Stupid question, if the search program finds a best answer, how does it show that's the best?
Not a stupid question at all, since nobody else had thought of an answer to that problem!
Ian's basic idea is to start by (WLOG) assuming each input stick contributes at most one piece to each output stick (clearly if that's not true then you can merge fragments until it is without making your score worse). So now you have an n × m matrix of fragment lengths. Next, we want to treat the problem as a linear-programming optimisation exercise, trying to maximise one variable ("smallest nonzero matrix entry") under a collection of linear constraints (matrix entries all positive, rows and columns sum to the right thing). The trouble with that is that it's not actually a *linear* problem, since the objective criterion is nonlinear (taking min of all the matrix entries isn't a linear function). But if you knew the adjacency matrix (that is, you knew which input sticks contribute *at all* to which output sticks) it would be, because you could introduce the minimum frag length itself as an extra variable, so that some matrix elements were constrained to zero and others were of the form (min_frag + some positive extra value). So Ian simply iterates over all possible adjacency matrices and appeals to a linear-programming library to solve that optimisation problem for each one, and once he's gone through all possible matrices then he knows he's considered every answer!
Of course, then you optimise as hard as you can (by e.g. iterating over adjacency matrices in a sensible order that means you see the best ones first, stopping as soon as your matrix gets too dense because then some stick has to be cut into enough pieces that the smallest won't beat your existing best answer, parallelising over all the CPUs you can find, etc). But that's the basic idea. |