Matpic
mathematics, in pictures
Generator

The vertex that maximises 3x₁ + 4x₂

A generator in the applied library, called 88 times across 17 essays. Below: what it draws with nothing chosen and at each mode an essay asks for, what it checks while drawing, and everywhere it is used.

polytope is one function. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page — so a figure here is the same figure a reader meets in an essay, and if the generator changes, this page changes with it.

With nothing chosen

The vertex that maximises 3x₁ + 4x₂. A two-variable linear program's feasible region, drawn from the exact intersection of every pair of its constraints, with the objective's contour lines and the optimal vertex marked.

Two triangles and three paths: the relaxation against the shortest tour

Two triangles and three paths: the relaxation against the shortest tour. k = 3: 12 cities; subtour relaxation 12 with 6 half-edges; shortest tour 14; ratio 1.167.

Degree two is not enough: the cut that forbids two loops

Degree two is not enough: the cut that forbids two loops. Two clusters of four; degree-only optimum 2.5012 as two loops; after 1 cut(s) 5.3008 = shortest tour.

Shortest tour over the relaxation's bound, as the paths lengthen

Shortest tour over the relaxation's bound, as the paths lengthen. k 1: 6/6 = 1.0000; k 2: 10/9 = 1.1111; k 3: 14/12 = 1.1667; k 4: 18/15 = 1.2000; k 5: 22/18 = 1.2222.

The gap on random cities: usually none, rarely more than a few per cent

The gap on random cities: usually none, rarely more than a few per cent. 40 random 15-city instances; 35 with ratio 1; mean 1.00046; worst 1.00542.

The weights the relaxation uses, on random cities

The weights the relaxation uses, on random cities. 40 instances of 15: 587 edges at 1, 26 at ½, 0 at other fractions; 4 instances fractional.

What it checks while it draws

Collected by running the family and recording what it asserted, not written here. The count is how many separate times the claim was put to the test while these drawings were made.

Where it is called

Every figure on this list is drawn by the same rule, so a change to the rule changes all of them at once. That is why the list is published.

Applied

A bound that may be off by a third

The shortest tour through a set of cities is hard to find, and a linear programme gives a lower bound for it in polynomial time: give every road a weight between nought and one, two at each city, at least two across every division of the map. On random cities the bound is almost always exact. On two triangles joined by three long paths it falls short by nearly a third, and whether a third is the worst it can ever do has been conjectured for decades and never proved.

 Applied

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.

 Applied

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.

 Applied

A signal both can see

Two choosers who randomise privately can reach a set of outcomes that is smaller, and worse, than the set they reach when a device draws one cell and whispers each of them their half of it. Nothing is enforced and nobody is bound, and the arrangement is stable anyway.

 Applied

A split nobody can walk away from

Every way of dividing what a group earns is a point of a triangle, and every coalition's threat to leave cuts a straight line across it. What survives all the cuts is the set of stable divisions — and for one three-player game there is nothing left.

 Applied

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.

 Applied

Prices at every corner

The duality theorem says a linear program's best value equals its dual's, and says nothing about how to find either. The simplex method finds both at once — it walks from corner to corner, and at each one asks the constraints that meet there for prices. A negative price names an edge that climbs; when none is negative, the prices are the proof.

 Applied

Prices the bidders raise

The cheapest assignment is certified by a price on every task, and those prices can be found without anyone in charge. Let each unassigned person bid for the task that suits them best at current prices, raise its price by a little more than it is worth to them over the next best, and wait. The bidding ends, and when the increment is small enough the prices it ends at are a proof of optimality.

 Applied

The corners are whole assignments

A table of shares can be written as a lottery over whole assignments, which one worked example shows. 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.

 Applied

The cube that takes every corner

The simplex method is fast on every program anybody meets in practice. In 1972 Victor Klee and George Minty squashed a cube so that the method, choosing the steepest edge each time, visits all of its corners — 2ⁿ − 1 moves in n variables, with the optimum one edge from the start.

 Applied

The lines the optimum lies under

Change the resources a linear program is given and its best value changes too, tracing a graph. Every solution of the dual is a straight line lying above that graph, and the graph is exactly the lowest of those lines — a bent roof of finitely many planks. Require the answer to be in whole numbers and the roof stays where it was while the graph falls away beneath it in steps, and the space between is the part of the problem no price can see.

 Applied

The prices nobody can break away from

When houses are sold to buyers who value them differently, there is a whole range of prices at which nobody wants to walk away, and it has a remarkable shape — one corner best for every buyer at once, one best for every seller at once, and the buyers' corner pays each buyer exactly what the market would lose without them.

 Applied

Tours within half again of the best

Nobody can find the shortest tour through many cities quickly, but a tour at most half as long again as the best can be built in a few steps: the shortest tree, a cheapest pairing of the cities where the tree branches oddly, an Euler circuit, and shortcuts. Nicos Christofides found it in 1976, and for forty-five years nobody could guarantee better. A strip of cities shows the half is really lost, and Laurence Wolsey's reading of the same argument shows it bounds the linear programme too.

 Applied

Two numbers that have to meet

Every linear program has a shadow — a second program built from the same numbers read the other way, whose minimum can never fall below the first's maximum. That much is a one-line calculation; the theorem is that the two numbers are always exactly equal.

 Applied

What a constraint is worth

A linear program and its dual reach the same number. What the dual's variables are is a separate question, and the answer converts a solution into a rate for every constraint — piecewise constant, zero on the constraints that are not doing any work.

 Applied

When one of the two numbers is missing

The duality theorem is usually quoted as an equality: a linear program and its dual reach the same number. That is one of four cases. A program can run away to infinity, or have no feasible point at all, and then its dual is forced into a matching failure. Every small program with coefficients from minus one to one has been classified, and the table has exactly four occupied cells out of nine.

 Applied

Where the corners stop being whole

The easy theory of assignment 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.

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