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### Paper:

TR97-033 | 1st August 1997 00:00

#### Parametrizing Above Guaranteed Values: MaxSat and MaxCut

TR97-033
Authors: Meena Mahajan, Venkatesh Raman
Publication: 10th September 1997 21:10
Keywords:

Abstract:

In this paper we investigate the parametrized
complexity of the problems MaxSat and MaxCut using the
framework developed by Downey and Fellows.

Let $G$ be an arbitrary graph having $n$ vertices and $m$
edges, and let $f$ be an arbitrary CNF formula with $m$
clauses on $n$ variables. We improve Cai and Chen's
$O(2^{2k}m)$ time algorithm for determining if at least $k$
clauses of of a $c$-CNF formula $f$ can be satisfied; our
algorithm runs in $O(|f| + k^2\phi^k)$ time for arbitrary
formulae and in $O(m + k\phi^k)$ time for $c$-CNF
formulae. We also give an algorithm for finding a cut of
size at least $k$; our algorithm runs in $O(m + n + k4^k)$
time.

Since it is known that $G$ has a cut of size at least
$\lceil \frac{m}{2} \rceil$ and that there exists an
assignment to the variables of $f$ that satisfies at least
$\lceil \frac{m}{2} \rceil$ clauses of $f$, we argue that
the standard parametrization of these problems is
unsuitable. Non-trivial situations arise only for large
parameter values, in which range the fixed-parameter
tractable algorithms are infeasible. A more meaningful
question in the parametrized setting is to ask whether
$\lceil \frac{m}{2} \rceil + k$ clauses can be satisfied, or
$\lceil \frac{m}{2} \rceil + k$ edges can be placed in a
cut. We show that these problems remain fixed-parameter
tractable even under this parametrization. Furthermore, for
upto logarithmic values of the parameter, our algorithms run
in polynomial time.

We also discuss the complexity of the parametrized versions
of these problems where all but $k$ clauses have to
satisfied or all but $k$ edges have to be in the cut.

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