Inapproximability of Unique Games in Fixed-Point Logic with Counting
We study the extent to which it is possible to approximate the optimal value of a Unique Games instance in Fixed-Point Logic with Counting (FPC). Formally, we prove lower bounds against the accuracy of FPC-interpretations that map Unique Games instances (encoded as relational structures) to rational...
Uložené v:
| Vydané v: | Logical methods in computer science Ročník 20, Issue 2 |
|---|---|
| Hlavný autor: | |
| Médium: | Journal Article |
| Jazyk: | English |
| Vydavateľské údaje: |
Logical Methods in Computer Science e.V
10.04.2024
|
| Predmet: | |
| ISSN: | 1860-5974, 1860-5974 |
| On-line prístup: | Získať plný text |
| Tagy: |
Pridať tag
Žiadne tagy, Buďte prvý, kto otaguje tento záznam!
|
| Abstract | We study the extent to which it is possible to approximate the optimal value
of a Unique Games instance in Fixed-Point Logic with Counting (FPC). Formally,
we prove lower bounds against the accuracy of FPC-interpretations that map
Unique Games instances (encoded as relational structures) to rational numbers
giving the approximate fraction of constraints that can be satisfied. We prove
two new FPC-inexpressibility results for Unique Games: the existence of a
$(1/2, 1/3 + \delta)$-inapproximability gap, and inapproximability to within
any constant factor. Previous recent work has established similar
FPC-inapproximability results for a small handful of other problems. Our
construction builds upon some of these ideas, but contains a novel technique.
While most FPC-inexpressibility results are based on variants of the
CFI-construction, ours is significantly different. We start with a graph of
very large girth and label the edges with random affine vector spaces over
$\mathbb{F}_2$ that determine the constraints in the two structures.
Duplicator's strategy involves maintaining a partial isomorphism over a minimal
tree that spans the pebbled vertices of the graph. |
|---|---|
| AbstractList | We study the extent to which it is possible to approximate the optimal value of a Unique Games instance in Fixed-Point Logic with Counting (FPC). Formally, we prove lower bounds against the accuracy of FPC-interpretations that map Unique Games instances (encoded as relational structures) to rational numbers giving the approximate fraction of constraints that can be satisfied. We prove two new FPC-inexpressibility results for Unique Games: the existence of a $(1/2, 1/3 + \delta)$-inapproximability gap, and inapproximability to within any constant factor. Previous recent work has established similar FPC-inapproximability results for a small handful of other problems. Our construction builds upon some of these ideas, but contains a novel technique. While most FPC-inexpressibility results are based on variants of the CFI-construction, ours is significantly different. We start with a graph of very large girth and label the edges with random affine vector spaces over $\mathbb{F}_2$ that determine the constraints in the two structures. Duplicator's strategy involves maintaining a partial isomorphism over a minimal tree that spans the pebbled vertices of the graph. We study the extent to which it is possible to approximate the optimal value of a Unique Games instance in Fixed-Point Logic with Counting (FPC). Formally, we prove lower bounds against the accuracy of FPC-interpretations that map Unique Games instances (encoded as relational structures) to rational numbers giving the approximate fraction of constraints that can be satisfied. We prove two new FPC-inexpressibility results for Unique Games: the existence of a $(1/2, 1/3 + \delta)$-inapproximability gap, and inapproximability to within any constant factor. Previous recent work has established similar FPC-inapproximability results for a small handful of other problems. Our construction builds upon some of these ideas, but contains a novel technique. While most FPC-inexpressibility results are based on variants of the CFI-construction, ours is significantly different. We start with a graph of very large girth and label the edges with random affine vector spaces over $\mathbb{F}_2$ that determine the constraints in the two structures. Duplicator's strategy involves maintaining a partial isomorphism over a minimal tree that spans the pebbled vertices of the graph. |
| Author | Tucker-Foltz, Jamie |
| Author_xml | – sequence: 1 givenname: Jamie surname: Tucker-Foltz fullname: Tucker-Foltz, Jamie |
| BookMark | eNpNkE1LAzEQhoNUsGp_gLcc9bCaTD52402qrYWCHuw5ZLNJTdkmdXeL7b9324o4l3l5YR6G5xINYooOoRtK7rkEVTzUa9tmQG7hkd0BAX6GhrSQJBMq54N_-QKN2nZF-mGMFiCH6HkWzWbTpF1YmzLUodvj5PEihq-tw1Ozdi0OEU_CzlXZewqxw_O0DBZ_h-4Tj9M2diEur9G5N3XrRr_7Ci0mLx_j12z-Np2Nn-aZBaA8g9xQZxilUBFqhM8LSWXpHDUlkUpx5ismwOYlURXxoj9SvpB5KUXJiQDDrtDsxK2SWelN0__c7HUyQR-L1Cy1abpga6e5dNyoygEVBRfKGun7wIRnObGmEj2Lnli2SW3bOP_Ho0QfreqDVQ1Eg2b6YJX9AMuMa-c |
