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Electronic Colloquium on Computational Complexity

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Reports tagged with counting:
TR99-032 | 7th July 1999
Cristopher Moore

Quantum Circuits: Fanout, Parity, and Counting

We propose definitions of $\QAC^0$, the quantum analog of the
classical class $\AC^0$ of constant-depth circuits with AND and OR
gates of arbitrary fan-in, and $\QACC^0[q]$, the analog of the class
$\ACC^0[q]$ where $\Mod_q$ gates are also allowed. We show that it is
possible to make a `cat' state on ... more >>>

TR05-099 | 9th September 2005
Leslie G. Valiant

Holographic Algorithms

Complexity theory is built fundamentally on the notion of efficient
reduction among computational problems. Classical
reductions involve gadgets that map solution fragments of one problem to
solution fragments of another in one-to-one, or
possibly one-to-many, fashion. In this paper we propose a new kind of
reduction that allows for gadgets ... more >>>

TR05-121 | 17th October 2005
Martin Dyer, Leslie Ann Goldberg, Michael S. Paterson

On counting homomorphisms to directed acyclic graphs

We give a dichotomy theorem for the problem of counting homomorphisms to
directed acyclic graphs. $H$ is a fixed directed acyclic graph.
The problem is, given an input digraph $G$, how many homomorphisms are there
from $G$ to $H$. We give a graph-theoretic classification, showing that
for some digraphs $H$, ... more >>>

TR08-087 | 31st July 2008
Tomas Feder, Carlos Subi

Nearly Tight Bounds on the Number of Hamiltonian Circuits of the Hypercube and Generalizations (revised)

It has been shown that for every perfect matching $M$ of the $d$-dimensional
$n$-vertex hypercube, $d\geq 2, n=2^d$, there exists a second perfect matching
$M'$ such that the union of $M$ and $M'$ forms a Hamiltonian circuit of the
$d$-dimensional hypercube. We prove a generalization of a special case of ... more >>>

TR13-133 | 23rd September 2013
Cassio P. de Campos, Georgios Stamoulis, Dennis Weyland

A Structured View on Weighted Counting with Relations to Quantum Computation and Applications

Revisions: 2

Weighted counting problems are a natural generalization of counting problems where a weight is associated with every computational path and the goal is to compute the sum of the weights of all paths (instead of computing the number of accepting paths). We present a structured view on weighted counting by ... more >>>

TR14-079 | 11th June 2014
Simon Straub, Thomas Thierauf, Fabian Wagner

Counting the Number of Perfect Matchings in $K_5$-free Graphs

Counting the number of perfect matchings in arbitrary graphs is a $\#$P-complete problem. However, for some restricted classes of graphs the problem can be solved efficiently. In the case of planar graphs, and even for $K_{3,3}$-free graphs, Vazirani showed that it is in NC$^2$. The technique there is to compute ... more >>>

TR16-024 | 22nd February 2016
Patrick Scharpfenecker, Jacobo Toran

Solution-Graphs of Boolean Formulas and Isomorphism

The solution graph of a Boolean formula on n variables is the subgraph of the hypercube Hn induced by the satisfying assignments of the formula. The structure of solution graphs has been the object of much research in recent years since it is important for the performance of SAT-solving procedures ... more >>>

TR19-033 | 20th February 2019
Ashish Dwivedi, Rajat Mittal, Nitin Saxena

Counting basic-irreducible factors mod $p^k$ in deterministic poly-time and $p$-adic applications

Finding an irreducible factor, of a polynomial $f(x)$ modulo a prime $p$, is not known to be in deterministic polynomial time. Though there is such a classical algorithm that {\em counts} the number of irreducible factors of $f\bmod p$. We can ask the same question modulo prime-powers $p^k$. The irreducible ... more >>>

TR20-177 | 12th October 2020
Lior Gishboliner, Yevgeny Levanzov, Asaf Shapira

Counting Subgraphs in Degenerate Graphs

We consider the problem of counting the number of copies of a fixed graph $H$ within an input graph $G$. This is one of the most well-studied algorithmic graph problems, with many theoretical and practical applications. We focus on solving this problem when the input $G$ has {\em bounded degeneracy}. ... more >>>

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