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Electronic Colloquium on Computational Complexity

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TR01-026 | 3rd April 2001 00:00

Approximation Hardness of Bounded Degree MIN-CSP and MIN-BISECTION



We consider bounded occurrence (degree) instances of a minimum
constraint satisfaction problem MIN-LIN2 and a MIN-BISECTION problem for
graphs. MIN-LIN2 is an optimization problem for a given system of linear
equations mod 2 to construct a solution that satisfies the minimum number
of them. E3-OCC-MIN-E3-LIN2 is the bounded occurrence (degree) problem
restricted as follows: each equation has exactly 3 variables and each
variable occurs in exactly 3 equations. Clearly, MIN-LIN2 is equivalent to
another well known problem, the Nearest Codeword problem, and
E3-OCC-MIN-E3-LIN2 to its bounded occurrence version. MIN-BISECTION is a
problem of finding a minimum bisection of a graph, while 3-MIN-BISECTION is
the MIN-BISECTION problem restricted to 3-regular graphs only. We show that,
somewhat surprisingly, these two restricted problems are exactly as hard to
approximate as their general versions. In particular, an approximation ratio
lower bound for E3-OCC-MIN-E3-LIN2 (bounded 3-occurrence 3-ary Nearest
Codeword problem) is equal to MIN-LIN2 (Nearest Codeword problem) lower bound
n^(Omega(1)/log log n). Moreover, an existence of a constant factor
approximation ratio (or a PTAS) for 3-MIN-BISECTION entails existence of a
constant approximation ratio (or a PTAS) for the general MIN-BISECTION.

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