Applications of Graph and Scheduling Theory for getting out of Ohio State

1
Applications of Graph and Scheduling Theoryfor
getting out of Ohio State

• Eugene Talagrand
• Mauktik Gandhi
• Jeff Mathew

2
The Problem

• Design and implement algorithms to solve
  scheduling problems with prerequisites.
• Schedule a certain amount of events in as short
  of a time as possible, while considering the fact
  that some events must occur before others, some
  events cannot occur at the same time due to
  conflicts, and that at any given time there is a
  bounded number of events that can be happening
  simultaneously

3
The Motivation

• This is a hard problem
• Hard in the theoretical sense
• Brute force approach
• O (life time of the universe2)
• 1055 operations
• A different class of projects
• Non conventional
• Very long and complex project

4
The Applications

• This isnt just a toy problem
• Similar problems are encountered in industry, for
  example building an airplane
• The plane cannot be painted before it is built

5
Existing work in the field

• The Crew Scheduling problem, studied in
  Operations Research, is similar to this one. It
  can be solved using Binary Programming and the
  Simplex Method. It is NP-Complete.
• Different data structures will only serve to
  reduce the complexity of a problem marginally.

6
The Goals

• Develop intelligent heuristics to find
  polynomial-time approximations for this problem
• Develop progressive solutions, meaning that given
  more time, the optimality of the solution
  improves accordingly

7
The Approach

• We took a three-step approach to solving this
  problem
• Develop an efficient data structure
• Must model every behavior
• Develop polynomial-time deterministic algorithms
  to reduce the size of the problem set
• Reduction was chosen over construction to make
  use of these algorithms
• Develop polynomial time heuristic algorithms to
  get closer to the optimal solution

8
The Data Structures

• A first attempt was made to work directly on
  graphs.
• Refined into a tape model.
• Bins represent quarters.
• Each bin initially contains all courses offered
  that quarter
• Bins are reduced based on prerequisites and
  conflicts
• How to resolve conflicts?

9
The Prerequisite Graph

• A prerequisite graph needs to be built
• Longest-path algorithms (NP-Complete) need to be
  as fast as possible constant time!
• Implementation akin to adjacency matrices, but
  every cell contains the directed longest path
  between two courses. Negative dependencies are
  also indexed (222 has a negative dependency on
  321)

10
The Branch-and-Bound Algorithms

• Conflict resolution gets easier as more courses
  are scheduled. How to start? (Edge effect)
• Pack the bins from the left
• Pack the bins from the right least amount of
  time through college can be minimum bounded
• Heuristics will help later on
• When a conflict is resolved, cascade the result

11
The Core Dilemma

• Many more deterministic algorithms have been
  considered
• For many, we hit the core dilemma the fragile
  balance between prerequisites and scheduling
  conflicts
• Long-term effects of scheduling

12
The Initial Heuristics

• For each course in a bin, assign a priority equal
  to the sum of the lengths of the longest paths to
  outgoing classes
• Pick the subset of courses within the credit-hour
  limit that maximizes the sum of priorities
• This is the subset-sum problem another
  NP-Complete problem! Fortunately, there already
  exists a polynomial time approximation
• Approximate the approximation
• Optimizations like aging effects to prevent
  starvation can be applied as well

13
The Future

• Develop a more formal model for the problem,
  allowing for more algebraic solutions
• Explore scheduling bottlenecks, and work towards
  them (such as tilting the heuristics towards
  them). These might be detected through adjacency
  matrix eigenvalues
• Exploit graph cut sets, to localize the effects
  of the heuristics