A Short Implicant of a CNF Formula with Many Satisfying Assignments
Abstract Consider any Boolean function %$F(X_1,\ldots ,X_N)%$ that has more than %$2^{-N^{\delta }}\cdot 2^N%$ satisfying assignments for some %$\delta %$, %$0<\delta <1%$, and that can be expressed by a CNF formula with at most %$N^d%$ clauses for some %$d>0%$. Then how many variables do w...
Ausführliche Beschreibung
Gespeichert in:
| Autor*in: |
Kane, Daniel [verfasserIn] Watanabe, Osamu [verfasserIn] |
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| Format: |
E-Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2016 |
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| Schlagwörter: |
Short satisfying partial assignment |
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| Übergeordnetes Werk: |
Enthalten in: Algorithmica - New York, NY : Springer, 1986, 76(2016), 4 vom: 01. Feb., Seite 1203-1223 |
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| Übergeordnetes Werk: |
volume:76 ; year:2016 ; number:4 ; day:01 ; month:02 ; pages:1203-1223 |
| Links: |
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| DOI / URN: |
10.1007/s00453-016-0125-z |
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| Katalog-ID: |
SPR006178251 |
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| 100 | 1 | |a Kane, Daniel |e verfasserin |4 aut | |
| 245 | 1 | 2 | |a A Short Implicant of a CNF Formula with Many Satisfying Assignments |
| 264 | 1 | |c 2016 | |
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| 520 | |a Abstract Consider any Boolean function %$F(X_1,\ldots ,X_N)%$ that has more than %$2^{-N^{\delta }}\cdot 2^N%$ satisfying assignments for some %$\delta %$, %$0<\delta <1%$, and that can be expressed by a CNF formula with at most %$N^d%$ clauses for some %$d>0%$. Then how many variables do we need to fix in order to satisfy F? We show that one can always find some “short” partial assignment on which F evaluates to 1 by fixing at most %$\alpha N%$ variables for some constant %$\alpha <1%$; that is, F has an implicant of size %$\le \alpha N%$. A lower bound for such %$\alpha %$ is also shown in terms of %$\delta %$ and d. We also discuss an algorithm for obtaining a short partial assignment. For any %$\delta %$ and %$\varepsilon %$ such that %$0<\delta +\varepsilon <1%$, we show a deterministic algorithm that finds a short partial assignment in %${\widetilde{O}}(2^{N^{\beta }})%$-time (By %${\widetilde{O}}(2^{N^{\beta }})%$ we mean %$O\bigl (2^{N^{\beta }}\cdot N^{O(1)}\bigr )%$.) for some %$\beta <1%$ and for any CNF formula with at most %$N^{1+\varepsilon }%$ clauses having more than %$2^{-N^{\delta }}\cdot 2^N%$ satisfying assignments. | ||
| 650 | 4 | |a Short satisfying partial assignment |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a Conjunctive normal form formula |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a Large satisfying assignment ratio |7 (dpeaa)DE-He213 | |
| 700 | 1 | |a Watanabe, Osamu |e verfasserin |4 aut | |
| 773 | 0 | 8 | |i Enthalten in |t Algorithmica |d New York, NY : Springer, 1986 |g 76(2016), 4 vom: 01. Feb., Seite 1203-1223 |w (DE-627)253389704 |w (DE-600)1458414-1 |x 1432-0541 |7 nnns |
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10.1007/s00453-016-0125-z doi (DE-627)SPR006178251 (SPR)s00453-016-0125-z-e DE-627 ger DE-627 rakwb eng 004 ASE 510 004 ASE 54.00 bkl Kane, Daniel verfasserin aut A Short Implicant of a CNF Formula with Many Satisfying Assignments 2016 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract Consider any Boolean function %$F(X_1,\ldots ,X_N)%$ that has more than %$2^{-N^{\delta }}\cdot 2^N%$ satisfying assignments for some %$\delta %$, %$0<\delta <1%$, and that can be expressed by a CNF formula with at most %$N^d%$ clauses for some %$d>0%$. Then how many variables do we need to fix in order to satisfy F? We show that one can always find some “short” partial assignment on which F evaluates to 1 