The rainbowness, rb(G), of a connected plane graph G is the minimum number k such that any colouring of vertices of the graph G using at least k colours involves a face all vertices of which have different colours. For a cubic polyhedral (i.e. 3-connected plane) graph G we prove that n 2 + α∗ 1 − 1 ≤ rb(G) ≤ n− α∗ 0 + 1 , where α∗ 0 and α ∗ 1 denote the independence number and the edge independ...

We present primal-dual decomposition algorithms for convex optimization problems with cost functions f(x) + g(Ax), where f and g have inexpensive proximal operators and A can be decomposed as a sum of two structured matrices. The methods are based on the Douglas–Rachford splitting algorithm applied to various splittings of the primal-dual optimality conditions. We discuss applications to image ...

| According to a remarkable re-interpretation of a theorem of E.M. Andreev (1970) by W.P. Thurston (1982), there is a unique (up to inversive transformations) packing of interior-disjoint circles in the plane, whose contact graph is any given polyhedral graph G, and such that an analagous \dual" circle packing simultaneously exists, whose contact graph is the planar dual graph G , and such that...

Consider a continuous function g ∈ L2(R) that is supported on [−1, 1] and generates a Gabor frame with translation parameter 1 and modulation parameter 0 < b < 2N 2N+1 for some N ∈ N. Under an extra condition on the zeroset of the window g we show that there exists a continuous dual window supported on [−N, N]. We also show that this result is optimal: indeed, if b > 2N 2N+1 then a dual window ...

For an edge-bicolored graph G where each edge is colored either red or blue, a vertex set S is a dual feedback vertex set if S hits all blue cycles and red cycles of G. In this paper, we show that a dual feedback vertex set of size at most k can be found in time O∗(ck1) and all minimal dual feedback vertex set of size at most k can be enumerated in time O(c +k 2 ) by compact representations for...

The dual magnetic mass of a hot quark-gluon plasma is computed in the lowest order of dual QCD, which predicts a well-defined (dual) gauge-invariant result for it. This is because, in dual QCD, electricity and magnetism are interchanged, so magnetic calculations in dual QCD are easy if the corresponding electric ones in ordinary QCD are easy, and vice versa. We obtain the (leading-order) numeri...

In this paper, we propose a new primal-dual algorithm for minimizing f(x) + g(x) + h(Ax), where f , g, and h are convex functions, f is differentiable with a Lipschitz continuous gradient, and A is a bounded linear operator. It has some famous primal-dual algorithms for minimizing the sum of two functions as special cases. For example, it reduces to the Chambolle-Pock algorithm when f = 0 and a...