On the geometry of $k$-SAT solutions: what more can PPZ and Sch\"oning's algorithms do?
Given a $k$-CNF formula and an integer $s$, we study algorithms that obtain $s$ solutions to the formula that are maximally dispersed. For $s=2$, the problem of computing the diameter of a $k$-CNF formula was initiated by Creszenzi and Rossi, who showed strong hardness results even for $k=2$. Assumi...
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Zusammenfassung: | Given a $k$-CNF formula and an integer $s$, we study algorithms that obtain
$s$ solutions to the formula that are maximally dispersed. For $s=2$, the
problem of computing the diameter of a $k$-CNF formula was initiated by
Creszenzi and Rossi, who showed strong hardness results even for $k=2$.
Assuming SETH, the current best upper bound [Angelsmark and Thapper '04] goes
to $4^n$ as $k \rightarrow \infty$. As our first result, we give exact
algorithms for using the Fast Fourier Transform and clique-finding that run in
$O^*(2^{(s-1)n})$ and $O^*(s^2 |\Omega_{F}|^{\omega \lceil s/3 \rceil})$
respectively, where $|\Omega_{F}|$ is the size of the solution space of the
formula $F$ and $\omega$ is the matrix multiplication exponent.
As our main result, we re-analyze the popular PPZ (Paturi, Pudlak, Zane '97)
and Sch\"{o}ning's ('02) algorithms (which find one solution in time
$O^*(2^{\varepsilon_{k}n})$ for $\varepsilon_{k} \approx 1-\Theta(1/k)$), and
show that in the same time, they can be used to approximate the diameter as
well as the dispersion ($s>2$) problems. While we need to modify Sch\"{o}ning's
original algorithm, we show that the PPZ algorithm, without any modification,
samples solutions in a geometric sense. We believe that this property may be of
independent interest.
Finally, we present algorithms to output approximately diverse, approximately
optimal solutions to NP-complete optimization problems running in time
$\text{poly}(s)O^*(2^{\varepsilon n})$ with $\varepsilon |
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DOI: | 10.48550/arxiv.2408.03465 |