Related papers: Base collapse of holographic algorithms
Holographic algorithms with matchgates are a novel approach to design polynomial time computation. It uses Kasteleyn's algorithm for perfect matchings, and more importantly a holographic reduction . The two fundamental parameters of a…
The theory of holographic algorithms introduced by Valiant represents a novel approach to achieving polynomial-time algorithms for seemingly intractable counting problems via a reduction to counting planar perfect matchings and a linear…
An essential problem in the design of holographic algorithms is to decide whether the required signatures can be realized by matchgates under a suitable basis transformation (SRP). For holographic algorithms on domain size 2, [1, 2, 4, 5]…
Holographic algorithms introduced by Valiant are composed of two ingredients: matchgates, which are gadgets realizing local constraint functions by weighted planar perfect matchings, and holographic reductions, which show equivalences among…
The theory of holographic algorithms, which are polynomial time algorithms for certain combinatorial counting problems, yields insight into the hierarchy of complexity classes. In particular, the theory produces algebraic tests for a…
Valiant introduced matchgate computation and holographic algorithms. A number of seemingly exponential time problems can be solved by this novel algorithmic paradigm in polynomial time. We show that, in a very strong sense, matchgate…
We provide a recursive description of the signatures realizable on the standard basis by a holographic algorithm. The description allows us to prove tight bounds on the size of planar matchgates and efficiently test for standard signatures.…
We prove a complexity classification theorem that classifies all counting constraint satisfaction problems ($\#$CSP) over Boolean variables into exactly three categories: (1) Polynomial-time tractable; (2) $\#$P-hard for general instances,…
It remains an open question whether the apparent additional power of quantum computation derives inherently from quantum mechanics, or merely from the flexibility obtained by "lifting" Boolean functions to linear operators and evaluating…
An ideal of a local polynomial ring can be described by calculating a standard basis with respect to a local monomial ordering. However standard basis algorithms are not numerically stable. Instead we can describe the ideal numerically by…
We consider the algorithmic decision problem that takes as input an $n$-vertex $k$-uniform hypergraph $H$ with minimum codegree at least $m-c$ and decides whether it has a matching of size $m$. We show that this decision problem is fixed…
Hypergraph partitioning is an NP-hard problem that occurs in many computer science applications where it is necessary to reduce large problems into a number of smaller, computationally tractable sub-problems. Current techniques use a…
Solving Algebra Problems with Geometry Diagrams (APGDs) is still a challenging problem because diagram processing is not studied as intensively as language processing. To work against this challenge, this paper proposes a hologram reasoning…
Observing certain patches in an image reduces the uncertainty of others. Their realization lowers the distribution entropy of each remaining patch feature, analogous to collapsing a particle's wave function in quantum mechanics. This…
In this paper, we consider dynamic matroids, where elements can be inserted to or deleted from the ground set over time. The independent sets change to reflect the current ground set. As matroids are central to the study of many…
This paper presents an automatic approach for the evaluation of the plastic load and failure modes of planar frames. The method is based on the generation of elementary collapse mechanisms and on their linear combination aimed at minimizing…
Finding cliques in random graphs and the closely related "planted" clique variant, where a clique of size t is planted in a random G(n,1/2) graph, have been the focus of substantial study in algorithm design. Despite much effort, the best…
Graph mining analyzes real-world graphs to find core substructures (connected subgraphs) in applications modeled as graphs. Substructure discovery is a process that involves identifying meaningful patterns, structures, or components within…
Holant problems are a general framework to study the algorithmic complexity of counting problems. Both counting constraint satisfaction problems and graph homomorphisms are special cases. All previous results of Holant problems are over the…
Problems in scientific computing, such as distributing large sparse matrix operations, have analogous formulations as hypergraph partitioning problems. A hypergraph is a generalization of a traditional graph wherein "hyperedges" may connect…