Undirected Determinant and Its Complexity
Abstract We view the determinant and permanent as functions on directed weighted graphs and introduce their analogues for undirected graphs. We prove that computing undirected determinants as well as permanents for planar graphs whose vertices have degree at most 4 is #P-complete. In the case of pla...
Ausführliche Beschreibung
Gespeichert in:
| Autor*in: |
Dziewa-Dawidczyk, Diana [verfasserIn] |
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| Format: |
E-Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2023 |
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| Schlagwörter: |
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| Anmerkung: |
© The Author(s) 2023 |
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| Übergeordnetes Werk: |
Enthalten in: Graphs and combinatorics - Tokyo : Springer-Verl. Tokyo, 1985, 39(2023), 4 vom: 06. Juli |
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| Übergeordnetes Werk: |
volume:39 ; year:2023 ; number:4 ; day:06 ; month:07 |
| Links: |
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| DOI / URN: |
10.1007/s00373-023-02671-7 |
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| Katalog-ID: |
SPR052166309 |
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| 100 | 1 | |a Dziewa-Dawidczyk, Diana |e verfasserin |0 (orcid)0000-0001-9486-1685 |4 aut | |
| 245 | 1 | 0 | |a Undirected Determinant and Its Complexity |
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| 520 | |a Abstract We view the determinant and permanent as functions on directed weighted graphs and introduce their analogues for undirected graphs. We prove that computing undirected determinants as well as permanents for planar graphs whose vertices have degree at most 4 is #P-complete. In the case of planar graphs whose vertices have degree at most 3, the computation of the undirected determinant remains #P-complete while computing the permanent can be reduced to the FKT algorithm, and therefore can be done in polynomial time. Computing the undirected permanent is a Holant problem and its complexity can be deduced from the existing literature. It is mentioned in the paper as a natural context but no new results in this direction are obtained. The concept of undirected determinant is new. Its introduction is motivated by the formal resemblance to the directed determinant, a property that may inspire generalizations of some of the many algorithms which compute the latter. For a sizable class of planar 3-regular graphs, we are able to compute the undirected determinant in polynomial time. | ||
| 650 | 4 | |a Computational complexity |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a Enumerative combinatorics |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a Planar graphs |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a Determinant |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a Permanent |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a Pfaffian orientation |7 (dpeaa)DE-He213 | |
| 700 | 1 | |a Przeździecki, Adam J. |0 (orcid)0000-0002-5414-6828 |4 aut | |
| 773 | 0 | 8 | |i Enthalten in |t Graphs and combinatorics |d Tokyo : Springer-Verl. Tokyo, 1985 |g 39(2023), 4 vom: 06. Juli |w (DE-627)30018381X |w (DE-600)1481435-3 |x 1435-5914 |7 nnns |
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| 856 | 4 | 0 | |u https://dx.doi.org/10.1007/s00373-023-02671-7 |z kostenfrei |3 Volltext |
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10.1007/s00373-023-02671-7 doi (DE-627)SPR052166309 (SPR)s00373-023-02671-7-e DE-627 ger DE-627 rakwb eng Dziewa-Dawidczyk, Diana verfasserin (orcid)0000-0001-9486-1685 aut Undirected Determinant and Its Complexity 2023 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s) 2023 Abstract We view the determinant and permanent as functions on directed weighted graphs and introduce their analogues for undirected graphs. We prove that computing undirected determinants as well as permanents for planar graphs whose vertices have degree at most 4 is #P-complete. In the case of planar graphs whose vertices have degree at most 3, the computation of the undirected determinant remains #P-complete while computing the permanent can be reduced to the FKT algorithm, and therefore can be done in polynomial time. Computing the undirected permanent is a Holant problem and its complexity can be deduced from the existing literature. It is mentioned in the paper as a natural context but no new results in this direction are obtained. The concept of undirected determinant is new. Its introduction is motivated by the formal resemblance to the directed determinant, a property that may inspire generalizations of some of the many algorithms which compute the latter. For a sizable class of planar 3-regular graphs, we are able to compute the undirected determinant in polynomial time. Computational complexity (dpeaa)DE-He213 Enumerative combinatorics (dpeaa)DE-He213 Planar graphs (dpeaa)DE-He213 Determinant (dpeaa)DE-He213 Permanent (dpeaa)DE-He213 Pfaffian orientation (dpeaa)DE-He213 Przeździecki, Adam J. (orcid)0000-0002-5414-6828 aut Enthalten in Graphs and combinatorics Tokyo : Springer-Verl. Tokyo, 1985 39(2023), 4 vom: 06. Juli (DE-627)30018381X (DE-600)1481435-3 1435-5914 nnns volume:39 year:2023 number:4 day:06 month:07 https://dx.doi.org/10.1007/s00373-023-02671-7 kostenfrei Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER 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_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 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_206 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 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_2008 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_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 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_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_4126 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 AR 39 2023 4 06 07 |
