Holographic algorithms: from art to science
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Holographic algorithms with unsymmetric signatures
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Holographic algorithms: guest column
ACM SIGACT News
Basis Collapse in Holographic Algorithms
Computational Complexity
Time-Space Tradeoffs for Counting NP Solutions Modulo Integers
Computational Complexity
Holographic algorithms: The power of dimensionality resolved
Theoretical Computer Science
Holant problems and counting CSP
Proceedings of the forty-first annual ACM symposium on Theory of computing
Classification of a Class of Counting Problems Using Holographic Reductions
COCOON '09 Proceedings of the 15th Annual International Conference on Computing and Combinatorics
On blockwise symmetric signatures for matchgates
Theoretical Computer Science
On symmetric signatures in holographic algorithms
STACS'07 Proceedings of the 24th annual conference on Theoretical aspects of computer science
Maximum edge-disjoint paths problem in planar graphs
TAMC'07 Proceedings of the 4th international conference on Theory and applications of models of computation
Journal of Symbolic Computation
Graph homomorphisms with complex values: a dichotomy theorem
ICALP'10 Proceedings of the 37th international colloquium conference on Automata, languages and programming
Holographic reduction: a domain changed application and its partial converse theorems
ICALP'10 Proceedings of the 37th international colloquium conference on Automata, languages and programming
Holographic algorithms: From art to science
Journal of Computer and System Sciences
The complexity of symmetric Boolean parity Holant problems
ICALP'11 Proceedings of the 38th international colloquim conference on Automata, languages and programming - Volume Part I
Guest column: complexity dichotomies of counting problems
ACM SIGACT News
Computational Complexity of Holant Problems
SIAM Journal on Computing
Some observations on holographic algorithms
LATIN'10 Proceedings of the 9th Latin American conference on Theoretical Informatics
Dichotomy for Holant problems of Boolean domain
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
A dichotomy theorem for the approximate counting of complex-weighted bounded-degree Boolean CSPs
Theoretical Computer Science
Holographic algorithms on domain size k2
TAMC'12 Proceedings of the 9th Annual international conference on Theory and Applications of Models of Computation
Approximate counting for complex-weighted Boolean constraint satisfaction problems
Information and Computation
Approximation complexity of complex-weighted degree-two counting constraint satisfaction problems
Theoretical Computer Science
On block-wise symmetric signatures for matchgates
FCT'07 Proceedings of the 16th international conference on Fundamentals of Computation Theory
Holographic algorithms: the power of dimensionality resolved
ICALP'07 Proceedings of the 34th international conference on Automata, Languages and Programming
A complete dichotomy rises from the capture of vanishing signatures: extended abstract
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
The complexity of planar boolean #CSP with complex weights
ICALP'13 Proceedings of the 40th international conference on Automata, Languages, and Programming - Volume Part I
The complexity of complex weighted Boolean #CSP
Journal of Computer and System Sciences
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We provide evidence for the proposition that the computational complexity of individual problems, and of whole complexity classes, hinge on the existence of certain solvable polynomial systems that are unlikely to be encountered other than by systematic explorations for them. We consider a minimalist version of Cook's 3CNF problem, namely that ofmonotone planar 3CNF formulae where each variable occurs twice. We show that counting the number of solutions of these modulo 2 is \oplusP-complete (hence NP-hard) but counting them modulo 7 is polynomial time computable (sic). We also show a similar dichtomy for a vertex cover problem. To derive completeness results we use a new holographic technique for proving completeness results in \oplusP for problems that are in P. For example, we can show in this way that \oplus2CNF, the parity problem for 2CNF, is \oplusP-complete. To derive efficient algorithms we use computer algebra systems to find appropriate holographic gates. In order to explore the limits of holographic techniques we define the notion of an elementary matchgrid algorithm to capture a natural but restricted use of them. We show that for the NP-complete general 3CNF problem no such elementary matchgrid algorithm can exist. We observe, however, that it remains open for many natural #Pcomplete problems whether such elementary matchgrid algorithms exist, and for the general CNF problem whether non-elementary matchgrid algorithms exist.