| ContentType | Journal Article |
| DBID | AAYXX CITATION DOA |
| DOI | 10.46298/lmcs-20(2:3)2024 |
| DatabaseName | CrossRef DOAJ Directory of Open Access Journals |
| DatabaseTitle | CrossRef |
| DatabaseTitleList | CrossRef |
| Database_xml | – sequence: 1 dbid: DOA name: DOAJ Directory of Open Access Journals url: https://www.doaj.org/ sourceTypes: Open Website |
| DeliveryMethod | fulltext_linktorsrc |
| Discipline | Computer Science |
| EISSN | 1860-5974 |
| ExternalDocumentID | oai_doaj_org_article_46e4a9de2158459ca6f58435f370cad5 10_46298_lmcs_20_2_3_2024 |
| GroupedDBID | .4S .DC 29L 2WC 5GY 5VS AAFWJ AAYXX ADBBV ADMLS ADQAK AENEX AFPKN ALMA_UNASSIGNED_HOLDINGS ARCSS BCNDV CITATION EBS EJD FRP GROUPED_DOAJ J9A KQ8 MK~ ML~ M~E OK1 OVT P2P TR2 TUS XSB |
| ID | FETCH-LOGICAL-c2214-27a1ea3112d01a5f78616bee1ab069943fd352c7b09d0f5c229f867b65b4052a3 |
| IEDL.DBID | DOA |
| ISSN | 1860-5974 |
| IngestDate | Fri Oct 03 12:53:36 EDT 2025 Sat Nov 29 06:21:52 EST 2025 |
| IsDoiOpenAccess | true |
| IsOpenAccess | true |
| IsPeerReviewed | true |
| IsScholarly | true |
| Language | English |
| LinkModel | DirectLink |
| MergedId | FETCHMERGED-LOGICAL-c2214-27a1ea3112d01a5f78616bee1ab069943fd352c7b09d0f5c229f867b65b4052a3 |
| ORCID | 0000-0001-9174-3341 |
| OpenAccessLink | https://doaj.org/article/46e4a9de2158459ca6f58435f370cad5 |
| ParticipantIDs | doaj_primary_oai_doaj_org_article_46e4a9de2158459ca6f58435f370cad5 crossref_primary_10_46298_lmcs_20_2_3_2024 |
| PublicationCentury | 2000 |
| PublicationDate | 2024-04-10 |
| PublicationDateYYYYMMDD | 2024-04-10 |
| PublicationDate_xml | – month: 04 year: 2024 text: 2024-04-10 day: 10 |
| PublicationDecade | 2020 |
| PublicationTitle | Logical methods in computer science |
| PublicationYear | 2024 |
| Publisher | Logical Methods in Computer Science e.V |
| Publisher_xml | – name: Logical Methods in Computer Science e.V |
| SSID | ssj0000331826 |
| Score | 2.30786 |
| Snippet | We study the extent to which it is possible to approximate the optimal value
of a Unique Games instance in Fixed-Point Logic with Counting (FPC). Formally,
we... We study the extent to which it is possible to approximate the optimal value of a Unique Games instance in Fixed-Point Logic with Counting (FPC). Formally, we... |
| SourceID | doaj crossref |
| SourceType | Open Website Index Database |
| SubjectTerms | computer science - logic in computer science |
| Title | Inapproximability of Unique Games in Fixed-Point Logic with Counting |
| URI | https://doaj.org/article/46e4a9de2158459ca6f58435f370cad5 |
| Volume | 20, Issue 2 |
| hasFullText | 1 |
| inHoldings | 1 |
| isFullTextHit | |
| isPrint | |
| journalDatabaseRights | – providerCode: PRVAON databaseName: DOAJ Directory of Open Access Journals customDbUrl: eissn: 1860-5974 dateEnd: 99991231 omitProxy: false ssIdentifier: ssj0000331826 issn: 1860-5974 databaseCode: DOA dateStart: 20040101 isFulltext: true titleUrlDefault: https://www.doaj.org/ providerName: Directory of Open Access Journals – providerCode: PRVHPJ databaseName: ROAD: Directory of Open Access Scholarly Resources customDbUrl: eissn: 1860-5974 dateEnd: 99991231 omitProxy: false ssIdentifier: ssj0000331826 issn: 1860-5974 databaseCode: M~E dateStart: 20040101 isFulltext: true titleUrlDefault: https://road.issn.org providerName: ISSN International Centre |
| link | http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwrV1LSwMxEA5SPHjxLdYXOXhQYTGvzW68aW1V0NKDSm8hmwcs2K20VerF326SXaWevHhZwpIN4Zts5ptk-AaAY81TxbnBSSaMS0IeYZJrhBLlAzeOlbEqnuk-32f9fj4cisFCqa-QE1bLA9fAnTNumRLGeteUs1RoxZ1v0NTRDGllonqpZz0LwVTcgykNxLm-xmSciPz8ZaSnfk2ckAt66iN-9ssRLej1R8fSWwerDSOEl_VMNsCSrTbB2ne1Bdj8fFvg-q6KAuDzclRra3_AsYNPUYAV3oRkV1hWsFfOrUkG47KawVBHWcNw0go7TUmIbfDU6z52bpOmBkKiCcEsIZnCVlHPigzCKnVZzjEvrMWqQFwIRp3xFEpnBRIGudR_JFzOs4KnhadiRNEd0KrGld0F0BMt6rc2wjwpYLlxwlHjjNJCk6LArGiDs29A5GstdSF9iBDRkwE9SZAkksqAXhtcBch-OgaV6vjC2042tpN_2W7vPwbZBythQuGGB6MD0JpN3uwhWNbvs3I6OYrLwj8fPrtfB6W8jw |
| linkProvider | Directory of Open Access Journals |
| openUrl | ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Inapproximability+of+Unique+Games+in+Fixed-Point+Logic+with+Counting&rft.jtitle=Logical+methods+in+computer+science&rft.au=Jamie+Tucker-Foltz&rft.date=2024-04-10&rft.pub=Logical+Methods+in+Computer+Science+e.V&rft.eissn=1860-5974&rft.volume=20%2C+Issue+2&rft_id=info:doi/10.46298%2Flmcs-20%282%3A3%292024&rft.externalDBID=DOA&rft.externalDocID=oai_doaj_org_article_46e4a9de2158459ca6f58435f370cad5 |
| thumbnail_l | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=1860-5974&client=summon |
| thumbnail_m | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=1860-5974&client=summon |
| thumbnail_s | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=1860-5974&client=summon |