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The Complexity of Finding Dense Subgraphs in Graphs with Large Cliques

dc.contributor.authorSeth, Cameron
dc.date.accessioned2021-08-25T16:14:56Z
dc.date.available2021-08-25T16:14:56Z
dc.date.issued2021-08-25
dc.date.submitted2021-08-15
dc.description.abstractThe GapDensest-k-Subgraph(d) problem (GapDkS(d)) is defined as follows: given a graph G and parameters k,d, distinguish between the case that G contains a k-clique, and the case that every k-subgraph of G has density at most d. GapDkS(d) is a natural relaxation of the standard Clique problem, which is known to be NP-complete. For d very close to 1, the GapDkS(d) problem is equivalent to the Clique problem, and when d is very close to 0 the GapDkS(d) problem can easily be solved in polynomial time. However, despite much work on both the algorithmic and hardness front, the exact k and d parameter values for which GapDkS(d) can be solved in polynomial time are still unknown. In particular, the best polynomial-time algorithms can solve GapDkS(d) when d is an inverse polynomial in the number of vertices n, but there have been no NP-hardness results beyond the trivial result. This thesis attempts to understand the GapDkS(d) problem better by studying the case when k is restricted to be linear in n (where n is the number of vertices in G). In particular, we survey the GapDkS(d) algorithms and hardness results that can be best applied to this restriction in an attempt to determine the threshold for when the problem becomes NP-hard. With some modifications to the algorithms and proofs, we produce algorithms and hardness results for the GapDkS(d) problem with k linear in n. In addition, we study the connection between GapDkS(d) and MaxClique, and show that despite having strong hardness results for MaxClique, reductions from MaxClique do not give strong hardness bounds for GapDkS(d).en
dc.identifier.urihttp://hdl.handle.net/10012/17249
dc.language.isoenen
dc.pendingfalse
dc.publisherUniversity of Waterlooen
dc.subjectapproximation algorithmsen
dc.subjectNP Hardnessen
dc.subjectdensest k-subgraphen
dc.titleThe Complexity of Finding Dense Subgraphs in Graphs with Large Cliquesen
dc.typeMaster Thesisen
uws-etd.degreeMaster of Mathematicsen
uws-etd.degree.departmentDavid R. Cheriton School of Computer Scienceen
uws-etd.degree.disciplineComputer Scienceen
uws-etd.degree.grantorUniversity of Waterlooen
uws-etd.embargo.terms0en
uws.contributor.advisorBlais, Eric
uws.contributor.affiliation1Faculty of Mathematicsen
uws.peerReviewStatusUnrevieweden
uws.published.cityWaterlooen
uws.published.countryCanadaen
uws.published.provinceOntarioen
uws.scholarLevelGraduateen
uws.typeOfResourceTexten

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