Rendezvous Search on the Edges of Vertex-Transitive Solids

A classic "rendezvous search" problem is the "astronaut problem," in which two agents are placed on a sphere and move around until they meet. Research focuses on finding an optimal strategy for both agents to use. We consider a model that utilizes discrete units of time, with movement along the edges of vertex-transitive solids. The search ends when the two agents can see each other. We first examine the five platonic solids, then look at several larger Archimedean solids for comparison. We compare the mean times to meet on the solids under an unbiased random walk strategy, and we alter assumptions and strategies in various versions of the search to see how certain changes affect the mean time to end. One version involves the possibility of waiting on any given turn under both biased and unbiased random strategies. We also examine multi-step strategies, which involve a random step and a predetermined sequence of directions. The calculations of expected meeting times all involve first-step Markov chain decompositions

Search-and-rescue rendezvous

We consider a new type of asymmetric rendezvous search problem in which player II needs to give player I a ‘gift’ which can be in the form of information or material. The gift can either be transfered upon meeting, as in traditional rendezvous, or it can be dropped off by player II at a location he passes, in the hope it will be found by player I. The gift might be a water bottle for a traveller lost in the desert; a supply cache for Captain Scott in the Antarctic; or important information (left as a gift). The common aim of the two players is to minimize the time taken for I to either meet II or find the gift. We find optimal agent paths and drop off times when the search region is a line, the initial distance between the players is known and one or both of the players can leave gifts. A novel and important technique introduced in this paper is the use of families of linear programs to solve this and previous rendezvous problems. Previously, the approach was to guess the answer and then prove it was optimal. Our work has applications to other forms of rendezvous on the line: we can solve the symmetric version (players must use the same strategy) with two gifts and we show that there are no asymmetric solutions to this two gifts problem. We also solve the GiftStart problem, where the gift or gifts must be dropped at the start of the game. Furthermore, we can solve the Minmax version of the game where the objective function is to minimize the maximum rendezvous time. This problem admits variations where players have 0, 1 or 2 gifts at disposal. In particular, we show that the classical Wait For Mommy strategy is optimal for this setting

Approximation to the Optimal Strategy in the Mozart Café Problem by Simultaneous Perturbation Stochastic Approximation

The rendezvous search problem is an old and classic problem in operations research. In this problem, two agents with unit speed are placed in some common region and they try to find each other in the least expected time with the assumption that they neither have devices for communication nor necessarily share the same coordinates/directions. In this thesis, we focus on one specific problem in this field, called the “Mozart Café problem,” in which two agents search for each other among n discrete locations (cafés). They can go to any café each day and will stay there the whole day waiting for the other to come, and they wish to minimize the expected time to rendezvous. Previous researchers have found and shown the optimal strategies for n = 2, 3 cases. In this study, we first present some preliminary work on a variant of the general Mozart Café problem on the n = 4 case in which each agent can leave a token in the initial café he visits saying that he would never come back. The optimal strategy on this variant provides a lower bound for the optimal expected rendezvous time in the general n = 4 case. Then we propose a novel modelling technique named k-Markovian modelling where the model parameters can be optimized by stochastic optimization algorithms. This also parameterizes this problem. The aims of this work are to provide a parameterization method and demonstrate the potential to approximate the optimal rendezvous search strategy

Hide-and-seek and other search games

In the game of hide-and-seek played between two players, a Hider picks a hiding place and a Searcher tries to find him in the least possible time. Since Isaacs had the idea of formulating this mathematically as a zero-sum game almost fifty years ago in his book, Differential Games, the theory of search games has been studied and developed extensively. In the classic model of search games on networks, first formalised by Gal in 1979, a Hider strategy is a point on the network and a Searcher strategy is a constant speed path starting from a designated point of the network. The Searcher wishes to minimise the time to find the Hider (the payoff), and the Hider wishes to maximise it. Gal solved this game for certain classes of networks: that is, he found optimal strategies and the payoff assuming best play on both sides. Here we study new formulations of search games, starting with a model proposed by Alpern where the speed of the Searcher depends on which direction he is traveling. We give a solution of this game on a class of networks called trees, generalising Gal's work. We also show how the game relates to another new model of search studied by Baston and Kikuta, where the Searcher must pay extra search costs to search the network's nodes (or vertices). We go on to study another new model of search called expanding search, which models coal mining. We solve this game on trees and also study the related problem where the Hider's strategy is known to the Searcher. We extend the expanding search game to consider what happens if there are several hidden objects and solve this game for certain classes of networks. Finally we study a game in which a squirrel hides nuts from a pilferer

Multi-Step Strategies for Rendezvous Search on the Edges of the Platonic Solids

The astronaut problem is an open problem in the field of rendezvous search. The premise is that two astronauts randomly land on a planet and want to find one another. Research explores what strategies accomplish this in the least expected time. To investigate this problem, we create a discrete model which takes place on the edges of the Platonic solids. Some baseline assumptions of the model are: (1) The agents can see all of the faces around them. (2) The agents travel along the edges from vertex to vertex and cannot jump. (3) The agents move at a rate of one edge length per unit time. We first explore an unbiased random walk strategy where the agents move in a random direction on each turn. We then explore multi-step strategies, which are strategies where both agents move randomly for one step, and then follow a pre-determined sequence. We compare the performance of multi-step strategies and the unbiased strategy for all of the solids. For the cube and octahedron, we are able to prove optimality of the “Left Strategy”, in which the agents move in a random direction for the first step and then turn left. For a dodechedron, we prove optimality of a multi-step strategy using a lower bound for the expected meeting time. For the icosahedron, we present results for a subset of the multi-step strategies. In an effort to find lower expected times, we explore mixed strategies. Mixed strategies incorporate an asymmetric case which under certain conditions can result in lower expected times. Due to the greater complexity of calculating the expected time of mixed strategy, we again utilize lower bounds to find bounds for the optimal expected time. Most of the calculations were done using first-step decompositions for Markov chains