Arrow's Impossibility Theorem
No ranked voting system satisfies a short list of obviously reasonable conditions. Applies directly to prioritization, ranking, and consensus UX.
Optimization
Gale–Shapley Stable Matching
A stable matching always exists and is findable in O(n^2), but it's optimal for the proposing side. Mechanism design in twenty lines.
Matroids and Greedy Optimality
The exact structural condition under which greedy is provably correct. Answers "when can I just be greedy?" once and for all.
LP Duality
Every optimization problem has a shadow problem whose optimum is the same number, and it certifies your answer: the only post in the series about proving your...
Max-Flow Min-Cut
The bottleneck equals the capacity, exactly. A duality result you can see, and the workhorse behind matching and segmentation.
Approximation Algorithms and Ratios
The positive side: 2-approximate vertex cover, PTAS, and where the approximability cliff sits.