Applications of Graph and Scheduling Theory

for getting out of Ohio Statefor getting out of Ohio State

Eugene TalagrandMauktik GandhiJeff Mathew

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

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

The Applications

This isn’t just a toy problem Similar problems are encountered in

industry, for example building an airplane The plane cannot be painted before it is

built

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.

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

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

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?

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)

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

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

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

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