| spelling |
10.1007/s00373-023-02671-7 doi (DE-627)SPR052166309 (SPR)s00373-023-02671-7-e DE-627 ger DE-627 rakwb eng Dziewa-Dawidczyk, Diana verfasserin (orcid)0000-0001-9486-1685 aut Undirected Determinant and Its Complexity 2023 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s) 2023 Abstract We view the determinant and permanent as functions on directed weighted graphs and introduce their analogues for undirected graphs. We prove that computing undirected determinants as well as permanents for planar graphs whose vertices have degree at most 4 is #P-complete. In the case of planar graphs whose vertices have degree at most 3, the computation of the undirected determinant remains #P-complete while computing the permanent can be reduced to the FKT algorithm, and therefore can be done in polynomial time. Computing the undirected permanent is a Holant problem and its complexity can be deduced from the existing literature. It is mentioned in the paper as a natural context but no new results in this direction are obtained. The concept of undirected determinant is new. Its introduction is motivated by the formal resemblance to the directed determinant, a property that may inspire generalizations of some of the many algorithms which compute the latter. For a sizable class of planar 3-regular graphs, we are able to compute the undirected determinant in polynomial time. Computational complexity (dpeaa)DE-He213 Enumerative combinatorics (dpeaa)DE-He213 Planar graphs (dpeaa)DE-He213 Determinant (dpeaa)DE-He213 Permanent (dpeaa)DE-He213 Pfaffian orientation (dpeaa)DE-He213 Przeździecki, Adam J. (orcid)0000-0002-5414-6828 aut Enthalten in Graphs and combinatorics Tokyo : Springer-Verl. Tokyo, 1985 39(2023), 4 vom: 06. Juli (DE-627)30018381X (DE-600)1481435-3 1435-5914 nnns volume:39 year:2023 number:4 day:06 month:07 https://dx.doi.org/10.1007/s00373-023-02671-7 kostenfrei Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER 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_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 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_206 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 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_2008 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_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 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_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_4126 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 AR 39 2023 4 06 07 |
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10.1007/s00373-023-02671-7 doi (DE-627)SPR052166309 (SPR)s00373-023-02671-7-e DE-627 ger DE-627 rakwb eng Dziewa-Dawidczyk, Diana verfasserin (orcid)0000-0001-9486-1685 aut Undirected Determinant and Its Complexity 2023 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s) 2023 Abstract We view the determinant and permanent as functions on directed weighted graphs and introduce their analogues for undirected graphs. We prove that computing undirected determinants as well as permanents for planar graphs whose vertices have degree at most 4 is #P-complete. In the case of planar graphs whose vertices have degree at most 3, the computation of the undirected determinant remains #P-complete while computing the permanent can be reduced to the FKT algorithm, and therefore can be done in polynomial time. Computing the undirected permanent is a Holant problem and its complexity can be deduced from the existing literature. It is mentioned in the paper as a natural context but no new results in this direction are obtained. The concept of undirected determinant is new. Its introduction is motivated by the formal resemblance to the directed determinant, a property that may inspire generalizations of some of the many algorithms which compute the latter. For a sizable class of planar 3-regular graphs, we are able to compute the undirected determinant in polynomial time. Computational complexity (dpeaa)DE-He213 Enumerative combinatorics (dpeaa)DE-He213 Planar graphs (dpeaa)DE-He213 Determinant (dpeaa)DE-He213 Permanent (dpeaa)DE-He213 Pfaffian orientation (dpeaa)DE-He213 Przeździecki, Adam J. (orcid)0000-0002-5414-6828 aut Enthalten in Graphs and combinatorics Tokyo : Springer-Verl. Tokyo, 1985 39(2023), 4 vom: 06. Juli (DE-627)30018381X (DE-600)1481435-3 1435-5914 nnns volume:39 year:2023 number:4 day:06 month:07 https://dx.doi.org/10.1007/s00373-023-02671-7 kostenfrei Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER 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_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 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_206 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 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_2008 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_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 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_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_4126 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 AR 39 2023 4 06 07 |
