The **rendezvous dilemma** is a logical dilemma, typically formulated in this way:

- Two people have a date in a park they have never been to before. Arriving separately in the park, they are both surprised to discover that it is a huge area and consequently they cannot find one another. In this situation each person has to choose between waiting in a fixed place in the hope that the other will find them, or else starting to look for the other in the hope that
*they*have chosen to wait somewhere.

If they both choose to wait, they will never meet. If they both choose to walk there are chances that they meet and chances that they do not. If one chooses to wait and the other chooses to walk, then there is a theoretical certainty that they will meet eventually; in practice, though, it may take too long for it to be guaranteed. The question posed, then, is: what strategies should they choose to maximize their probability of meeting?

Examples of this class of problems are known as **rendezvous problems**. These problems were first introduced informally by Steve Alpern in 1976,^{ [1] } and he formalised the continuous version of the problem in 1995.^{ [2] } This has led to much recent research in rendezvous search.^{ [3] } Even the symmetric rendezvous problem played in *n* discrete locations (sometimes called the *Mozart Cafe Rendezvous Problem*)^{ [4] } has turned out to be very difficult to solve, and in 1990 Richard Weber and Eddie Anderson conjectured the optimal strategy.^{ [5] } Only recently^{[ when? ]} has the conjecture been proved for *n* = 3 by Richard Weber.^{ [6] } This was the first non-trivial symmetric rendezvous search problem to be fully solved. Note that the corresponding asymmetric rendezvous problem has a simple optimal solution: one player stays put and the other player visits a random permutation of the locations.

As well as being problems of theoretical interest, rendezvous problems include real-world problems with applications in the fields of synchronization, operating system design, operations research, and even search and rescue operations planning.

The **deterministic rendezvous problem** is a variant of the rendezvous problem where the players, or *robots*, must find each other by following a deterministic sequence of instructions. Although each robot follows the same instruction sequence, a unique label assigned to each robot is used for symmetry breaking.^{ [7] }

In mathematics and computer science, an **algorithm** is a finite sequence of well-defined, computer-implementable instructions, typically to solve a class of problems or to perform a computation. Algorithms are always unambiguous and are used as specifications for performing calculations, data processing, automated reasoning, and other tasks.

The **travelling salesman problem ** asks the following question: "Given a list of cities and the distances between each pair of cities, what is the shortest possible route that visits each city exactly once and returns to the origin city?" It is an NP-hard problem in combinatorial optimization, important in theoretical computer science and operations research.

In graph theory, the **shortest path problem** is the problem of finding a path between two vertices in a graph such that the sum of the weights of its constituent edges is minimized.

A **greedy algorithm** is any algorithm that follows the problem-solving heuristic of making the locally optimal choice at each stage. In many problems, a greedy strategy does not usually produce an optimal solution, but nonetheless a greedy heuristic may yield locally optimal solutions that approximate a globally optimal solution in a reasonable amount of time.

In graph theory, **graph coloring** is a special case of graph labeling; it is an assignment of labels traditionally called "colors" to elements of a graph subject to certain constraints. In its simplest form, it is a way of coloring the vertices of a graph such that no two adjacent vertices are of the same color; this is called a **vertex coloring**. Similarly, an edge coloring assigns a color to each edge so that no two adjacent edges are of the same color, and a **face coloring** of a planar graph assigns a color to each face or region so that no two faces that share a boundary have the same color.

In computer science, **2-satisfiability**, **2-SAT** or just **2SAT** is a computational problem of assigning values to variables, each of which has two possible values, in order to satisfy a system of constraints on pairs of variables. It is a special case of the general Boolean satisfiability problem, which can involve constraints on more than two variables, and of constraint satisfaction problems, which can allow more than two choices for the value of each variable. But in contrast to those more general problems, which are NP-complete, 2-satisfiability can be solved in polynomial time.

In graph theory, an **edge coloring** of a graph is an assignment of "colors" to the edges of the graph so that no two incident edges have the same color. For example, the figure to the right shows an edge coloring of a graph by the colors red, blue, and green. Edge colorings are one of several different types of graph coloring. The **edge-coloring problem** asks whether it is possible to color the edges of a given graph using at most k different colors, for a given value of k, or with the fewest possible colors. The minimum required number of colors for the edges of a given graph is called the **chromatic index** of the graph. For example, the edges of the graph in the illustration can be colored by three colors but cannot be colored by two colors, so the graph shown has chromatic index three.

The **El Farol bar problem** is a problem in game theory. Every Thursday night, a fixed population want to go have fun at the El Farol Bar, unless it's too crowded.