by fixing at most %$\alpha N%$ variables for some constant %$\alpha <1%$; that is, F has an implicant of size %$\le \alpha N%$. A lower bound for such %$\alpha %$ is also shown in terms of %$\delta %$ and d. We also discuss an algorithm for obtaining a short partial assignment. For any %$\delta %$ and %$\varepsilon %$ such that %$0<\delta +\varepsilon <1%$, we show a deterministic algorithm that finds a short partial assignment in %${\widetilde{O}}(2^{N^{\beta }})%$-time (By %${\widetilde{O}}(2^{N^{\beta }})%$ we mean %$O\bigl (2^{N^{\beta }}\cdot N^{O(1)}\bigr )%$.) for some %$\beta <1%$ and for any CNF formula with at most %$N^{1+\varepsilon }%$ clauses having more than %$2^{-N^{\delta }}\cdot 2^N%$ satisfying assignments. Short satisfying partial assignment (dpeaa)DE-He213 Conjunctive normal form formula (dpeaa)DE-He213 Large satisfying assignment ratio (dpeaa)DE-He213 Watanabe, Osamu verfasserin aut Enthalten in Algorithmica New York, NY : Springer, 1986 76(2016), 4 vom: 01. Feb., Seite 1203-1223 (DE-627)253389704 (DE-600)1458414-1 1432-0541 nnns volume:76 year:2016 number:4 day:01 month:02 pages:1203-1223 https://dx.doi.org/10.1007/s00453-016-0125-z lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER SSG-OPC-MAT SSG-OPC-ASE GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_101 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_267 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 54.00 ASE AR 76 2016 4 01 02 1203-1223 |
| spelling |
10.1007/s00453-016-0125-z doi (DE-627)SPR006178251 (SPR)s00453-016-0125-z-e DE-627 ger DE-627 rakwb eng 004 ASE 510 004 ASE 54.00 bkl Kane, Daniel verfasserin aut A Short Implicant of a CNF Formula with Many Satisfying Assignments 2016 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract Consider any Boolean function %$F(X_1,\ldots ,X_N)%$ that has more than %$2^{-N^{\delta }}\cdot 2^N%$ satisfying assignments for some %$\delta %$, %$0<\delta <1%$, and that can be expressed by a CNF formula with at most %$N^d%$ clauses for some %$d>0%$. Then how many variables do we need to fix in order to satisfy F? We show that one can always find some “short” partial assignment on which F evaluates to 1 by fixing at most %$\alpha N%$ variables for some constant %$\alpha <1%$; that is, F has an implicant of size %$\le \alpha N%$. A lower bound for such %$\alpha %$ is also shown in terms of %$\delta %$ and d. We also discuss an algorithm for obtaining a short partial assignment. For any %$\delta %$ and %$\varepsilon %$ such that %$0<\delta +\varepsilon <1%$, we show a deterministic algorithm that finds a short partial assignment in %${\widetilde{O}}(2^{N^{\beta }})%$-time (By %${\widetilde{O}}(2^{N^{\beta }})%$ we mean %$O\bigl (2^{N^{\beta }}\cdot N^{O(1)}\bigr )%$.) for some %$\beta <1%$ and for any CNF formula with at most %$N^{1+\varepsilon }%$ clauses having more than %$2^{-N^{\delta }}\cdot 2^N%$ satisfying assignments. Short satisfying partial assignment (dpeaa)DE-He213 Conjunctive normal form formula (dpeaa)DE-He213 Large satisfying assignment ratio (dpeaa)DE-He213 Watanabe, Osamu verfasserin aut Enthalten in Algorithmica New York, NY : Springer, 1986 76(2016), 4 vom: 01. Feb., Seite 1203-1223 (DE-627)253389704 (DE-600)1458414-1 1432-0541 nnns volume:76 year:2016 number:4 day:01 month:02 pages:1203-1223 https://dx.doi.org/10.1007/s00453-016-0125-z lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER SSG-OPC-MAT SSG-OPC-ASE GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_101 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_267 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 54.00 ASE AR 76 2016 4 01 02 1203-1223 |
