7,531 research outputs found
Automorphism Groups of Geometrically Represented Graphs
We describe a technique to determine the automorphism group of a
geometrically represented graph, by understanding the structure of the induced
action on all geometric representations. Using this, we characterize
automorphism groups of interval, permutation and circle graphs. We combine
techniques from group theory (products, homomorphisms, actions) with data
structures from computer science (PQ-trees, split trees, modular trees) that
encode all geometric representations.
We prove that interval graphs have the same automorphism groups as trees, and
for a given interval graph, we construct a tree with the same automorphism
group which answers a question of Hanlon [Trans. Amer. Math. Soc 272(2), 1982].
For permutation and circle graphs, we give an inductive characterization by
semidirect and wreath products. We also prove that every abstract group can be
realized by the automorphism group of a comparability graph/poset of the
dimension at most four
New developments in the theory of Groebner bases and applications to formal verification
We present foundational work on standard bases over rings and on Boolean
Groebner bases in the framework of Boolean functions. The research was
motivated by our collaboration with electrical engineers and computer
scientists on problems arising from formal verification of digital circuits. In
fact, algebraic modelling of formal verification problems is developed on the
word-level as well as on the bit-level. The word-level model leads to Groebner
basis in the polynomial ring over Z/2n while the bit-level model leads to
Boolean Groebner bases. In addition to the theoretical foundations of both
approaches, the algorithms have been implemented. Using these implementations
we show that special data structures and the exploitation of symmetries make
Groebner bases competitive to state-of-the-art tools from formal verification
but having the advantage of being systematic and more flexible.Comment: 44 pages, 8 figures, submitted to the Special Issue of the Journal of
Pure and Applied Algebr
Expanding Einstein-Yang-Mills by Yang-Mills in CHY frame
Using the Cachazo-He-Yuan (CHY) formalism, we prove a recursive expansion of
tree level single trace Einstein-Yang-Mills (EYM) amplitudes with arbitrary
number of gluons and gravitons, which is valid for general spacetime dimensions
and any helicity configurations. The recursion is written in terms of
fewer-graviton EYM amplitudes and pure Yang-Mills (YM) amplitudes, which can be
further carried out until we reach an expansion in terms of pure YM amplitudes
in Kleiss-Kuijf (KK) basis. Our expansion then generates naturally a spanning
tree structure rooted on gluons whose vertices are gravitons. We further
propose a set of graph theoretical rules based on spanning trees that evaluate
directly the pure YM expansion coefficients.Comment: 36 pages, 3 captioned figures; v2: more details added, revised and
published versio
Ackermannian and Primitive-Recursive Bounds with Dickson's Lemma
Dickson's Lemma is a simple yet powerful tool widely used in termination
proofs, especially when dealing with counters or related data structures.
However, most computer scientists do not know how to derive complexity upper
bounds from such termination proofs, and the existing literature is not very
helpful in these matters.
We propose a new analysis of the length of bad sequences over (N^k,\leq) and
explain how one may derive complexity upper bounds from termination proofs. Our
upper bounds improve earlier results and are essentially tight
A Parallel Branch and Bound Algorithm for the Maximum Labelled Clique Problem
The maximum labelled clique problem is a variant of the maximum clique
problem where edges in the graph are given labels, and we are not allowed to
use more than a certain number of distinct labels in a solution. We introduce a
new branch-and-bound algorithm for the problem, and explain how it may be
parallelised. We evaluate an implementation on a set of benchmark instances,
and show that it is consistently faster than previously published results,
sometimes by four or five orders of magnitude.Comment: Author-final version. Accepted to Optimization Letter
Termination of rewrite relations on -terms based on Girard's notion of reducibility
In this paper, we show how to extend the notion of reducibility introduced by
Girard for proving the termination of -reduction in the polymorphic
-calculus, to prove the termination of various kinds of rewrite
relations on -terms, including rewriting modulo some equational theory
and rewriting with matching modulo , by using the notion of
computability closure. This provides a powerful termination criterion for
various higher-order rewriting frameworks, including Klop's Combinatory
Reductions Systems with simple types and Nipkow's Higher-order Rewrite Systems
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