CAPTURE TIME IN VARIANTS OF COPS & ROBBERS GAMES A Thesis Submitted to the Faculty ...

144 Pages·2013·0.8 MB·English

CAPTURE TIME IN VARIANTS OF COPS & ROBBERS GAMES
A Thesis
Submitted to the Faculty
in partial fulfillment of the requirements for the
degree of
Doctor of Philosophy
in
Mathematics
by
Natasha Komarov
DARTMOUTH COLLEGE
Hanover, New Hampshire
July 30, 2013
Examining Committee:
Peter Winkler, Chair
Richard Nowakowski
Sergi Elizalde
Peter Doyle
F. Jon Kull, Ph.D.
Dean of Graduate Studies
Copyright by
Natasha Komarov
2013
Abstract
We examine variations of cops and robbers games on graphs. Our goals are to intro-
duce some randomness into their study, and to estimate (expected) capture time. We
show that a cop chasing a random walker can capture him in expected time n+o(n).
We also discuss games in which the players move in the dark (showing that a cop can
capture an immobile hider in time n on any graph and any robber in time n on K )
n
and in which the players suffer various restrictions on their movements. Finally, we
consider open problems, including the idea of a patrolling scheme—that is, a plan for
the “beat” a cop ought to walk on a graph in order to maximize the danger for the
robber of committing a crime at any given location.
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Acknowledgements
I want to thank all of the people that made it possible for me to get through graduate
school. Firstofall, thankyoutomyparents: withoutyourlifelongsupportIprobably
wouldn’t have been in graduate school to begin with (an obvious understatement—I
would not have been here in many senses of the phrase without you). Thank you Asa:
you have always kept things running so smoothly while I worried about this thesis
and the work it contains that it’s taken me a few years to understand how amazing of
a partner you actually are; you’ve also changed for the better my entire mental layout
about life, math, and everything. Thank you Misha and Oliver, for being there.
Thank you to my advisor, Peter Winkler: you’ve been a great advisor, and I
don’t think there’s anything more for which I could have asked. Thank you also
to the rest of my thesis defense committee: Richard Nowakowski, Peter Doyle, and
Sergi Elizalde. And finally, thank you to all of the people I’ve spoken with over the
years who have given me ideas, directly or indirectly, that have fed into my work. In
particular, the work in Chapter 2 benefited from conversations at Microsoft Research
inRedmond, WashingtonwithOmerAngel, AnderHolroyd, RussLyons, YuvalPeres,
and David Wilson.
iii
Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
1 Introduction 1
1.1 History of Pursuit and Evasion Games . . . . . . . . . . . . . . . . . 3
1.2 Assumptions and Notation . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.1 General assumptions . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Cops & Drunks 6
2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 The Smarter Cop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2.1 Intuition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2.2 Gross Progress . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2.3 Fine Progress . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3 Generalizations and Variations . . . . . . . . . . . . . . . . . . . . . . 24
3 Cops in the Dark: Depth First Pursuit 26
3.1 Cops & Sitters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
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3.1.1 Player Strategies . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.1.2 Value of the Cop vs. Sitter Game . . . . . . . . . . . . . . . . 33
3.2 Cops & Robbers on K . . . . . . . . . . . . . . . . . . . . . . . . . . 42
n
3.2.1 Expected Capture Time . . . . . . . . . . . . . . . . . . . . . 43
3.3 Depth First Pursuit on a Binary Tree . . . . . . . . . . . . . . . . . . 47
4 Hunters & Moles 52
4.1 A Characterization of Hunter-Win Graphs . . . . . . . . . . . . . . . 52
4.1.1 Some Hunter-Win Graphs . . . . . . . . . . . . . . . . . . . . 53
4.1.2 Mole-win Graphs . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.1.3 Characterization . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.2 Optimality of the Hunter Strategy . . . . . . . . . . . . . . . . . . . . 63
4.3 Hunter vs. Sneakier Mole . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.4 Appendix to Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.4.1 Huntervs.Mole: AnAlgebraicApproachandHunter-WinStrat-
egy Generation . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.4.2 Hunter vs. Mole: Winning Strategy Generating Code . . . . . 78
5 Cops & Gamblers 81
5.1 Cop vs. gambler on P . . . . . . . . . . . . . . . . . . . . . . . . . . 82
n
5.2 Cop vs. gambler on a tree . . . . . . . . . . . . . . . . . . . . . . . . 83
5.3 Cop vs. gambler on a general graph . . . . . . . . . . . . . . . . . . . 84
5.4 Cop vs. unknown gambler . . . . . . . . . . . . . . . . . . . . . . . . 86
6 Capture Time in (Traditional) Cops & Robbers 96
6.1 Undirected Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
v
6.2 Directed Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6.3 Graph Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6.4 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
7 Speculation and Future Directions 106
7.1 Summary of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
7.2 Further Investigations into Capture Time . . . . . . . . . . . . . . . . 109
7.2.1 Cop vs. Visible Robber . . . . . . . . . . . . . . . . . . . . . . 109
7.2.2 Cop vs. Invisible Robber . . . . . . . . . . . . . . . . . . . . . 110
7.3 Patrolling Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
7.3.1 Example: α-momentum cop on C . . . . . . . . . . . . . . . 113
4
7.3.2 Topology on the space of legal walks . . . . . . . . . . . . . . 117
7.3.3 Patrolling schemes . . . . . . . . . . . . . . . . . . . . . . . . 118
7.3.4 Optimal patrolling schemes . . . . . . . . . . . . . . . . . . . 122
7.3.5 Example: Optimal patrolling scheme on K . . . . . . . . . . 127
3
References 130
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List of Tables
3.1 A table of the cop’s and robber’s heights during the first half of a DFP
on a tree of height 4 (where p = P(capture at t)). . . . . . . . . . . . 50
t
6.1 A table of the robber’s moves . . . . . . . . . . . . . . . . . . . . . . 101
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List of Figures
2.1 The Ladder to the Basement . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 At least 4 steps are required to secure a useful bound on a random
walk’s progress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3 Our (alleged) counterexample G . . . . . . . . . . . . . . . . . . . . . 19
4.1 A diagram of the mole’s choices on C given any hunter move sequence 58
n
4.2 The graph S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.3 A diagram of the mole’s choices given any hunter move sequence . . . 61
4.4 A diagram of the mole’s choices after the hunter’s first arrival at the
loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.5 An example of a lobster with a loop, separated into L and R . . . . . 69
4.6 The graphs G , G , and G . . . . . . . . . . . . . . . . . . . . . . . 72
1 2 3
4.7 A diagram of the mole’s choices in G given any hunter move sequence 74
1
4.8 A diagram of the mole’s choices in G given any hunter move sequence 75
2
4.9 A diagram of the mole’s choices in G given any hunter move sequence 75
3
6.1 H: A cop-win graph with a unique corner . . . . . . . . . . . . . . . 99
6.2 H(11) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
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6.3 The ring digraph R(7) . . . . . . . . . . . . . . . . . . . . . . . . . . 102
6.4 A digraph with one cop edge . . . . . . . . . . . . . . . . . . . . . . . 104
7.1 The Ladder to the Basement . . . . . . . . . . . . . . . . . . . . . . . 109
7.2 The graph B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
12,4
7.3 The cycle C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
4
7.4 Graph of the functions W (α), W (α), and W (α). . . . . . . . . . . . 116
1 2 3
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