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10.1007/s00453-016-0125-z doi (DE-627)SPR006178251 (SPR)s00453-016-0125-z-e DE-627 ger DE-627 rakwb eng 004 ASE 510 004 ASE 54.00 bkl Kane, Daniel verfasserin aut A Short Implicant of a CNF Formula with Many Satisfying Assignments 2016 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract Consider any Boolean function %$F(X_1,\ldots ,X_N)%$ that has more than %$2^{-N^{\delta }}\cdot 2^N%$ satisfying assignments for some %$\delta %$, %$0<\delta <1%$, and that can be expressed by a CNF formula with at most %$N^d%$ clauses for some %$d>0%$. Then how many variables do we need to fix in order to satisfy F? We show that one can always find some “short” partial assignment on which F evaluates to 1 by fixing at most %$\alpha N%$ variables for some constant %$\alpha <1%$; that is, F has an implicant of size %$\le \alpha N%$. A lower bound for such %$\alpha %$ is also shown in terms of %$\delta %$ and d. We also discuss an algorithm for obtaining a short partial assignment. For any %$\delta %$ and %$\varepsilon %$ such that %$0<\delta +\varepsilon <1%$, we show a deterministic algorithm that finds a short partial assignment in %${\widetilde{O}}(2^{N^{\beta }})%$-time (By %${\widetilde{O}}(2^{N^{\beta }})%$ we mean %$O\bigl (2^{N^{\beta }}\cdot N^{O(1)}\bigr )%$.) for some %$\beta <1%$ and for any CNF formula with at most %$N^{1+\varepsilon }%$ clauses having more than %$2^{-N^{\delta }}\cdot 2^N%$ satisfying assignments. Short satisfying partial assignment (dpeaa)DE-He213 Conjunctive normal form formula (dpeaa)DE-He213 Large satisfying assignment ratio (dpeaa)DE-He213 Watanabe, Osamu verfasserin aut Enthalten in Algorithmica New York, NY : Springer, 1986 76(2016), 4 vom: 01. Feb., Seite 1203-1223 (DE-627)253389704 (DE-600)1458414-1 1432-0541 nnns volume:76 year:2016 number:4 day:01 month:02 pages:1203-1223 https://dx.doi.org/10.1007/s00453-016-0125-z lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER SSG-OPC-MAT SSG-OPC-ASE GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_101 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_267 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 54.00 ASE AR 76 2016 4 01 02 1203-1223 |
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10.1007/s00453-016-0125-z doi (DE-627)SPR006178251 (SPR)s00453-016-0125-z-e DE-627 ger DE-627 rakwb eng 004 ASE 510 004 ASE 54.00 bkl Kane, Daniel verfasserin aut A Short Implicant of a CNF Formula with Many Satisfying Assignments 2016 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract Consider any Boolean function %$F(X_1,\ldots ,X_N)%$ that has more than %$2^{-N^{\delta }}\cdot 2^N%$ satisfying assignments for some %$\delta %$, %$0<\delta <1%$, and that can be expressed by a CNF formula with at most %$N^d%$ clauses for some %$d>0%$. Then how many variables do we need to fix in order to satisfy F? We show that one can always find some “short” partial assignment on which F evaluates to 1 by fixing at most %$\alpha N%$ variables for some constant %$\alpha <1%$; that is, F has an implicant of size %$\le \alpha N%$. A lower bound for such %$\alpha %$ is also shown in terms of %$\delta %$ and d. We also discuss an algorithm for obtaining a short partial assignment. For any %$\delta %$ and %$\varepsilon %$ such that %$0<\delta +\varepsilon <1%$, we show a deterministic algorithm that finds a short partial assignment in %${\widetilde{O}}(2^{N^{\beta }})%$-time (By %${\widetilde{O}}(2^{N^{\beta }})%$ we mean %$O\bigl (2^{N^{\beta }}\cdot N^{O(1)}\bigr )%$.) for some %$\beta <1%$ and for any CNF formula with at most %$N^{1+\varepsilon }%$ clauses having more than %$2^{-N^{\delta }}\cdot 2^N%$ satisfying assignments. Short satisfying partial assignment (dpeaa)DE-He213 Conjunctive normal form formula (dpeaa)DE-He213 Large satisfying assignment ratio (dpeaa)DE-He213 Watanabe, Osamu verfasserin aut Enthalten in Algorithmica New York, NY : Springer, 1986 76(2016), 4 vom: 01. Feb., Seite 1203-1223 (DE-627)253389704 (DE-600)1458414-1 1432-0541 nnns volume:76 year:2016 number:4 day:01 month:02 pages:1203-1223 https://dx.doi.org/10.1007/s00453-016-0125-z lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER SSG-OPC-MAT SSG-OPC-ASE GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_101 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_267 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 54.00 ASE AR 76 2016 4 01 02 1203-1223 |
