Dagstuhl Seminar Proceedings, Volume 10441
Dagstuhl Seminar Proceedings
DagSemProc
https://www.dagstuhl.de/dagpub/1862-4405
https://dblp.org/db/series/dagstuhl
1862-4405
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
10441
2011
https://drops.dagstuhl.de/entities/volume/DagSemProc-volume-10441
10441 Abstracts Collection – Exact Complexity of NP-hard Problems
A decade before NP-completeness became the
lens through which Computer Science views computationally hard
problems, beautiful algorithms were discovered that are much better
than exhaustive search, for example
Bellman's 1962 dynamic programming treatment of the Traveling Salesman problem
and Ryser's 1963 inclusion--exclusion formula for the permanent.
Complexity
Algorithms
NP-hard Problems
Exponential Time
SAT
Graphs
1-22
Regular Paper
Thore
Husfeldt
Thore Husfeldt
Dieter
Kratsch
Dieter Kratsch
Ramamohan
Paturi
Ramamohan Paturi
Gregory B.
Sorkin
Gregory B. Sorkin
10.4230/DagSemProc.10441.1
Creative Commons Attribution 4.0 International license
https://creativecommons.org/licenses/by/4.0/legalcode
Listing all maximal cliques in sparse graphs in near-optimal time
The degeneracy of an $n$-vertex graph $G$ is the smallest number $d$ such that every subgraph of $G$ contains a vertex of degree at most $d$. We show that there exists a nearly-optimal fixed-parameter tractable algorithm for enumerating all maximal cliques, parametrized by degeneracy. To achieve this result, we modify the classic Bron--Kerbosch algorithm and show that it runs in time $O(dn3^{d/3})$. We also provide matching upper and lower bounds showing that the largest possible number of maximal cliques in an $n$-vertex graph with degeneracy $d$ (when $d$ is a multiple of 3 and $nge d+3$) is $(n-d)3^{d/3}$. Therefore, our algorithm matches the $Theta(d(n-d)3^{d/3})$ worst-case output size of the problem whenever $n-d=Omega(n)$.
Clique
backtracking
degeneracy
worst-case optimality
1-14
Regular Paper
David
Eppstein
David Eppstein
Maarten
Löffler
Maarten Löffler
Darren
Strash
Darren Strash
10.4230/DagSemProc.10441.2
Creative Commons Attribution 4.0 International license
https://creativecommons.org/licenses/by/4.0/legalcode