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10.1007/s00373-023-02671-7 doi (DE-627)SPR052166309 (SPR)s00373-023-02671-7-e DE-627 ger DE-627 rakwb eng Dziewa-Dawidczyk, Diana verfasserin (orcid)0000-0001-9486-1685 aut Undirected Determinant and Its Complexity 2023 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s) 2023 Abstract We view the determinant and permanent as functions on directed weighted graphs and introduce their analogues for undirected graphs. We prove that computing undirected determinants as well as permanents for planar graphs whose vertices have degree at most 4 is #P-complete. In the case of planar graphs whose vertices have degree at most 3, the computation of the undirected determinant remains #P-complete while computing the permanent can be reduced to the FKT algorithm, and therefore can be done in polynomial time. Computing the undirected permanent is a Holant problem and its complexity can be deduced from the existing literature. It is mentioned in the paper as a natural context but no new results in this direction are obtained. The concept of undirected determinant is new. Its introduction is motivated by the formal resemblance to the directed determinant, a property that may inspire generalizations of some of the many algorithms which compute the latter. For a sizable class of planar 3-regular graphs, we are able to compute the undirected determinant in polynomial time. Computational complexity (dpeaa)DE-He213 Enumerative combinatorics (dpeaa)DE-He213 Planar graphs (dpeaa)DE-He213 Determinant (dpeaa)DE-He213 Permanent (dpeaa)DE-He213 Pfaffian orientation (dpeaa)DE-He213 Przeździecki, Adam J. (orcid)0000-0002-5414-6828 aut Enthalten in Graphs and combinatorics Tokyo : Springer-Verl. Tokyo, 1985 39(2023), 4 vom: 06. Juli (DE-627)30018381X (DE-600)1481435-3 1435-5914 nnns volume:39 year:2023 number:4 day:06 month:07 https://dx.doi.org/10.1007/s00373-023-02671-7 kostenfrei Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER 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_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 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_206 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 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_2008 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_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 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_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_4126 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 AR 39 2023 4 06 07 |
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10.1007/s00373-023-02671-7 doi (DE-627)SPR052166309 (SPR)s00373-023-02671-7-e DE-627 ger DE-627 rakwb eng Dziewa-Dawidczyk, Diana verfasserin (orcid)0000-0001-9486-1685 aut Undirected Determinant and Its Complexity 2023 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s) 2023 Abstract We view the determinant and permanent as functions on directed weighted graphs and introduce their analogues for undirected graphs. We prove that computing undirected determinants as well as permanents for planar graphs whose vertices have degree at most 4 is #P-complete. In the case of planar graphs whose vertices have degree at most 3, the computation of the undirected determinant remains #P-complete while computing the permanent can be reduced to the FKT algorithm, and therefore can be done in polynomial time. Computing the undirected permanent is a Holant problem and its complexity can be deduced from the existing literature. It is mentioned in the paper as a natural context but no new results in this direction are obtained. The concept of undirected determinant is new. Its introduction is motivated by the formal resemblance to the directed determinant, a property that may inspire generalizations of some of the many algorithms which compute the latter. For a sizable class of planar 3-regular graphs, we are able to compute the undirected determinant in polynomial time. Computational complexity (dpeaa)DE-He213 Enumerative combinatorics (dpeaa)DE-He213 Planar graphs (dpeaa)DE-He213 Determinant (dpeaa)DE-He213 Permanent (dpeaa)DE-He213 Pfaffian orientation (dpeaa)DE-He213 Przeździecki, Adam J. (orcid)0000-0002-5414-6828 aut Enthalten in Graphs and combinatorics Tokyo : Springer-Verl. Tokyo, 1985 39(2023), 4 vom: 06. Juli (DE-627)30018381X (DE-600)1481435-3 1435-5914 nnns volume:39 year:2023 number:4 day:06 month:07 https://dx.doi.org/10.1007/s00373-023-02671-7 kostenfrei Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER 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_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 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_206 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 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_2008 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_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 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_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_4126 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 AR 39 2023 4 06 07 |
