Strategy for working on algo/theory qualifiers.

Most of this is common sense.
The exam is NOT testing you on brilliant ideas, nor memorization of
recipes, nor extensive background knowledge. Most nonbasic definitions
are explicitly provided.

The exam is testing you on:
(1) basic maturity in thinking about algorithms and
complexity -- i.e, facility with basic
algorithm and design and analysis techniques, basic complexity concepts
(2) logical clarity and minimality of thought process
which includes:
*Ability to understand the problem;
*Ability to precisely formulate/reformulate problems;
*Lack of clutter with inessential details;
*Splitting up problem;
*Formulating conjectures and claims;
*Step by step systematic approach.

To expand more on this..

0) Overall:
Think logically clearly which generally means minimally, without clutter.
Don't enmesh yourself in statements,
properties and details that are inessential. Not only do they
waste time, they actively prevent you from thinking clearly and making
essential observations.
WRITE DOWN your thought process step by step, but in a clear way.
This is absolutely essential for partial credit.

1) Understand the problem. What are the givens, what is required?
Again think minimally: don't try to give solutions beyond
what is required.

2) (Re)formulate the problem (if possible) into a form you know

3) Split up the problem into parts, helps you both think and write down
your solution step by step.

4) Work through some example input instances (in case of algo qual)

5) How long does the brute force algorithm take (in case of algo qual)
Usually, the problem requires something better than the brute force algo.
What are the givens that the brute force algorithm is not taking
advantage of? What is the jump we need to get beyond the brute force
algorithm?

6) If a desired complexity is specified (in case of algo qual)
what is the structure of the algo that will give this type of complexity?

7) Work through standard design techniques

8) Work through standard analysis techniques. If complexity claim
depends on some key conjectures, write them down and then try to
prove them. (This again is what I mean by thinking clearly, splitting
problem into parts etc..)

9) Make sure your algorithm is correct. If correctness depends on
some key conjectures, write them down and later try to prove them.

10) Last, but not least:
critically check your claims and conjectures. If you are doubtful
about them, indicate it. If you are sure they are wrong (you have a
counterexample) but think they can be modified to become correct,
indicate roughly how?