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Ladder

Assignment — the ladder

5 distinct arguments against one idea, from the one that introduces it to the one that assumes the rest.
  1. A table of shares written as a lottery over 3 whole assignments. A doubly stochastic table of shares, and beneath it the permutation matrices and weights that add up to it exactly, each drawn as a grid with one marked cell per row.

    A lottery over whole assignments

    A table of shares in which every person's shares add to one task and every task is exactly covered is never anything more than a mixture of whole assignments — and finding the mixture is a matter of taking one complete assignment out at a time.

    rung 1 · applied
  2. The 6 corners, and nothing in between. The 6 permutation matrices of size 3, drawn as grids. A search over every table of shares on a fine grid finds these and only these as corners of the set.

    The corners are whole assignments

    A table of shares can be written as a lottery over whole assignments, which the anchor's first rung demonstrates on one example. The general statement is that the corners of the set of such tables are exactly the whole assignments, and that single fact is why the whole subject is easy.

    rung 2 · applied
  3. A cheapest assignment, and the proof that it is cheapest. A 4 by 4 cost table with the cheapest assignment marked, and a row price and column price beside each. Every used cell's two prices add to its cost, and the prices total the assignment's cost.

    A price for every person and task

    The cheapest assignment can be found without comparing it to any other. Attach a number to each person and each task so that no pair's two numbers exceed its cost, and if the numbers add to an assignment's total, that assignment is cheapest — proved, by an argument that never mentions the alternatives.

    rung 3 · applied
  4. One table of shares, two different lotteries. A doubly stochastic table decomposed into whole assignments twice, by two different orders, giving two mixtures that reconstruct the same shares.

    One table, two lotteries

    A table of shares says what fraction of each task each person does. It does not say how — the same table is a mixture of whole assignments in many different ways, and the differences are exactly what the people being assigned would care about.

    rung 4 · applied
  5. The corner that is a half on every edge. A 3-vertex graph beside a table of the 5 corners of its matching relaxation. 4 are whole and one assigns a half to every edge.

    Where the corners stop being whole

    Everything on this ladder rests on one property — the relaxation of the assignment problem has whole-numbered corners. Add a single edge that closes an odd cycle and the property fails, a corner appears with a half in every coordinate, and the problem changes character completely.

    rung 5 · applied

All ladders