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Dziewa-Dawidczyk, Diana |
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Dziewa-Dawidczyk, Diana misc Computational complexity misc Enumerative combinatorics misc Planar graphs misc Determinant misc Permanent misc Pfaffian orientation Undirected Determinant and Its Complexity |
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Undirected Determinant and Its Complexity Computational complexity (dpeaa)DE-He213 Enumerative combinatorics (dpeaa)DE-He213 Planar graphs (dpeaa)DE-He213 Determinant (dpeaa)DE-He213 Permanent (dpeaa)DE-He213 Pfaffian orientation (dpeaa)DE-He213 |
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Elektronische Aufsätze Aufsätze Elektronische Ressource |
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Undirected Determinant and Its Complexity |
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Undirected Determinant and Its Complexity |
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Dziewa-Dawidczyk, Diana |
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undirected determinant and its complexity |
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Undirected Determinant and Its Complexity |
| abstract |
Abstract We view the determinant and permanent as functions on directed weighted graphs and introduce their analogues for undirected graphs. We prove that computing undirected determinants as well as permanents for planar graphs whose vertices have degree at most 4 is #P-complete. In the case of planar graphs whose vertices have degree at most 3, the computation of the undirected determinant remains #P-complete while computing the permanent can be reduced to the FKT algorithm, and therefore can be done in polynomial time. Computing the undirected permanent is a Holant problem and its complexity can be deduced from the existing literature. It is mentioned in the paper as a natural context but no new results in this direction are obtained. The concept of undirected determinant is new. Its introduction is motivated by the formal resemblance to the directed determinant, a property that may inspire generalizations of some of the many algorithms which compute the latter. For a sizable class of planar 3-regular graphs, we are able to compute the undirected determinant in polynomial time. © The Author(s) 2023 |
| abstractGer |
Abstract We view the determinant and permanent as functions on directed weighted graphs and introduce their analogues for undirected graphs. We prove that computing undirected determinants as well as permanents for planar graphs whose vertices have degree at most 4 is #P-complete. In the case of planar graphs whose vertices have degree at most 3, the computation of the undirected determinant remains #P-complete while computing the permanent can be reduced to the FKT algorithm, and therefore can be done in polynomial time. Computing the undirected permanent is a Holant problem and its complexity can be deduced from the existing literature. It is mentioned in the paper as a natural context but no new results in this direction are obtained. The concept of undirected determinant is new. Its introduction is motivated by the formal resemblance to the directed determinant, a property that may inspire generalizations of some of the many algorithms which compute the latter. For a sizable class of planar 3-regular graphs, we are able to compute the undirected determinant in polynomial time. © The Author(s) 2023 |
| abstract_unstemmed |
Abstract We view the determinant and permanent as functions on directed weighted graphs and introduce their analogues for undirected graphs. We prove that computing undirected determinants as well as permanents for planar graphs whose vertices have degree at most 4 is #P-complete. In the case of planar graphs whose vertices have degree at most 3, the computation of the undirected determinant remains #P-complete while computing the permanent can be reduced to the FKT algorithm, and therefore can be done in polynomial time. Computing the undirected permanent is a Holant problem and its complexity can be deduced from the existing literature. It is mentioned in the paper as a natural context but no new results in this direction are obtained. The concept of undirected determinant is new. Its introduction is motivated by the formal resemblance to the directed determinant, a property that may inspire generalizations of some of the many algorithms which compute the latter. For a sizable class of planar 3-regular graphs, we are able to compute the undirected determinant in polynomial time. © The Author(s) 2023 |
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Undirected Determinant and Its Complexity |
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https://dx.doi.org/10.1007/s00373-023-02671-7 |
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Przeździecki, Adam J. |
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| doi_str |
10.1007/s00373-023-02671-7 |
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2024-07-04T01:35:44.216Z |
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