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Discussion

Carlos, Dave and Yolanda were fascinated by the discussion on complexity. Zori was less enthusiastic but even she sensed that the question of which problems could be solved quickly had practical implications.

Discussion

Carlos, Dave and Yolanda were fascinated by the discussion on complexity. Zori was less enthusiastic but even she sensed that the question of which problems could be solved quickly had practical implications. She could even predict that people could earn a nice income solving problems faster and more accurately than their competition.

Bob remarked, I'm not sure I understand what's being talked about here. I don't see why it can't be the case that all problems can be solved. Maybe we just don't know how to do it. Xing said, Any finite problem can be solved. There is always a way to list all the possibilities, compare them one by one and take the best one as the answer. Alice joined in, Well, a problem might take a long time just because it is big. For example, suppose you are given two DVD's, each completely full with the data for a large integer. How are you possibly going to multiply them together, even with a large computer and fancy software. Carlos then offered, But I think there are really hard problems that any algorithm will take a long time to solve and not just because the input size is large. At this point, I don't know how to formulate such a problem but I suspect that they exist.

Symbols used here

n!
factorial
n × (n−1) × … × 1; the number of orderings of n things. 0! = 1.
\binom{n}{k}
binomial coefficient, "n choose k"
Number of k-element subsets of n things: n!/(k!(n−k)!).
\sum_{k=1}^{n} a_k
summation
Add a_k for k = 1 up to n.
\prod_{k=1}^{n} a_k
product
Multiply a_k for k = 1 up to n.
\emptyset,\ |A|
empty set, cardinality
The set with no elements; the number of elements of A.

Questions people ask

Permutation or combination?

Ask whether order matters. A lock code is a permutation (order matters); a hand of cards is a combination (it does not).

What is a graph in this sense?

Dots (vertices) joined by lines (edges) — not a plot. Road maps, social networks and molecules are graphs; questions like "is there a route" and "how few colours" are graph theory.

Zama wena

Parts of this page are adapted from Keller & Trotter, Applied Combinatorics (CC BY-SA 4.0). Condensed and re-explained here; errors are ours.

Okuningi Combinatorics & Graph Theory

The counting principlesPigeonhole principle and inclusion–exclusionBinomial coefficients and Pascal's triangleRecurrences and generating functionsGraphs: vertices, edges, degreesPaths, cycles, trees, Euler and HamiltonColouring and planar graphs