Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1007.1726

Vsep-New Heuristic and Exact Algorithms for Graph Automorphism Group Computation

Stoicho D. Stoichev
View original source ↗Read source document ↗Export metadata ↓

Abstract

One exact and two heuristic algorithms for determining the generators, orbits and order of the graph automorphism group are presented. A basic tool of these algorithms is the well-known individualization and refinement procedure. A search tree is used in the algorithms - each node of the tree is a partition. All nonequivalent discreet partitions derivative of the selected vertices are stored in a coded form. A new strategy is used in the exact algorithm: if during its execution some of the searched or intermediate variables obtain a wrong value then the algorithm continues from a new start point losing some of the results determined so far. The algorithms has been tested on one of the known benchmark graphs and shows lower running times for some graph families. The heuristic versions of the algorithms are based on determining some number of discreet partitions derivative of each vertex in the selected cell of the initial partition and comparing them for an automorphism - their search trees are reduced. The heuristic algorithms are almost exact and are many times faster than the exact one. The experimental tests exhibit that the worst-cases running time of the exact algorithm is exponential but it is polynomial for the heuristic algorithms. Several cell selectors are used. Some of them are new. We also use a chooser of cell selector for choosing the optimal cell selector for the manipulated graph. The proposed heuristic algorithms use two main heuristic procedures that generate two different forests of search trees.

Explore related subjects

cs.DScs.DMmath.CO

Keep this discovery

BibTeXRIS

Stoicho D. Stoichev. 2010-07-10. Vsep-New Heuristic and Exact Algorithms for Graph Automorphism Group Computation. https://arxiv.org/abs/1007.1726

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Quasi-Monte Carlo Beyond Hardy-Krause II: $(1 + \varepsilon)n$ Samples Suffice

Numerical integration studies how well one can estimate the integral of a function $f$ over $[0,1)^d$ using $n$ sample points. The two classical methods, Monte Carlo (MC) and quasi-Monte Carlo (QMC), have complementary strengths and weaknesses, and a fundamental question is to design an approach that combines the benefits of both. Recently, building on the transference principle in discrepancy theory, Bansal and Jiang~\cite{BJ25a} gave a randomized QMC method that bridges MC and QMC guarantees using only i.i.d.\ samples. Their method also goes beyond the classical Koksma--Hlawka inequality: it achieves integration error $\widetilde{O}_d(\sigma_{\mathsf{SO}}(f)/n)$, where the smoothed-out variation $\sigma_{\mathsf{SO}}(f)$ can be substantially smaller than the Hardy--Krause variation that governs the classical bound. However, their algorithm requires $n^2$ i.i.d.\ samples as input, and this quadratic blowup is inherent to any method based on the transference principle. In this work, we bypass the quadratic blowup: for any constant $\varepsilon > 0$, we show that $(1+\varepsilon)n$ i.i.d.\ samples suffice to both obtain the beyond-Hardy--Krause guarantee of~\cite{BJ25a}, resolving an open problem posed there, and to produce low-discrepancy point sequences. Our algorithms are variants of the online Haar-thinning method of Dwivedi, Feldheim, Gurel-Gurevich, and Ramdas~\cite{DFG+19}.

cs.DS↗
arXiv2026

Single-Exponential Algorithms and a Polynomial Kernel for Strong Connectivity Augmentation

Strong Connectivity Augmentation (SCA) asks whether a directed acyclic graph can be made strongly connected by adding at most $k$ prescribed links whose total weight is within a given budget. Klinkby, Misra, and Saurabh (SODA 2021) gave an $O^*(2^{O(k\log k)})$-time algorithm and asked whether the problem admits a single-exponential parameterized algorithm and a polynomial kernel. We answer both questions affirmatively: SCA can be solved in $O^*(9^k)$ time and admits a polynomial kernel with $O(k^4)$ vertices and $O(k^{16})$ bits. For unweighted SCA, we obtain $O^*(4^k)$ time and a kernel with $O(k^3)$ vertices. Our algorithms are based on a particularly simple reduction to Strongly Connected Spanning Subgraph with two edge costs.

cs.DS↗