Discrete Mathematics
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
Exact algorithms for NP-hard problems: a survey
Combinatorial optimization - Eureka, you shrink!
Measure and conquer: a simple O(20.288n) independent set algorithm
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Efficient Exact Algorithms through Enumerating Maximal Independent Sets and Other Techniques
Theory of Computing Systems
Computational complexity and the classification of finite simple groups
SFCS '83 Proceedings of the 24th Annual Symposium on Foundations of Computer Science
Exact computation of maximum induced forest
SWAT'06 Proceedings of the 10th Scandinavian conference on Algorithm Theory
On the complexity of uniformly mixed nash equilibria and related regular subgraph problems
FCT'05 Proceedings of the 15th international conference on Fundamentals of Computation Theory
Finding a minimum feedback vertex set in time O(1.7548n)
IWPEC'06 Proceedings of the Second international conference on Parameterized and Exact Computation
Enumerating maximal independent sets with applications to graph colouring
Operations Research Letters
The complexity of uniform Nash equilibria and related regular subgraph problems
Theoretical Computer Science
A measure & conquer approach for the analysis of exact algorithms
Journal of the ACM (JACM)
Feedback vertex sets in tournaments
ESA'10 Proceedings of the 18th annual European conference on Algorithms: Part I
Covering and packing in linear space
Information Processing Letters
Exact algorithm for the maximum induced planar subgraph problem
ESA'11 Proceedings of the 19th European conference on Algorithms
Maximum regular induced subgraphs in 2P3-free graphs
Theoretical Computer Science
Feedback Vertex Sets in Tournaments
Journal of Graph Theory
Branch and recharge: exact algorithms for generalized domination
WADS'07 Proceedings of the 10th international conference on Algorithms and Data Structures
Finding a maximum induced degenerate subgraph faster than 2n
IPEC'12 Proceedings of the 7th international conference on Parameterized and Exact Computation
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Given a graph G = (V,E) on n vertices, the Maximum γ-Regular Induced Subgraph (M-γ-RIS) problems ask for a maximum sized subset of vertices R⊆V such that the induced subgraph on R, G[R], is γ-regular. We give an $\mathcal{O}(c^n)$ time algorithm for these problems for any fixed constant γ, where c is a positive constant strictly less than 2, solving a well known open problem. These algorithms are then generalized to solve counting and enumeration version of these problems in the same time. An interesting consequence of the enumeration algorithm is, that it shows that the number of maximal γ-regular induced subgraphs for a fixed constant γ on any graph on n vertices is upper bounded by o(2n). We then give combinatorial lower bounds on the number of maximal γ-regular induced subgraphs possible on a graph on n vertices and also give matching algorithmic upper bounds. We use the techniques and results obtained in the paper to obtain an improved exact algorithm for a special case of Induced Subgraph Isomorphism that is Induced γ-Regular Subgraph Isomorphism, where γ is a constant. All the algorithms in the paper are simple but their analyses are not. Some of the upper bound proofs or algorithms require a new and different measure than the usual number of vertices or edges to measure the progress of the algorithm, and require solving an interesting system of polynomials.