Michael Hoffmann

AUTOMATIC SEMIGROUPS WITH SUBSEMIGROUPS OF FINITE REES INDEX

The notion of automaticity has been widely studied in groups and some progress has been made in understanding this notion in the wider context of semigroups. The purpose of this paper is to study t...

Efficient Update Strategies for Geometric Computing with Uncertainty

We consider the problems of computing maximal points and the convex hull of a set of points in two dimensions, when the points are “in motion.” We assume that the point locations (or trajectories) are not known precisely and determining these values exactly is feasible, but expensive. In our model the algorithm only knows areas within which each of the input points lie, and is required to identify the maximal points or points on the convex hull correctly by updating some points (i.e., determining their location exactly). We compare the number of points updated by the algorithm on a given instance to the minimum number of points that must be updated by a nondeterministic strategy in order to compute the answer provably correctly. We give algorithms for both of the above problems that always update at most three times as many points as the nondeterministic strategy, and show that this is the best possible. Our model is similar to that in [3] and [5].

On Temporal Graph Exploration

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 temporal graphs that are connected at each step. For such temporal graphs with n nodes, we show that it is \(\mathbf {NP}\)-hard to approximate TEXP with ratio \(O(n^{1-\varepsilon })\) for any \(\varepsilon >0\). We also provide an explicit construction of temporal graphs that require \(\Theta (n^2)\) steps to be explored. We then consider TEXP under the assumption that the underlying graph (i.e. the graph that contains all edges that are present in the temporal graph in at least one step) belongs to a specific class of graphs. Among other results, we show that temporal graphs can be explored in \(O(n^{1.5}k^2\log n)\) steps if the underlying graph has treewidth k and in \(O(n\log ^3 n)\) steps if the underlying graph is a \(2 \times n\) grid. We also show that sparse temporal graphs with regularly present edges can always be explored in O(n) steps.

network discovery and verification

Consider the problem of discovering (or verifying) the edges and non-edges of a network, modeled as a connected undirected graph, using a minimum number of queries. A query at a vertex v discovers (or verifies) all edges and non-edges whose endpoints have different distance from v. In the network discovery problem, the edges and non-edges are initially unknown, and the algorithm must select the next query based only on the results of previous queries. We study the problem using competitive analysis and give a randomized on-line algorithm with competitive ratio

Biautomatic semigroups

O(\sqrt{nlogn})

Query-competitive algorithms for cheapest set problems under uncertainty

for graphs with n vertices. We also show that no deterministic algorithm can have competitive ratio better than 3. In the network verification problem, the graph is known in advance and the goal is to compute a minimum number of queries that verify all edges and non-edges. This problem has previously been studied as the problem of placing landmarks in a graph or determining the metric dimension of a graph. We show that there is no approximation algorithm for this problem with ratio o(log n) unless

Minimum Spanning Tree Verification Under Uncertainty

\mathcal{P} = \mathcal{nP}

Singular artin monoids of finite coxeter type are automatic

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Query-Competitive Algorithms for Cheapest Set Problems under Uncertainty

We consider biautomatic semigroups. There are two different definitions of a biautomatic structure for a group in the literature; whilst these definitions are not equivalent, the idea of a biautomatic group is well defined, in that a group possesses one type of biautomatic structure if and only if it possesses the other. However the two definitions give rise to different notions of biautomaticity for semigroups and we study these ideas in this paper. In particular, we settle the question as to whether automaticity and biautomaticity are equivalent for semigroups by giving examples of semigroups which are automatic but not biautomatic.

Network discovery and verification with distance queries

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 that makes at most d ? OPT + d queries, where d is the maximum cardinality of any given set and OPT is the optimal number of queries needed to identify a cheapest set. For the minimum multi-cut problem in trees with d terminal pairs, we give an algorithm that makes at most d ? OPT + 1 queries. For the problem of computing a minimum-weight base of a given matroid, we give an algorithm that makes at most 2 ? OPT queries, generalizing a known result for the minimum spanning tree problem. For each of the above algorithms we give matching lower bounds. We also settle the complexity of the verification version of the general cheapest set problem and the minimum multi-cut problem in trees under uncertainty.