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Incoming links: Conditional expectations cut example
Conditional expectations cut example 2026-10-18
Consider the following original five-vertex simple graph, where denotes the undirected edge :Its seven edges give initial expected value . Use vertex order and choose zero whenever the two child conditional expectations tie.
The complete decision ledger is below. A prefix lists fixed labels in vertex order; all later labels remain hypothetically independent fair bits. Each candidate entry is the conditional expectation of the final cut of a graph, including the contribution of unresolved edges.
| Prefix before choice | Next vertex | Set to 0 | Set to 1 | Chosen label | Retained conditional expectation |
|---|---|---|---|---|---|
| Empty | |||||
For example, after , edge is fixed crossing and the other six edges are unresolved, giving . Setting fixes crossing and noncrossing, leaving four unresolved edges: . Setting instead fixes crossing and noncrossing, giving the same conditional expectation.
Conditional expected cut values along every chosen branch
. At each step, both candidate values are computed by the conditional expectation cut formula. The highlighted cell is chosen; ties choose zero. Below, the retained conditional expectation follows , starting before any label is fixed. The last value is the actual size of the resulting cut of a graph, rather than an average over remaining choices. Details.
The final labels are , so the two sides are and . The five crossing edges are ; do not cross. This exceeds , and hence also the rounded-up guarantee of four edges.
It is not a maximum cut. Labels cross all edges except , giving six. Seven are impossible because the triangle in a graph cannot have all three edges crossing a two-sided cut of a graph. An independent enumeration of all complete label assignments gives the following distribution, counting an assignment and its label-swapped counterpart separately:
| Size of cut of a graph | Number of label assignments |
|---|---|
The weighted sum is , so the average is , agreeing with linearity of expectation. Choosing in the earlier tie discards the remaining size-six completion while preserving the average of four. The method protects an expected value guarantee; it need not preserve the best possible completion.
Method of conditional probabilities Created 2026-10-06 Updated 2026-10-18
The method of conditional probabilities turns an expected value guarantee into a deterministic construction. For a maximization objective function, fix one choice at a time, always keeping the conditional expectation of the final value at least as large as before. Once every choice is fixed, that conditional expectation is the actual value. The useful ingredient is a way to evaluate the conditional expectations efficiently.
No actual random choices are needed during this construction. The probability distribution describes hypothetical completions used in the analysis; the algorithm compares their conditional expectations and commits deterministically. For maximum cut, this converts the random cut lower bound into a cut of a graph containing at least half the edges. A maximum cut can still be larger, as the conditional expectations cut example shows.