Overview

  1. Review of women pessimal proof of GS
  2. Scheduling (Greedy algorithms)

Proof that women get their worst possible match under GS

graph LR
M' --- W
M --- W
M --- W'

Assume that $W$ … complete later

Scheduling

|
|
|
|                      -------         <------ conflicting/overlapping jobs (must choose one in advance)
|     -------       ------
|
|
------------------------------- time

Every job has a start time $s_i$ and end time $f_i$.

Definition: Two jobs $i,j$ are compatible $\mathrm{iff}$ $f_i \leq s_j$ or $f_j \leq s_i$.

Problem 1

Given a set of jobs, maximize number of accepted jobs.

         1
|-----------------|
    2        3
|-------||--------|
   4     5     6
|----||----||-----|

Obviously, choosing jobs 4, 5, 6 would maximize.

Claim: We can solve this problem using a greedy algorithm.

  1. Use a simple rule to pick a job $i$
  2. Cross out all jobs that are incompatible with $i$
  3. Repeat

Which rule to use to pick a job?