Dattorro Convex Optimization of Eternity II, Jon Dattoro

http://www.convexoptimization.com/wikimization/index.php/Dattorro_Convex_Optimization_of_Eternity_II

An Eternity II puzzle (http://www.eternityii.com/) problem
formulation A*x=b is discussed thoroughly in section 4.6.0.0.15 of
the book Convex Optimization & Euclidean Distance Geometry which
is freely available. That A matrix is obtained by presolving a
sparse 864,593-by-1,048,576 system.  The 3 problems in this set
contains three successive reductions, each equivalent to that
larger system:

    * EternityII_E: a 11077-by-262144 system E*x=tau, where tau is
        11077-by-1.  This is the million column Eternity II matrix
        having redundant rows and columns removed analytically.
        In the UF Collection, E is the Problem.A matrix, and tau
        is Problem.b.  All entries in E are from the set {-1,0,1,2}.
        tau is binary.

    * EternityII_Etilde: a 10054-by-204304 system Etilde*x=tautilde
        with tautilde of size 10054-by-1.  The system has columns
        removed corresponding to some known zero variables
        (removal produced dependent rows).  In the UF Collection,
        Etilde is the Problem.A matrix, and tautilde is Problem.b.
        All entries in Etilde are from the set {-1,0,1}.
        tautilde is binary.

    * EternityII_A: a 7362-by-150638 system A*x=b, where b is
        7362-by-1.  This system has columns removed not in
        smallest face (containing tautilde) of polyhedral cone
        K = { Etilde*x | x >= 0 }.

The following linear program is a very difficult problem that
remains unsolved:

    maximize_x z'*x, subject to A*x=b and x >= 0

The matrix A in the EternityII_A problem is sparse, having only
782,087 nonzeros.  All entries of A are integers from the set
{-1,0,1}.  The vector b is binary, with only 358 nonzeros.

Direction vector z is determined by Convex Iteration: 

    maximize_z z'*x^{star}
    subject to 0 <= z <= 1 and z'*1 = 256

(for a vector x, x >= 0 means all(x>=0) in MATLAB notation).

These two problems are iterated to find a minimal cardinality
solution x.  Constraint A*x=b bounds the variable from above by 1.
Any minimal cardinality solution is binary and solves the Eternity
II puzzle. The Eternity II puzzle is solved when
z^{star}'*x^{star} = 256.

Minimal cardinality of this Eternity II problem is equal to number
of puzzle pieces, 256.

Comment: The technique, convex iteration, requires no modification
(and works very well) when applied instead to mixed integer
programming (MIP, not discussed in book). There is no modification
to the linear program statement here except 256 variables,
corresponding to the largest entries of iterate x^{star}, are
declared binary.