17312370
A note on short cycles in a hypercube
How many edges can a quadrilateral-free subgraph of a hypercube have? This question was raised by Paul Erdos about 27 years ago. His conjecture that such a subgraph asymptotically has at most half the edges of a hypercube is still unresolved. Let f(n,C"l) be the largest number of edges in a subgraph of a hypercube Q"n containing no cycle of length l. It is known that f(n,C"l)=o(|E(Q"n)|), when l=4k, k>=2 and that f(n,C"6)>=13|E(Q"n)|. It is an open question to determine f(n,C"l) for l=4k+2, k>=2. Here, we give a general upper bound for f(n,C"l) when l=4k+2 and provide a coloring of E(Q"n) by four colors containing no induced monochromatic C"1"0.
119154847
How many random edges make a dense graph hamiltonian
This paper investigates the number of random edges required to add to an arbitrary dense graph in order to make the resulting graph hamiltonian with high probability. Adding $\Theta(n)$ random edges is both necessary and sufficient to ensure this for all such dense graphs. If, however, the original graph contains no large independent set, then many fewer random edges are required. We prove a similar result for directed graphs.
2243848
Reducing Epistasis in Combinatorial Problems by Expansive Coding
This paper describes a new technique for tackling highly epistatic combinatorial optimization problems. Rather than having a simple representation, simple operators, a simple fitness function, but a highly epistatic search space, this technique is intended to spread the problem’s complexity more evenly. Using our new technique, known as expansive coding, the representation, operators and fitness function become more complicated, but the search space becomes less epistatic, and therefore easier for a GA to tackle. In effect, the combinatorial task is changed to a function optimization one. We demonstrate how this technique can be applied in the field of arithmetic algorithm design/electronic circuit simplification. In the design of a multiplier for quaternion numbers, consistently good results are obtained.
94046
Rapidly mixing Markov chains for sampling contingency tables with a constant number of rows
We consider the problem of sampling almost uniformly from the set of contingency tables with given row and column sums, when the number of rows is a constant. (2002) have recently given a fully polynomial randomized approximation scheme (fpras) for the related counting problem, which only employs Markov chain methods indirectly. But they leave open the question as to whether a natural Markov chain on such tables mixes rapidly. Here we answer this question in the affirmative, and hence provide a very different proof of the main result of Cryan and Dyer. We show that the "2 /spl times/ 2 heat-bath" Markov chain is rapidly mixing. We prove this by considering first a heat-bath chain operating on a larger window. Using techniques developed by Morris and Sinclair (2002) (see also Morris (2002)) for the multidimensional knapsack problem, we show that this chain mixes rapidly. We then apply the comparison method of Diaconis and Saloff-Coste (1993) to show that the 2 /spl times/ 2 chain is rapidly mixing. As part of our analysis, we give the first proof that the 2 /spl times/ 2 chain mixes in time polynomial in the input size when both the number of rows and the number of columns is constant.
1359860
Skewed Mirror Symmetry in the 3D Reconstruction of Polyhedral Models
We aim to reconstruct three-dimensional polyhedra from axonometric line drawings. Existence of mirror symmetry in polyhedra can assist the reconstruction process. We present a new approach for determining planes of mirror symmetry of such polyhedral objects based on prior detection of their planar faces and any axes of symmetry of these faces. The axes are obtained from skewed facial symmetries, for which we also give a new method of determination.
1385691
Skewed Mirror Symmetry for Depth Estimation in 3D Line-Drawings
We aim to reconstruct three-dimensional polyhedral solids from axonometric-like line drawings. A new approach is proposed to make use of planes of mirror symmetry detected in such sketches. Taking account of mirror symmetry of such polyhedra can significantly improve the reconstruction process. Applying symmetry as a regularity in optimisation-based reconstruction is shown to be adequate by itself, without the need for other inflation techniques or regularities. Furthermore, symmetry can be used to reduce the size of the reconstruction problem, leading to a reduction in computing time.
750888
Detecting approximate incomplete symmetries in discrete point sets
Motivated by the need to detect design intent in approximate boundary representation models, we give an algorithm to detect incomplete symmetries of discrete points, giving the models' potential local symmetries at various automatically detected tolerances. Here, incomplete symmetry is defined as a set of incomplete cycles which are constructed by, e.g., a set of consecutive vertices of an approximately regular polygon, induced by a single isometry. All seven 3D elementary isometries are considered for symmetry detection. Incomplete cycles are first found using a tolerance-controlled point expansion approach. Subsequently, these cycles are clustered for incomplete symmetry detection. The resulting clusters have welldefined, unambiguous approximate symmetries suitable for design intent detection, as demonstrated experimentally.