$\{0,1\}$-solutions for integer equations via lattice base reduction?

Question

I would like to find $\{0,1\}$-solutions of a system of equations of the form $$\left\{\begin{array}{c}\sum_{i\in I_1}x_i=1\\\sum_{i\in I_2}x_i=1\\\vdots\\\sum_{i\in I_k}x_i=1\end{array}\right.$$

where $x_1,\ldots,x_n\in\{0,1\}$ are the variables and $I_1,I_2,\ldots,I_k$ are given subsets of $\{1,2,\ldots,n\}$.

I have now been solving this system by reducing its matrix over $\mathbb{Q}$ and then iterating over all possible combinations for the free variables in $\{0,1\}$, and verifying if the other variables are also in $\{0,1\}$ then. However that is likely not the best way. So my questions:

1. Several people have mentioned that "Lattice base reduction" provides a good solution to this. Unfortunately, when looking that up, I fail to see how that would make this system of equations easier to solve. What is the link?
2. Are there any other methods that I should be aware of for solving this kind of problems? Bonus points if not all constant coefficients have to be $1$, then I could add a few more equations to reduce my search space.

Answer 1

Your equation system (coefficients, right hand side, variables are all Boolean) has the structure of an exact cover problem. Based on Algorithm X, there are specialized solvers for this problem, most prominently Donald Knuth's dancing links, and his newer version dancing cells, where the data structure got adjusted to better fit modern processor architecture.

For implementations, look for DLX (dancing links) and SSXCC (dancing cells) Knuth's homepage. Another implementation of dancing links is libexact.

With very high chance, these algorithms will be much faster than any LLL based approach. However, when coefficients / boundaries for variables in your Diophantine linear equation system are larger, LLL is usually the way of choice.

Answer 2

Your question is very similar to Sudoku type of problems.

Sudoku puzzles can be reduced to the system of equations that has the same structure as yours. In Sudoku puzzles, there should be one and only one 1 in each row, column and box; there should be one and only one 2 in each row, column and box etc. You can write these conditions as a system above. Let $x_{n, r, c}$ be 1 if the number $n$ is in row $r$, column $c$, and $0$ otherwise; $n, r, c$ $\in$ $\{1, 2, \dots, 9\}$. You will have 3 types of subsets.
The row equations are: $\forall r, n \in \{1, 2, \dots, 9\}$ will correspond to subsets $$ I_{r, n} = \bigcup_{\forall c \in \{1, \dots, 9\}} \{n, r, c\}$$ Column equations are: $\forall c, n \in \{1, 2, \dots, 9\}$ will correspond to subsets $$ I_{c, n} = \bigcup_{\forall r \in \{1, \dots, 9\}} \{n, r, c\}$$ Finally, box equations are: $\forall R, C \in \{0, 3, 6\}, n \in \{1, 2, \dots, 9\}$ will correspond to subsets $$ I_{R, C, n} = \bigcup_{\forall r, c \in \{1, 2, 3\}} \{n, R + r, C + c\}$$

All sudoku algorighm I have seen can be used for general problem - the one you state above. See http://en.wikipedia.org/wiki/Sudoku_solving_algorithms and http://www.math.cornell.edu/~mec/Summer2009/meerkamp/Site/Solving_any_Sudoku_II.html among others. Just replace "rows, columns and boxes" by "sets $I_\cdot$", and you can apply them right away. And, I believe, right hand equation values can be different from 1, still, the same algorithms will apply.

Hope this helps!

Answer 3

To elaborrate on the "Lattice base reduction". Given a system where $a_{n,m} \in \mathbb{R}$

$$\left\{\begin{array}{c}\sum_{i\in I_1} a_{1,i} x_i=1\\\sum_{i\in I_2}a_{2,i} x_i=1\\\vdots\\\sum_{i\in a_{k,i} I_k}x_i=1\end{array}\right.$$

Latice basis reduction will give you $k$ different nearly orthogonal vectors whose integer weighted combinations give you the entire lattice. In your case, you have an additional constraint $x_i \in \{0,1\}$. This implies the constraint $0 \leq x_i \leq 1$. However lattices can not contain inequalities. Thus a lattice approach is not appropriate or helpful.

When being faced with such a problem i would trust a mature, competitve software package which (not to cast shade at Donald Knuth) is developed by large group of people and used by a large group of people. There are specialized solver for this "exact-cover" type of problem matching that criteria.

However for flexibilities sake i'd encourage you to use a more general language called Mixed-Integer Linear Programming, which are able to solve and optimize arbitrary linear inequalities involving integers and real numbers. This will also allow you to express logical constraints if that helps in your application.

Software packages i can recommend for that are:

• Gurobi, which is a commercial solver but has some licensing options which might work for you.
• SCIP, which is a free open source solver
• HiGHS, which is a free open source solver

All of those should be able detect (some(?) of) the special structure of the problem and exploit it. What is nice about them is these competing software packages share an underlying mathematics and can be fed using the same modeling languages such as JuMP and input formats like the .nl file format.

It could also be that Google's OR-Tools handles those problems well but this uses a different paradigm.