On an optimization problem with nested constraints

AbstractWe describe algorithms for solving the integer programming problem maximise ∑j=1n⨍j(xj),subject to ∑jϵSixj⩽bi, i=1,…,m,xj⩾0, j=1,…,n, where the ⨍i are concave nondecreasing and the Si form a nested collection of sets. For the general problem, we present an algorithm of time-complexity O(n log2 n log b), where b is less than the largest of the bi. We also examine the case in which all ⨍i are identical and give an algorithm requiring O(n + m log m) time. Both algorithms use only O(n) space

Spatial Mixing of Coloring Random Graphs

We study the strong spatial mixing (decay of correlation) property of proper $q$-colorings of random graph $G(n, d/n)$ with a fixed $d$. The strong spatial mixing of coloring and related models have been extensively studied on graphs with bounded maximum degree. However, for typical classes of graphs with bounded average degree, such as $G(n, d/n)$, an easy counterexample shows that colorings do not exhibit strong spatial mixing with high probability. Nevertheless, we show that for $q\ge\alpha d+\beta$ with $\alpha>2$ and sufficiently large $\beta=O(1)$, with high probability proper $q$-colorings of random graph $G(n, d/n)$ exhibit strong spatial mixing with respect to an arbitrarily fixed vertex. This is the first strong spatial mixing result for colorings of graphs with unbounded maximum degree. Our analysis of strong spatial mixing establishes a block-wise correlation decay instead of the standard point-wise decay, which may be of interest by itself, especially for graphs with unbounded degree

A Dichotomy Theorem for Homomorphism Polynomials

In the present paper we show a dichotomy theorem for the complexity of polynomial evaluation. We associate to each graph H a polynomial that encodes all graphs of a fixed size homomorphic to H. We show that this family is computable by arithmetic circuits in constant depth if H has a loop or no edge and that it is hard otherwise (i.e., complete for VNP, the arithmetic class related to #P). We also demonstrate the hardness over the rational field of cut eliminator, a polynomial defined by B\"urgisser which is known to be neither VP nor VNP-complete in the field of two elements, if VP is not equal to VNP (VP is the class of polynomials computable by arithmetic circuit of polynomial size)

Finding Short Paths on Polytopes by the Shadow Vertex Algorithm

We show that the shadow vertex algorithm can be used to compute a short path between a given pair of vertices of a polytope P = {x : Ax \leq b} along the edges of P, where A \in R^{m \times n} is a real-valued matrix. Both, the length of the path and the running time of the algorithm, are polynomial in m, n, and a parameter 1/delta that is a measure for the flatness of the vertices of P. For integer matrices A \in Z^{m \times n} we show a connection between delta and the largest absolute value Delta of any sub-determinant of A, yielding a bound of O(Delta^4 m n^4) for the length of the computed path. This bound is expressed in the same parameter Delta as the recent non-constructive bound of O(Delta^2 n^4 \log (n Delta)) by Bonifas et al. For the special case of totally unimodular matrices, the length of the computed path simplifies to O(m n^4), which significantly improves the previously best known constructive bound of O(m^{16} n^3 \log^3(mn)) by Dyer and Frieze

A Polynomial Number of Random Points does not Determine the Volume of a Convex Body

We show that there is no algorithm which, provided a polynomial number of random points uniformly distributed over a convex body in R^n, can approximate the volume of the body up to a constant factor with high probability

Exponential Time Complexity of Weighted Counting of Independent Sets

We consider weighted counting of independent sets using a rational weight x: Given a graph with n vertices, count its independent sets such that each set of size k contributes x^k. This is equivalent to computation of the partition function of the lattice gas with hard-core self-repulsion and hard-core pair interaction. We show the following conditional lower bounds: If counting the satisfying assignments of a 3-CNF formula in n variables (#3SAT) needs time 2^{\Omega(n)} (i.e. there is a c>0 such that no algorithm can solve #3SAT in time 2^{cn}), counting the independent sets of size n/3 of an n-vertex graph needs time 2^{\Omega(n)} and weighted counting of independent sets needs time 2^{\Omega(n/log^3 n)} for all rational weights x\neq 0. We have two technical ingredients: The first is a reduction from 3SAT to independent sets that preserves the number of solutions and increases the instance size only by a constant factor. Second, we devise a combination of vertex cloning and path addition. This graph transformation allows us to adapt a recent technique by Dell, Husfeldt, and Wahlen which enables interpolation by a family of reductions, each of which increases the instance size only polylogarithmically.Comment: Introduction revised, differences between versions of counting independent sets stated more precisely, minor improvements. 14 page

Quantum Computing with NMR

A review of progress in NMR quantum computing and a brief survey of the literatureComment: Commissioned by Progress in NMR Spectroscopy (95 pages, no figures

Monte-Carlo sampling of energy-constrained quantum superpositions in high-dimensional Hilbert spaces

Recent studies into the properties of quantum statistical ensembles in high-dimensional Hilbert spaces have encountered difficulties associated with the Monte-Carlo sampling of quantum superpositions constrained by the energy expectation value. A straightforward Monte-Carlo routine would enclose the energy constrained manifold within a larger manifold, which is easy to sample, for example, a hypercube. The efficiency of such a sampling routine decreases exponentially with the increase of the dimension of the Hilbert space, because the volume of the enclosing manifold becomes exponentially larger than the volume of the manifold of interest. The present paper explores the ways to optimise the above routine by varying the shapes of the manifolds enclosing the energy-constrained manifold. The resulting improvement in the sampling efficiency is about a factor of five for a 14-dimensional Hilbert space. The advantage of the above algorithm is that it does not compromise on the rigorous statistical nature of the sampling outcome and hence can be used to test other more sophisticated Monte-Carlo routines. The present attempts to optimise the enclosing manifolds also bring insights into the geometrical properties of the energy constrained manifold itself.Comment: 9 pages, 7 figures, accepted for publication in European Physical Journal

Virus infection and grazing exert counteracting influences on survivorship of native bunchgrass seedlings competing with invasive exotics

1.  Invasive annual grasses introduced by European settlers have largely displaced native grassland vegetation in California and now form dense stands that constrain the establishment of native perennial bunchgrass seedlings. Bunchgrass seedlings face additional pressures from both livestock grazing and barley and cereal yellow dwarf viruses (B/CYDVs), which infect both young and established grasses throughout the state. 2.  Previous work suggested that B/CYDVs could mediate apparent competition between invasive exotic grasses and native bunchgrasses in California. 3.  To investigate the potential significance of virus-mediated mortality for early survivorship of bunchgrass seedlings, we compared the separate and combined effects of virus infection, competition and simulated grazing in a field experiment. We infected two species of young bunchgrasses that show different sensitivity to B/CYDV infection, subjected them to competition with three different densities of exotic annuals crossed with two clipping treatments, and monitored their growth and first-year survivorship. 4.  Although virus infection alone did not reduce first-year survivorship, it halved the survivorship of bunchgrasses competing with exotics. Within an environment in which competition strongly reduces seedling survivorship (as in natural grasslands), virus infection therefore has the power to cause additional seedling mortality and alter patterns of establishment. 5.  Surprisingly, clipping did not reduce bunchgrass survivorship further, but rather doubled it and disproportionately increased survivorship of infected bunchgrasses. 6.  Together with previous work, these findings show that B/CYDVs can be potentially powerful elements influencing species interactions in natural grasslands. 7.  More generally, our findings demonstrate the potential significance of multitrophic interactions in virus ecology. Although sometimes treated collectively as plant ‘predators’, viruses and herbivores may exert influences that are distinctly different, even counteracting