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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creator | Austrin, Per Bercea, Ioana O Goswami, Mayank Limaye, Nutan Srinivasan, Adarsh |
description | 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 $\varepsilon |
doi_str_mv | 10.48550/arxiv.2408.03465 |
format | Article |
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$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 $\varepsilon<1$ for several
problems such as Minimum Hitting Set and Feedback Vertex Set. For these
problems, all existing exact methods for finding optimal diverse solutions have
a runtime with at least an exponential dependence on the number of solutions
$s$. Our methods find bi-approximations with polynomial dependence on $s$.</description><identifier>DOI: 10.48550/arxiv.2408.03465</identifier><language>eng</language><subject>Computer Science - Computational Complexity ; Computer Science - Data Structures and Algorithms</subject><creationdate>2024-07</creationdate><rights>http://creativecommons.org/licenses/by/4.0</rights><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>228,230,780,885</link.rule.ids><linktorsrc>$$Uhttps://arxiv.org/abs/2408.03465$$EView_record_in_Cornell_University$$FView_record_in_$$GCornell_University$$Hfree_for_read</linktorsrc><backlink>$$Uhttps://doi.org/10.48550/arXiv.2408.03465$$DView paper in arXiv$$Hfree_for_read</backlink></links><search><creatorcontrib>Austrin, Per</creatorcontrib><creatorcontrib>Bercea, Ioana O</creatorcontrib><creatorcontrib>Goswami, Mayank</creatorcontrib><creatorcontrib>Limaye, Nutan</creatorcontrib><creatorcontrib>Srinivasan, Adarsh</creatorcontrib><title>On the geometry of $k$-SAT solutions: what more can PPZ and Sch\"oning's algorithms do?</title><description>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 $\varepsilon<1$ for several
problems such as Minimum Hitting Set and Feedback Vertex Set. For these
problems, all existing exact methods for finding optimal diverse solutions have
a runtime with at least an exponential dependence on the number of solutions
$s$. Our methods find bi-approximations with polynomial dependence on $s$.</description><subject>Computer Science - Computational Complexity</subject><subject>Computer Science - Data Structures and Algorithms</subject><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2024</creationdate><recordtype>article</recordtype><sourceid>GOX</sourceid><recordid>eNpjYJA0NNAzsTA1NdBPLKrILNMzMjGw0DMwNjEz5WQI989TKMlIVUhPzc9NLSmqVMhPU1DJVtENdgxRKM7PKS3JzM8rtlIoz0gsUcjNL0pVSE7MUwgIiFJIzEtRCE7OiFHKz8vMS1cvVkjMSc8vyizJyC1WSMm352FgTUvMKU7lhdLcDPJuriHOHrpgJ8QXFGXmJhZVxoOcEg92ijFhFQDJQT0f</recordid><startdate>20240728</startdate><enddate>20240728</enddate><creator>Austrin, Per</creator><creator>Bercea, Ioana O</creator><creator>Goswami, Mayank</creator><creator>Limaye, Nutan</creator><creator>Srinivasan, Adarsh</creator><scope>AKY</scope><scope>GOX</scope></search><sort><creationdate>20240728</creationdate><title>On the geometry of $k$-SAT solutions: what more can PPZ and Sch\"oning's algorithms do?</title><author>Austrin, Per ; Bercea, Ioana O ; Goswami, Mayank ; Limaye, Nutan ; Srinivasan, Adarsh</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-arxiv_primary_2408_034653</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2024</creationdate><topic>Computer Science - Computational Complexity</topic><topic>Computer Science - Data Structures and Algorithms</topic><toplevel>online_resources</toplevel><creatorcontrib>Austrin, Per</creatorcontrib><creatorcontrib>Bercea, Ioana O</creatorcontrib><creatorcontrib>Goswami, Mayank</creatorcontrib><creatorcontrib>Limaye, Nutan</creatorcontrib><creatorcontrib>Srinivasan, Adarsh</creatorcontrib><collection>arXiv Computer Science</collection><collection>arXiv.org</collection></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext_linktorsrc</fulltext></delivery><addata><au>Austrin, Per</au><au>Bercea, Ioana O</au><au>Goswami, Mayank</au><au>Limaye, Nutan</au><au>Srinivasan, Adarsh</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>On the geometry of $k$-SAT solutions: what more can PPZ and Sch\"oning's algorithms do?</atitle><date>2024-07-28</date><risdate>2024</risdate><abstract>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 $\varepsilon<1$ for several
problems such as Minimum Hitting Set and Feedback Vertex Set. For these
problems, all existing exact methods for finding optimal diverse solutions have
a runtime with at least an exponential dependence on the number of solutions
$s$. Our methods find bi-approximations with polynomial dependence on $s$.</abstract><doi>10.48550/arxiv.2408.03465</doi><oa>free_for_read</oa></addata></record> |
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subjects | Computer Science - Computational Complexity Computer Science - Data Structures and Algorithms |
title | On the geometry of $k$-SAT solutions: what more can PPZ and Sch\"oning's algorithms do? |
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