In probability theory, the **multi-armed bandit problem** is a problem in which a fixed limited set of resources must be allocated between competing (alternative) choices in a way that maximizes their expected gain, when each choice's properties are only partially known at the time of allocation, and may become better understood as time passes or by allocating resources to the choice. This is a classic reinforcement learning problem that exemplifies the exploration–exploitation tradeoff dilemma. The name comes from imagining a gambler at a row of slot machines, who has to decide which machines to play, how many times to play each machine and in which order to play them, and whether to continue with the current machine or try a different machine. The multi-armed bandit problem also falls into the broad category of stochastic scheduling.

**Václav (Vašek) Chvátal** (Czech: [ˈvaːtslaf ˈxvaːtal] is a Professor Emeritus in the Department of Computer Science and Software Engineering at Concordia University in Montreal, Quebec, Canada. He has published extensively on topics in graph theory, combinatorics, and combinatorial optimization.

In game theory, a **princess and monster game** is a pursuit-evasion game played by two players in a region. The game was devised by Rufus Isaacs and published in his book *Differential Games* (1965) as follows:

The monster searches for the princess, the time required being the payoff. They are both in a totally dark room, but they are each cognizant of its boundary. Capture means that the distance between the princess and the monster is within the capture radius, which is assumed to be small in comparison with the dimension of the room. The monster, supposed highly intelligent, moves at a known speed. We permit the princess full freedom of locomotion.

The **odds-algorithm** is a mathematical method for computing optimal strategies for a class of problems that belong to the domain of optimal stopping problems. Their solution follows from the *odds-strategy*, and the importance of the odds-strategy lies in its optimality, as explained below.

In the study of graph coloring problems in mathematics and computer science, a **greedy coloring** or **sequential coloring** is a coloring of the vertices of a graph formed by a greedy algorithm that considers the vertices of the graph in sequence and assigns each vertex its first available color. Greedy colorings can be found in linear time, but they do not in general use the minimum number of colors possible.

A **search game** is a two-person zero-sum game which takes place in a set called the search space. The searcher can choose any continuous trajectory subject to a maximal velocity constraint. It is always assumed that neither the searcher nor the hider has any knowledge about the movement of the other player until their distance apart is less than or equal to the discovery radius and at this very moment capture occurs. As mathematical models, search games can be applied to areas such as hide-and-seek games that children play or representations of some tactical military situations. The area of search games was introduced in the last chapter of Rufus Isaacs' classic book "Differential Games" and has been developed further by Shmuel Gal and Steve Alpern. The princess and monster game deals with a moving target.

In computational complexity theory, the **linear search problem** is an optimal search problem introduced by Richard E. Bellman..

**Shmuel Gal** is a mathematician and professor of statistics at the University of Haifa in Israel.

In computer science, one approach to the dynamic optimality problem on online algorithms for binary search trees involves reformulating the problem geometrically, in terms of augmenting a set of points in the plane with as few additional points as possible in order to avoid rectangles with only two points on their boundary.

In computer science, an **optimal binary search tree **, sometimes called a **weight-balanced binary tree**, is a binary search tree which provides the smallest possible search time for a given sequence of accesses. Optimal BSTs are generally divided into two types: static and dynamic.

- ↑ Alpern, Steve (1976),
*Hide and Seek Games*, Seminar, Institut fur Hohere Studien, Wien, 26 July. - ↑ Alpern, Steve (1995), "The rendezvous search problem",
*SIAM Journal on Control and Optimization*,**33**(3): 673–683, doi:10.1137/S0363012993249195, MR 1327232 - ↑ Alpern, Steve; Gal, Shmuel (2003),
*The Theory of Search Games and Rendezvous*, International Series in Operations Research & Management Science,**55**, Boston, MA: Kluwer Academic Publishers, ISBN 0-7923-7468-1, MR 2005053 . - ↑ Alpern, Steve (2011), "Rendezvous search games", in Cochran, James J. (ed.),
*Wiley Encyclopedia of Operations Research and Management Science*, Wiley, doi:10.1002/9780470400531.eorms0720 . - ↑ Anderson, E. J.; Weber, R. R. (1990), "The rendezvous problem on discrete locations",
*Journal of Applied Probability*,**27**(4): 839–851, doi:10.2307/3214827, MR 1077533 . - ↑ Weber, Richard (2012), "Optimal symmetric Rendezvous search on three locations" (PDF),
*Mathematics of Operations Research*,**37**(1): 111–122, doi:10.1287/moor.1110.0528, MR 2891149 . - ↑ Ta-Shma, Amnon; Zwick, Uri (April 2014). "Deterministic rendezvous, treasure hunts, and strongly universal exploration sequences".
*ACM Transactions on Algorithms*.**10**(3). 12.

This page is based on this Wikipedia article

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.