Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Tighter Hard Instances for PPSZ   We construct uniquely satisfiable $k$-CNF formulas that are hard for the
algorithm PPSZ. Firstly, we construct graph-instances on which "weak PPSZ" has
savings of at most $(2 + \epsilon) / k$; the saving of an algorithm on an input
formula with $n$ variables is the largest $\gamma$ such that the algorithm
succeeds (i.e. finds a satisfying assignment) with probability at least $2^{ -
(1 - \gamma) n}$. Since PPSZ (both weak and strong) is known to have savings of
at least $\frac{\pi^2 + o(1)}{6k}$, this is optimal up to the constant factor.
In particular, for $k=3$, our upper bound is $2^{0.333\dots n}$, which is
fairly close to the lower bound $2^{0.386\dots n}$ of Hertli [SIAM J.
Comput.'14]. We also construct instances based on linear systems over
$\mathbb{F}_2$ for which strong PPSZ has savings of at most
$O\left(\frac{\log(k)}{k}\right)$. This is only a $\log(k)$ factor away from
the optimal bound. Our constructions improve previous savings upper bound of
$O\left(\frac{\log^2(k)}{k}\right)$ due to Chen et al. [SODA'13].

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The categories of this paper are: acc-phys
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The categories of this paper are: chem-ph
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The categories of this paper are: eess
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The categories of this paper are: mtrl-th
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The categories of this paper are: nucl-th
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The categories of this paper are: nlin
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The categories of this paper are: gr-qc
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The categories of this paper are: cond-mat
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The categories of this paper are: q-fin
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The categories of this paper are: atom-ph
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The categories of this paper are: astro-ph
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The categories of this paper are: supr-con
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The categories of this paper are: hep-th
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The categories of this paper are: dg-ga
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The categories of this paper are: math
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The categories of this paper are: nucl-ex