Counting the number of independent sets in chordal graphs
Journal of Discrete Algorithms
Efficient algorithms for counting parameterized list H-colorings
Journal of Computer and System Sciences
Approximately Counting Embeddings into Random Graphs
APPROX '08 / RANDOM '08 Proceedings of the 11th international workshop, APPROX 2008, and 12th international workshop, RANDOM 2008 on Approximation, Randomization and Combinatorial Optimization: Algorithms and Techniques
The parameterized complexity of probability amplification
Information Processing Letters
Parameterized approximation of dominating set problems
Information Processing Letters
Finding, minimizing, and counting weighted subgraphs
Proceedings of the forty-first annual ACM symposium on Theory of computing
The union of minimal hitting sets: Parameterized combinatorial bounds and counting
Journal of Discrete Algorithms
Balanced families of perfect hash functions and their applications
ACM Transactions on Algorithms (TALG)
Quantifying systemic evolutionary changes by color coding confidence-scored PPI networks
WABI'09 Proceedings of the 9th international conference on Algorithms in bioinformatics
Finding and counting vertex-colored subtrees
MFCS'10 Proceedings of the 35th international conference on Mathematical foundations of computer science
Exact parameterized multilinear monomial counting via k-layer subset convolution and k-disjoint
COCOON'11 Proceedings of the 17th annual international conference on Computing and combinatorics
Approximate counting of cycles in streams
ESA'11 Proceedings of the 19th European conference on Algorithms
H-DB: a hybrid quantitative-structural sql optimizer
Proceedings of the 20th ACM international conference on Information and knowledge management
Cyclic pattern kernels revisited
PAKDD'05 Proceedings of the 9th Pacific-Asia conference on Advances in Knowledge Discovery and Data Mining
Linear-time counting algorithms for independent sets in chordal graphs
WG'05 Proceedings of the 31st international conference on Graph-Theoretic Concepts in Computer Science
Parameterized counting algorithms for general graph covering problems
WADS'05 Proceedings of the 9th international conference on Algorithms and Data Structures
Faster algorithms for finding and counting subgraphs
Journal of Computer and System Sciences
The parameterized complexity of maximality and minimality problems
IWPEC'06 Proceedings of the Second international conference on Parameterized and Exact Computation
Randomized approximations of parameterized counting problems
IWPEC'06 Proceedings of the Second international conference on Parameterized and Exact Computation
Parameterized approximation problems
IWPEC'06 Proceedings of the Second international conference on Parameterized and Exact Computation
On the effective enumerability of NP problems
IWPEC'06 Proceedings of the Second international conference on Parameterized and Exact Computation
The parameterized complexity of enumerating frequent itemsets
IWPEC'06 Proceedings of the Second international conference on Parameterized and Exact Computation
Solving #SAT using vertex covers
SAT'06 Proceedings of the 9th international conference on Theory and Applications of Satisfiability Testing
Static analysis and optimization of semantic web queries
PODS '12 Proceedings of the 31st symposium on Principles of Database Systems
Estimating entity importance via counting set covers
Proceedings of the 18th ACM SIGKDD international conference on Knowledge discovery and data mining
What's next? future directions in parameterized complexity
The Multivariate Algorithmic Revolution and Beyond
Parameterized complexity of weighted satisfiability problems
SAT'12 Proceedings of the 15th international conference on Theory and Applications of Satisfiability Testing
The parameterized complexity of k-edge induced subgraphs
ICALP'12 Proceedings of the 39th international colloquium conference on Automata, Languages, and Programming - Volume Part I
Balanced families of perfect hash functions and their applications
ICALP'07 Proceedings of the 34th international conference on Automata, Languages and Programming
A randomized approximation algorithm for parameterized 3-D matching counting problem
COCOON'07 Proceedings of the 13th annual international conference on Computing and Combinatorics
Parameterized top-K algorithms
Theoretical Computer Science
The fine classification of conjunctive queries and parameterized logarithmic space complexity
Proceedings of the 32nd symposium on Principles of database systems
Tractable counting of the answers to conjunctive queries
Journal of Computer and System Sciences
Counting matchings of size k is # W[1]-hard
ICALP'13 Proceedings of the 40th international conference on Automata, Languages, and Programming - Volume Part I
The complexity of weighted counting for acyclic conjunctive queries
Journal of Computer and System Sciences
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We develop a parameterized complexity theory for counting problems. As the basis of this theory, we introduce a hierarchy of parameterized counting complexity classes #W$[t]$, for $t\ge 1$, that corresponds to Downey and Fellows's W-hierarchy [R. G. Downey and M. R. Fellows, Parameterized Complexity, Springer-Verlag, New York, 1999] and we show that a few central W-completeness results for decision problems translate to \#W-completeness results for the corresponding counting problems. Counting complexity gets interesting with problems whose decision version is tractable, but whose counting version is hard. Our main result states that counting cycles and paths of length k in both directed and undirected graphs, parameterized by k, is #W$[1]-complete. This makes it highly unlikely that these problems are fixed-parameter tractable, even though their decision versions are fixed-parameter tractable. More explicitly, our result shows that most likely there is no $f(k) \cdot n^c$-algorithm for counting cycles or paths of length k in a graph of size n for any computable function $f: \mathbb{N} \to \mathbb{N}$ and constant c, even though there is a $2^{O(k)} \cdot n^{2.376}$ algorithm for finding a cycle or path of length k [N. Alon, R. Yuster, and U. Zwick, J. ACM, 42 (1995), pp. 844--856].