eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Dagstuhl Seminar Proceedings
1862-4405
2011-01-27
10441
1
22
10.4230/DagSemProc.10441.1
article
10441 Abstracts Collection – Exact Complexity of NP-hard Problems
Husfeldt, Thore
Kratsch, Dieter
Paturi, Ramamohan
Sorkin, Gregory B.
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.
https://drops.dagstuhl.de/storage/16dagstuhl-seminar-proceedings/dsp-vol10441/DagSemProc.10441.1/DagSemProc.10441.1.pdf
Complexity
Algorithms
NP-hard Problems
Exponential Time
SAT
Graphs
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Dagstuhl Seminar Proceedings
1862-4405
2011-01-27
10441
1
14
10.4230/DagSemProc.10441.2
article
Listing all maximal cliques in sparse graphs in near-optimal time
Eppstein, David
Löffler, Maarten
Strash, Darren
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 $n-d=Omega(n)$.
https://drops.dagstuhl.de/storage/16dagstuhl-seminar-proceedings/dsp-vol10441/DagSemProc.10441.2/DagSemProc.10441.2.pdf
Clique
backtracking
degeneracy
worst-case optimality