A strong inapproximability gap for a generalization of minimum bisection

As a problem with similar properties to minimum bisection, we consider the following: given a homogeneous system of linear equations over Z₂, with exactly k variables in each equation, find a balanced assignment that minimizes the number of satisfied equations. A balanced assignment is one which contains an equal number of 0s and 1s. When k=2, this is the minimum bisection problem. We consider the case k=3. In this case, it is NP-complete to determine whether the object function is zero [U. Feige, (2003)], so the problem is not approximable at all. However, we prove that it is NP-hard to determine distinguish between the cases that all but a fraction ε of the equations can be satisfied and that at least a fraction 1/4-ε of all equations cannot be satisfied. A similar result for minimum bisection would imply that the problem is hard to approximate within any constant. For the problem of approximating the maximum number of equations satisfied by a balanced assignment, this implies that the problem is NP-hard to approximate within 4/3-ε, for any ε>0.