes it difficult to predict the critical path of the scheduled workflow. To address this diffic...

In this chapter the author explains a clever approach to determining the winner of an election, and in so doing he demonstrates the following lessons: the most straightforward solution to a problem is not always the fastest one; often there are clever algorithms that can solve the same task with much less effort; it is not always easy to see whethe...

Broadcasting is a fundamental operation underlying different routing, multicasting and address resolution protocols. Broadcasting in a network requires that all the nodes in the network receive the broadcast packet. Mobility in the network induces link failures which cause some nodes to lose the broadcast packets. Objective of all broadcasting prot...

For $t>0$ and $gge 0$, a vertex-weighted graph of total weight $W$ is emph$(t,g)$-trimmable if it contains a vertex-induced subgraph of total weight at least $(1-1/t)W$ and with no simple path of more than $g$ edges. A family of graphs is emphtrimmable if for every constant $t>0$, there is a constant $gge 0$ such that every vertex-weighted graph in...

Compliance with the Gao-Rexford conditions [1] is perhaps the most realistic explanation of Internet routing stability, although BGP is renowned to be prone to oscillations. Informally, the Gao-Rexford conditions assume that (i) the business relationships between Internet Service Providers (ISPs) yield a hierarchy, (ii) each ISP behaves in a ration...

Advance reservation can ensure the future availability of the Grid's heterogeneous resources and help a scheduler to produce better schedules. We propose a new resource mapping technique for jobs of a Grid workflow in an advance reservation environment. Using the Dynamic Critical Path for Grid (DCP-G) scheduling algorithm's job selection method, ou...

The problem of scheduling wireless transmissions under signal to interference-plus-noise ratio (SINR) constraints has received increasing attention recently. While previous work has considered the unicast case where each transmission has one sender and one receiver, we consider the setting of multicast requests where each transmission has one sende...

If a set K of nodes in a wireless network want to set up a routing structure that allows them to communicate with each other, one possible approach is to use a Steiner tree that spans all the nodes in K. However, a tree can be disconnected by the failure of a single link, and so it is desirable to employ other routing structures that are fault-tole...

Flooding is an obligatory technique to broadcast messages within mobile ad hoc networks (MANETs). Simple flooding mechanisms cause the broadcast storm problem and swamp the network with a high number of unnecessary retransmissions, thus resulting in increased packet collisions and contention. This degrades the network performance enormously. There...

We present a (4 + ε)-approximation algorithm for the problem of computing a minimum-weight dominating set in unit disk graphs, where ε is an arbitrarily small constant. The previous best known approximation ratio was 5 + ε. The main result of this paper is a 4-approximation algorithm for the problem restricted to constant-size areas. To obtain the...

We consider the problem of discovering properties (such as the diameter) of an unknown network G=(V,E) with a minimum number of queries. Initially, only the vertex set V of the network is known. Information about the edges and non-edges of the network can be obtained by querying nodes of the network. A query at a node q∈V returns the union of all s...

This paper addresses the problem of exploring the fault tolerance potential of the routing primitive called path splicing. This routing mechanism has been recently introduced in order to improve the reliability level of networks. The idea is to provide for each destination node in a network several different routing trees, called slices, by running...

For a given number L, an L-length-bounded edge-cut (node-cut, respectively) in a graph G with source s and sink t is a set C of edges (nodes, respectively) such that no s-t-path of length at most L remains in the graph after removing the edges (nodes, respectively) in C. An L-length-bounded flow is a flow that can be decomposed into flow paths of l...

We study the planar version of Minimum-Weight Set Cover, where one has to cover a given set of points with a minimum-weight subset of a given set of planar objects. For the unit-weight case, one PTAS (on disks) is known. For arbitrary weights however, the problem appears much harder, and in particular no PTASs are known. We present the first PTAS f...

Consider the dynamic program h(n)=min 1≤j≤n a(n,j), where a(n,j) is some formula that may (online) or may not (offline) depend on the previously computed ...

Classical measures of network connectivity are the number of disjoint paths between a pair of nodes and the size of a minimum cut. For standard graphs, these measures can be computed efficiently using network flow techniques. However, in the Internet on the level of autonomous systems (ASs), referred to as AS-level Internet, routing policies impose...

Multicast communication in a wireless ad-hoc network can be established using a tree that spans the multicast sender and receivers as well as other intermediate nodes. If the network is modelled as a graph, the multicast tree is a Steiner tree, the multicast sender and receivers correspond to terminals, and other nodes participating in the tree are...

We consider online coloring of intervals with bandwidth in a setting where colors have variable capacities. Whenever the algorithm opens a new color, it must choose the capacity for that color and cannot change it later. A set of intervals can be assigned the same color a of capacity C a if the sum of bandwidths of intervals at each point does not...

We consider all-optical networks that use wavelength-division multiplexing and employ wavelength conversion at specific nodes in order to maximize their capacity usage. We investigate the effect of allowing reroutings on the number of necessary wavelength converters. We disprove a claim of Wilfong and Winkler [G. Wilfong, P. Winkler, Ring routing a...

The online removable square packing problem is a two-dimen-sional version of the online removable Knapsack problem. For a sequence of squares with side length at most 1, we are requested to pack a subset of them into a unit square bin in an online fashion ...

For intersection graphs of disks and other fat objects, polynomial-time approximation schemes are known for the independent set and vertex cover problems, but the existing techniques were not able to deal with the dominating set problem except in the special case of unit-size objects. We present approximation algorithms and inapproximability result...

For $t,g>0$, a vertex-weighted graph of total weight $W$ is $(t,g)$-trimmable if it contains a vertex-induced subgraph of total weight at least $(1-1/t)W$ and with no simple path of more than $g$ edges. A family of graphs is trimmable if for each constant $t>0$, there is a constant $g=g(t)$ such that every vertex-weighted graph in the family is $(t...

We consider the minimum spanning tree problem in a setting where information about the edge weights of the given graph is uncertain. Initially, for each edge $e$ of the graph only a set $A_e$, called an uncertainty area, that contains the actual edge weight $w_e$ is known. The algorithm can `update' $e$ to obtain the edge weight $w_e \in A_e$. The...

We consider the minimum spanning tree problem in a setting where information about the edge weights of the given graph is uncertain. Initially, for each edge e of the graph only a set Ae, called an uncertainty area, that contains the actual edge weight we is known. The algorithm can ‘update ’ e to obtain the edge weight we ∈ Ae. The task is to outp...

Broadcast Scheduling is a popular method for disseminating information in response to client requests. There are n pages of information, and clients request pages at different times. However, multiple clients can have their requests satisfied by a single broadcast of the requested page. In this article, we consider several related broadcast schedul...

We present the first study on the approximability of geometric versions of the unique coverage problem and the minimum membership set cover problem. In the former problem, one is given a family of sets of elements from some universe and aims to select sets that maximize the number of elements contained in precisely one selected set. Unique Coverage...