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10.1007/s00453-016-0125-z doi (DE-627)SPR006178251 (SPR)s00453-016-0125-z-e DE-627 ger DE-627 rakwb eng 004 ASE 510 004 ASE 54.00 bkl Kane, Daniel verfasserin aut A Short Implicant of a CNF Formula with Many Satisfying Assignments 2016 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract Consider any Boolean function %$F(X_1,\ldots ,X_N)%$ that has more than %$2^{-N^{\delta }}\cdot 2^N%$ satisfying assignments for some %$\delta %$, %$0<\delta <1%$, and that can be expressed by a CNF formula with at most %$N^d%$ clauses for some %$d>0%$. Then how many variables do we need to fix in order to satisfy F? We show that one can always find some “short” partial assignment on which F evaluates to 1 by fixing at most %$\alpha N%$ variables for some constant %$\alpha <1%$; that is, F has an implicant of size %$\le \alpha N%$. A lower bound for such %$\alpha %$ is also shown in terms of %$\delta %$ and d. We also discuss an algorithm for obtaining a short partial assignment. For any %$\delta %$ and %$\varepsilon %$ such that %$0<\delta +\varepsilon <1%$, we show a deterministic algorithm that finds a short partial assignment in %${\widetilde{O}}(2^{N^{\beta }})%$-time (By %${\widetilde{O}}(2^{N^{\beta }})%$ we mean %$O\bigl (2^{N^{\beta }}\cdot N^{O(1)}\bigr )%$.) for some %$\beta <1%$ and for any CNF formula with at most %$N^{1+\varepsilon }%$ clauses having more than %$2^{-N^{\delta }}\cdot 2^N%$ satisfying assignments. Short satisfying partial assignment (dpeaa)DE-He213 Conjunctive normal form formula (dpeaa)DE-He213 Large satisfying assignment ratio (dpeaa)DE-He213 Watanabe, Osamu verfasserin aut Enthalten in Algorithmica New York, NY : Springer, 1986 76(2016), 4 vom: 01. Feb., Seite 1203-1223 (DE-627)253389704 (DE-600)1458414-1 1432-0541 nnns volume:76 year:2016 number:4 day:01 month:02 pages:1203-1223 https://dx.doi.org/10.1007/s00453-016-0125-z lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER SSG-OPC-MAT SSG-OPC-ASE GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_101 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_267 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 54.00 ASE AR 76 2016 4 01 02 1203-1223 |
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English |
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Enthalten in Algorithmica 76(2016), 4 vom: 01. Feb., Seite 1203-1223 volume:76 year:2016 number:4 day:01 month:02 pages:1203-1223 |
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Enthalten in Algorithmica 76(2016), 4 vom: 01. Feb., Seite 1203-1223 volume:76 year:2016 number:4 day:01 month:02 pages:1203-1223 |
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findex.gbv.de |
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Short satisfying partial assignment Conjunctive normal form formula Large satisfying assignment ratio |
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Algorithmica |
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Kane, Daniel @@aut@@ Watanabe, Osamu @@aut@@ |
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<?xml version="1.0" encoding="UTF-8"?