I have these equations and I want to find one solution for them
0<=x1+x2+x3+x4+x5+x6<=3
0<=x7+x8<=2
2<=x1+x2+x3+x4<=4
2<=x3+x4+x5<=3
2<=x6+x7+x8<=3
the values of xi is 0 or 1 ( xi is binary variable)
is there any algorithm for solving this kind of equation and similar ones
Are you sure your problem is not a binary integer programming?
If you just want to solve this in-equation with such small amount of variables, brute-force search may just work....Construct 2^8 8*1 vectors, and verify if every vector satisfy your in-equation (you can write your equation in matrix form for sure).
If you just want ONE solution....you can even do it by hand: 10101011
But the general solution is not easy. Check this post. To solve the binary integer linear equation in polynomial time, there is one paper that you may take some time digging.
EDIT: update from @Ben Voigt
branch-and-bound is typically effective for efficiently solving (large) integer (incl. binary) and mixed-integer problems. Of course this problem is too small to be worth the overhead -- exhaustive search is quite adequate.
Regardless of the algorithm you may use, I would do some preprocessing to reduce the complexity. The sum of binary variables is always >=0:
x1+x2+x3+x4+x5+x6<=3
x7+x8<=2
2<=x1+x2+x3+x4<=4
2<=x3+x4+x5<=3
2<=x6+x7+x8<=3
The sum of binary variables is always <= num of variables:
x1+x2+x3+x4+x5+x6<=3
2<=x1+x2+x3+x4
2<=x3+x4+x5
2<=x6+x7+x8