Finding bounds for 21-variable inequalities

Question

I have the following inequalities on 21 variables:

http://pastebin.com/raw.php?i=FTU970Em

When I run "Reduce[ineq,Integers]" on this, Mathematica hangs for a long time.

That makes sense: there are MANY sets of values for x[1]..x[21] that satisfy the inequalities.

All I really want is bounds for each variable (eg, "2 <= x[i] <= 7" for each i).

How can I get this efficiently w/ Mathematica? Is there a better program for this?

Note: this is part of the larger project:

http://stackoverflow.com/questions/3799748

The entire hideous list of inequalities: http://pastebin.com/CyX9f70J

Running "Reduce[ineq,Integers]" on the above yields "false", so I've probably incorrectly translated: http://conquerclub.barrycarter.info/ONEOFF/7460216.html

Answer 1

Are there many sets of values which satisfy the inequalities ?

I ran the following commands through Mathematica:

In[14]:= ineqs = {x0 == 3, x1 >= 1, x1 <= x0, x2 >= 1, x2 <= x1, 
       x3 >= 1, x3 <= x2, x4 >= 1, x4 <= x3, x5 <= x4 + 3, x5 >= 1, 
       x6 >= 1, x6 <= x5, x7 >= 1, x7 <= x6, x8 >= 1, x8 <= x7, x9 >= 1, 
       x9 <= x8, x10 >= 1, x10 <= x9, x11 >= 1, x11 <= x10, x12 >= 1, 
       x12 <= x11, x13 >= 1, x13 <= x12, x14 <= x13 + 4, x14 >= 1, 
       x15 >= 1, x15 <= x14, x16 >= 1, x16 <= x15, x17 <= x16 + 6, 
       x17 >= 1, x18 >= 1, x18 <= x17, x19 >= 1, x19 <= x18, x20 >= 1, 
       x20 <= x19, x21 >= 1, x21 <= x20, x21 == 1};

In[15]:= vars = 
      Union[{x0, x1, x1, x2, x2, x3, x3, x4, x4, x5, x5, x6, x6, x7, x7, 
        x8, x8, x9, x9, x10, x10, x11, x11, x12, x12, x13, x13, x14, x14, 
        x15, x15, x16, x16, x17, x17, x18, x18, x19, x19, x20, x20, x21, 
        x21, x21}];

In[16]:= FindInstance[ineqs, vars]

and got the result:

Out[16]= {{x0 -> 3, x1 -> 1, x10 -> 1, x11 -> 1, x12 -> 1, x13 -> 1, 
  x14 -> 1, x15 -> 1, x16 -> 1, x17 -> 1, x18 -> 1, x19 -> 1, x2 -> 1,
   x20 -> 1, x21 -> 1, x3 -> 1, x4 -> 1, x5 -> 1, x6 -> 1, x7 -> 1, 
  x8 -> 1, x9 -> 1}}

I haven't been able to persuade Mathematica to provide another set of assignments and a little work with pencil and paper doesn't point me towards other sets of assignments. But it's late here, I may have missed something obvious.

Answer 2

OK, it turns out that solving this particular set of equations is easy, once you rewrite some of them slightly:

x5 <= x4 + 3 becomes x5 - 3 <= x4 
x6 <= x5 becomes x6 - 3 <= x5 - 3 

and so on until:

x13 <= x12 becomes x13 - 3 <= x12 - 3 
x14 <= x13 + 4 becomes x14 - 7 <= x13 -3 

By doing this, {x0, x1, x2, x3, x4, x5-3, x6-3, ..., x13-3, x14-7, ..., x21} becomes a strictly decreasing sequence of integers starting at 3 and ending at 1.

In fact, any sequence w/ that property works, since xi>=1 is trivally satisfied.

However, while this works to solve this particular set of inequalities, it doesn't work in general, so I don't consider it a complete solution.

Answer 3

I second the CLP(FD) suggestion given in the other thread. Using SWI-Prolog 5.10:

:- use_module(library(clpfd)).

vars([X0,X1,X2,X3,X4,X5,X6,X7,X8,X9,X10,X11,X12,X13,X14,X15,X16,X17,X18,
      X19,X20,X21]) :-
        X0 #= 3, X1 #>= 1, X1 #=< X0, X2 #>= 1, X2 #=< X1,
        X3 #>= 1, X3 #=< X2, X4 #>= 1, X4 #=< X3, X5 #=< X4 + 3,
        X5 #>= 1, X6 #>= 1, X6 #=< X5, X7 #>= 1, X7 #=< X6,
        X8 #>= 1, X8 #=< X7, X9 #>= 1, X9 #=< X8, X10 #>= 1,
        X10 #=< X9, X11 #>= 1, X11 #=< X10, X12 #>= 1, X12 #=< X11,
        X13 #>= 1, X13 #=< X12, X14 #=< X13 + 4, X14 #>= 1, X15 #>= 1,
        X15 #=< X14, X16 #>= 1, X16 #=< X15, X17 #=< X16 + 6, X17 #>= 1,
        X18 #>= 1, X18 #=< X17, X19 #>= 1, X19 #=< X18, X20 #>= 1,
        X20 #=< X19, X21 #>= 1, X21 #=< X20, X21 #= 1.

Example queries:

?- vars(Vs), maplist(fd_dom, Vs, Ds).
Ds = [3..3, 1..3, 1..3, 1..3, 1..3, 1..6, 1..6, 1..6, ... .. ...|...]

?- vars(Vs), label(Vs).
Vs = [3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] ;
Vs = [3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1] ;
Vs = [3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1] ;
etc.