><collection xmlns="http://www.loc.gov/MARC21/slim"><record><leader>01000caa a22002652 4500</leader><controlfield tag="001">SPR006178251</controlfield><controlfield tag="003">DE-627</controlfield><controlfield tag="005">20220110185403.0</controlfield><controlfield tag="007">cr uuu---uuuuu</controlfield><controlfield tag="008">201002s2016 xx |||||o 00| ||eng c</controlfield><datafield tag="024" ind1="7" ind2=" "><subfield code="a">10.1007/s00453-016-0125-z</subfield><subfield code="2">doi</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(DE-627)SPR006178251</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(SPR)s00453-016-0125-z-e</subfield></datafield><datafield tag="040" ind1=" " ind2=" "><subfield code="a">DE-627</subfield><subfield code="b">ger</subfield><subfield code="c">DE-627</subfield><subfield code="e">rakwb</subfield></datafield><datafield tag="041" ind1=" " ind2=" "><subfield code="a">eng</subfield></datafield><datafield tag="082" ind1="0" ind2="4"><subfield code="a">004</subfield><subfield code="q">ASE</subfield></datafield><datafield tag="082" ind1="0" ind2="4"><subfield code="a">510</subfield><subfield code="a">004</subfield><subfield code="q">ASE</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">54.00</subfield><subfield code="2">bkl</subfield></datafield><datafield tag="100" ind1="1" ind2=" "><subfield code="a">Kane, Daniel</subfield><subfield code="e">verfasserin</subfield><subfield code="4">aut</subfield></datafield><datafield tag="245" ind1="1" ind2="2"><subfield code="a">A Short Implicant of a CNF Formula with Many Satisfying Assignments</subfield></datafield><datafield tag="264" ind1=" " ind2="1"><subfield code="c">2016</subfield></datafield><datafield tag="336" ind1=" " ind2=" "><subfield code="a">Text</subfield><subfield code="b">txt</subfield><subfield code="2">rdacontent</subfield></datafield><datafield tag="337" ind1=" " ind2=" "><subfield code="a">Computermedien</subfield><subfield code="b">c</subfield><subfield code="2">rdamedia</subfield></datafield><datafield tag="338" ind1=" " ind2=" "><subfield code="a">Online-Ressource</subfield><subfield code="b">cr</subfield><subfield code="2">rdacarrier</subfield></datafield><datafield tag="520" ind1=" " ind2=" "><subfield code="a">Abstract Consider any Boolean function %$F(X_1,\ldots ,X_N)%$ that has more than %$2^{-N^{\delta }}\cdot 2^N%$ satisfying assignments for some %$\delta %$, %$0<\delta <1%$, and that can be expressed by a CNF formula with at most %$N^d%$ clauses for some %$d>0%$. Then how many variables do we need to fix in order to satisfy F? We show that one can always find some “short” partial assignment on which F evaluates to 1 by fixing at most %$\alpha N%$ variables for some constant %$\alpha <1%$; that is, F has an implicant of size %$\le \alpha N%$. A lower bound for such %$\alpha %$ is also shown in terms of %$\delta %$ and d. We also discuss an algorithm for obtaining a short partial assignment. For any %$\delta %$ and %$\varepsilon %$ such that %$0<\delta +\varepsilon <1%$, we show a deterministic algorithm that finds a short partial assignment in %${\widetilde{O}}(2^{N^{\beta }})%$-time (By %${\widetilde{O}}(2^{N^{\beta }})%$ we mean %$O\bigl (2^{N^{\beta }}\cdot N^{O(1)}\bigr )%$.) for some %$\beta <1%$ and for any CNF formula with at most %$N^{1+\varepsilon }%$ clauses having more than %$2^{-N^{\delta }}\cdot 2^N%$ satisfying assignments.</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Short satisfying partial assignment</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Conjunctive normal form formula</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Large satisfying assignment ratio</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="700" ind1="1" ind2=" "><subfield code="a">Watanabe, Osamu</subfield><subfield code="e">verfasserin</subfield><subfield code="4">aut</subfield></datafield><datafield tag="773" ind1="0" ind2="8"><subfield code="i">Enthalten in</subfield><subfield code="t">Algorithmica</subfield><subfield code="d">New York, NY : Springer, 1986</subfield><subfield code="g">76(2016), 4 vom: 01. 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Kane, Daniel |
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short implicant of a cnf formula with many satisfying assignments |
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A Short Implicant of a CNF Formula with Many Satisfying Assignments |
| abstract |
