A meeting scheduling problem respecting time and space

Abstract

We consider the problem of determining suitable meeting times and locations for a group of participants wishing to schedule a new meeting subject to already scheduled meetings possibly held at a number of different locations. Each participant must be able to reach the new meeting location, attend for the entire duration, and reach the next meeting location on time. In particular, we give two solutions to the problem instance where each participant has two scheduled meetings separated by a free time interval. We present an O(n logn) algorithm for n participants obtained by purely geometrical arguments. Our second approach uses the concept of LP-type problems and leads to a randomized algorithm with expected running time O(n). We also consider a graph-based model where participants belong to different groups and can travel along the edges of a graph. For the meeting, only one member out of each group is required. The resulting problem can be solved using furthest color Voronoi diagrams on graphs.

APPENDIX - Table of symbols

• M: = {M ₁,...,M _n } the set of participants; M _i the ith participant

• $l_{i}^{\,pre}$ previous meeting location

• $t_{i}^{\,pre}$ previous meeting finishing time

• $l_{i}^{\,sub}$ subsequent meeting location

• $t_{i}^{\,sub}$ subsequent meeting starting time

• S(x) the arrival time of the last participants from their previous meetings

• E(x) the departure time of the first participants to their subsequent meetings

• f(x) the time interval between E(x) and S(x)

• PRE _i lower vertical cone model for M _i to travel after previous meeting

• SUB _i upper vertical cone model for M _i to travel before subsequent meeting

• $\left(\left(l_{i}^{\,pre}\right)_X,\left(l_{i}^{\,pre}\right)_Y,t_{i}^{\,pre}\right)$ apex of PRE _i

• $\left(\left(l_{i}^{\,sub}\right)_X,\left(l_{i}^{\,sub}\right)_Y,t_{i}^{\,sub}\right)$ apex of SUB _i

• PRE intersection of PRE _i ∀ 1 ≤ i ≤ n

• SUB intersection of SUB _i ∀ 1 ≤ i ≤ n

• F intersection of PRE and SUB

• V ^pre the furthest Voronoi diagram of the discs cut from PRE _i at a time plane, ∀ 1 ≤ i ≤ n

• V ^sub the furthest Voronoi diagram of the discs cut from SUB _i at a time plane, ∀ 1 ≤ i ≤ n

• D _i a circle cut from PRE _i or SUB _i at a time plane

• l′ the earliest meeting possible point

• l′′ the latest meeting possible point

• m′ the bottommost point of PRE

• m′′ the topmost point of SUB

• S′ the projection of the convex shape of PRE cut by a plane H′ onto the XY plane

• S′′ the projection of the convex shape of SUB cut by a plane H′ onto the XY plane

• (f,S) a function to be lexicographically maximized for the LP-type problem

• x _i a location in the XY plane

• (x ₁, S(x ₁)) a point in 3-d space

• (x ₁, E(x ₁)) a point in 3-d space

• B _S the line segment from (x ₁, S(x ₁)) to (x ₂, S(x ₂))

• B _E the line segment from (x ₁, E(x ₁)) to (x ₂, E(x ₂))

• M′ subset of M

• OPT _M′ the lexicographically optimum pair (f(x),S(x)) for a meeting of all participants in M′

• f| _M′(x) the maximum duration for a meeting at location x of all participants in M′

• X(OPT _M′) the unique location where the optimum meeting is held.