21,574 research outputs found
Recognizing Graph Theoretic Properties with Polynomial Ideals
Many hard combinatorial problems can be modeled by a system of polynomial
equations. N. Alon coined the term polynomial method to describe the use of
nonlinear polynomials when solving combinatorial problems. We continue the
exploration of the polynomial method and show how the algorithmic theory of
polynomial ideals can be used to detect k-colorability, unique Hamiltonicity,
and automorphism rigidity of graphs. Our techniques are diverse and involve
Nullstellensatz certificates, linear algebra over finite fields, Groebner
bases, toric algebra, convex programming, and real algebraic geometry.Comment: 20 pages, 3 figure
Combinatorial algorithm for counting small induced graphs and orbits
Graphlet analysis is an approach to network analysis that is particularly
popular in bioinformatics. We show how to set up a system of linear equations
that relate the orbit counts and can be used in an algorithm that is
significantly faster than the existing approaches based on direct enumeration
of graphlets. The algorithm requires existence of a vertex with certain
properties; we show that such vertex exists for graphlets of arbitrary size,
except for complete graphs and , which are treated separately. Empirical
analysis of running time agrees with the theoretical results
- …