Abstract Consider any Boolean function %$F(X_1,\ldots ,X_N)%$ that has more than %$2^{-N^{\delta }}\cdot 2^N%$ satisfying assignments for some %$\delta %$, %$0<\delta <1%$, and that can be expressed by a CNF formula with at most %$N^d%$ clauses for some %$d>0%$. Then how many variables do we need to fix in order to satisfy F? We show that one can always find some “short” partial assignment on which F evaluates to 1 by fixing at most %$\alpha N%$ variables for some constant %$\alpha <1%$; that is, F has an implicant of size %$\le \alpha N%$. A lower bound for such %$\alpha %$ is also shown in terms of %$\delta %$ and d. We also discuss an algorithm for obtaining a short partial assignment. For any %$\delta %$ and %$\varepsilon %$ such that %$0<\delta +\varepsilon <1%$, we show a deterministic algorithm that finds a short partial assignment in %${\widetilde{O}}(2^{N^{\beta }})%$-time (By %${\widetilde{O}}(2^{N^{\beta }})%$ we mean %$O\bigl (2^{N^{\beta }}\cdot N^{O(1)}\bigr )%$.) for some %$\beta <1%$ and for any CNF formula with at most %$N^{1+\varepsilon }%$ clauses having more than %$2^{-N^{\delta }}\cdot 2^N%$ satisfying assignments. |
| abstractGer |
Abstract Consider any Boolean function %$F(X_1,\ldots ,X_N)%$ that has more than %$2^{-N^{\delta }}\cdot 2^N%$ satisfying assignments for some %$\delta %$, %$0<\delta <1%$, and that can be expressed by a CNF formula with at most %$N^d%$ clauses for some %$d>0%$. Then how many variables do we need to fix in order to satisfy F? We show that one can always find some “short” partial assignment on which F evaluates to 1 by fixing at most %$\alpha N%$ variables for some constant %$\alpha <1%$; that is, F has an implicant of size %$\le \alpha N%$. A lower bound for such %$\alpha %$ is also shown in terms of %$\delta %$ and d. We also discuss an algorithm for obtaining a short partial assignment. For any %$\delta %$ and %$\varepsilon %$ such that %$0<\delta +\varepsilon <1%$, we show a deterministic algorithm that finds a short partial assignment in %${\widetilde{O}}(2^{N^{\beta }})%$-time (By %${\widetilde{O}}(2^{N^{\beta }})%$ we mean %$O\bigl (2^{N^{\beta }}\cdot N^{O(1)}\bigr )%$.) for some %$\beta <1%$ and for any CNF formula with at most %$N^{1+\varepsilon }%$ clauses having more than %$2^{-N^{\delta }}\cdot 2^N%$ satisfying assignments. |
| abstract_unstemmed |
Abstract Consider any Boolean function %$F(X_1,\ldots ,X_N)%$ that has more than %$2^{-N^{\delta }}\cdot 2^N%$ satisfying assignments for some %$\delta %$, %$0<\delta <1%$, and that can be expressed by a CNF formula with at most %$N^d%$ clauses for some %$d>0%$. Then how many variables do we need to fix in order to satisfy F? We show that one can always find some “short” partial assignment on which F evaluates to 1 by fixing at most %$\alpha N%$ variables for some constant %$\alpha <1%$; that is, F has an implicant of size %$\le \alpha N%$. A lower bound for such %$\alpha %$ is also shown in terms of %$\delta %$ and d. We also discuss an algorithm for obtaining a short partial assignment. For any %$\delta %$ and %$\varepsilon %$ such that %$0<\delta +\varepsilon <1%$, we show a deterministic algorithm that finds a short partial assignment in %${\widetilde{O}}(2^{N^{\beta }})%$-time (By %${\widetilde{O}}(2^{N^{\beta }})%$ we mean %$O\bigl (2^{N^{\beta }}\cdot N^{O(1)}\bigr )%$.) for some %$\beta <1%$ and for any CNF formula with at most %$N^{1+\varepsilon }%$ clauses having more than %$2^{-N^{\delta }}\cdot 2^N%$ satisfying assignments. |
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| container_issue |
4 |
| title_short |
A Short Implicant of a CNF Formula with Many Satisfying Assignments |
| url |
https://dx.doi.org/10.1007/s00453-016-0125-z |
| remote_bool |
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| author2 |
Watanabe, Osamu |
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| doi_str |
10.1007/s00453-016-0125-z |
| up_date |
2024-07-03T21:21:50.994Z |
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| score |
7.4010143 |