Parameterized and Approximation Algorithms for the Load Coloring Problem

Let $c, k$ be two positive integers and let $G=(V,E)$ be a graph. The $(c,k)$-Load Coloring Problem (denoted $(c,k)$-LCP) asks whether there is a $c$-coloring $\varphi: V \rightarrow [c]$ such that for every $i \in [c]$, there are at least $k$ edges with both endvertices colored $i$. Gutin and Jones (IPL 2014) studied this problem with $c=2$. They showed $(2,k)$-LCP to be fixed parameter tractable (FPT) with parameter $k$ by obtaining a kernel with at most $7k$ vertices. In this paper, we extend the study to any fixed $c$ by giving both a linear-vertex and a linear-edge kernel. In the particular case of $c=2$, we obtain a kernel with less than $4k$ vertices and less than $8k$ edges. These results imply that for any fixed $c\ge 2$, $(c,k)$-LCP is FPT and that the optimization version of $(c,k)$-LCP (where $k$ is to be maximized) has an approximation algorithm with a constant ratio for any fixed $c\ge 2$

The radiating part of circular sources

An analysis is developed linking the form of the sound field from a circular source to the radial structure of the source, without recourse to far-field or other approximations. It is found that the information radiated into the field is limited, with the limit fixed by the wavenumber of source multiplied by the source radius (Helmholtz number). The acoustic field is found in terms of the elementary fields generated by a set of line sources whose form is given by Chebyshev polynomials of the second kind, and whose amplitude is found to be given by weighted integrals of the radial source term. The analysis is developed for tonal sources, such as rotors, and, for Helmholtz number less than two, for random disk sources. In this case, the analysis yields the cross-spectrum between two points in the acoustic field. The analysis is applied to the problems of tonal radiation, random source radiation as a model problem for jet noise, and to noise cancellation, as in active control of noise from rotors. It is found that the approach gives an accurate model for the radiation problem and explicitly identifies those parts of a source which radiate.Comment: Submitted to Journal of the Acoustical Society of Americ

Acceptability, feasibility and challenges of implementing an HIV prevention intervention for people living with HIV/AIDS among healthcare providers in Mozambique: Results of a qualitative study

Despite the Mozambique government’s efforts to curb human immunodeficiency virus (HIV)/acquired immune deficiency syndrome (AIDS), national prevalence is 11.5% and support is needed to expand HIV-related services and improve program quality. Positive prevention (PP) programs, which prioritize HIV prevention with people living with HIV and AIDS (PLHIV), have been recognized as an important intervention for preventing new HIV infections. To address this, an evidence-based PP training intervention was implemented with HIV healthcare providers in Mozambique. This study focuses on the acceptability and feasibility of a PP intervention in HIV clinics from the healthcare provider perspective. In depth interviews were conducted with 31 healthcare providers from three provinces who participated in PP trainings in Mozambique. Interview data were coded using content analysis. Study data suggest that healthcare providers found PP acceptable, feasible to implement in their HIV work in clinic settings, and valued this strategy to improve HIV prevention. The PP training also led providers to feel more comfortable counseling their patients about prevention, with a more holistic approach that included HIV testing, treatment and encouraging PLHIV to live positively. While overall acceptance of the PP training was positive, several barriers to feasibility surfaced in the data. Patient-level barriers included resistance to disclosing HIV status due to fear of stigma and discrimination, difficulty negotiating for condom use, difficulty engaging men in testing and treatment, and the effects of poverty on accessing care. Providers also identified work environment barriers including high patient load, time constraints, and frequent staff turnover. Recognizing PP as an important intervention, healthcare providers should be trained to provide comprehensive prevention, care and treatment for PLHIV. Further work is needed to explore the complex social dynamics and cultural challenges such as gender inequalities, stigma and discrimination which hinder the full impact of PP interventions in this context.Keywords: positive prevention, feasibility and acceptability, healthcare provider, HIV/AID

Average value of solutions for the bipartite boolean quadratic programs and rounding algorithms

We consider domination analysis of approximation algorithms for the bipartite boolean quadratic programming problem (BBQP) with m+n variables. A closed-form formula is developed to compute the average objective function value A of all solutions in O(mn) time. However, computing the median objective function value of the solutions is shown to be NP-hard. Also, we show that any solution with objective function value no worse than A dominates at least 2 m+n-2 solutions and this bound is the best possible. Further, we show that such a solution can be identified in O(mn) time and hence the domination ratio of this algorithm is at least 14. We then show that for any fixed natural numbers a and b such that η=ab > 1, no polynomial time approximation algorithm exists for BBQP with domination ratio larger than 1-2(1-η)η(m+n), unless P = NP. It is shown that some powerful local search algorithms can get trapped at a local maximum with objective function value less than A. One of our approximation algorithms has an interesting rounding property which provides a data dependent lower bound on the optimal objective function value. A new integer programming formulation of BBQP is also given and computational results with our rounding algorithms are reported

Optimal General Matchings

Given a graph $G=(V,E)$ and for each vertex $v \in V$ a subset $B(v)$ of the set $\{0,1,\ldots, d_G(v)\}$, where $d_G(v)$ denotes the degree of vertex $v$ in the graph $G$, a $B$-factor of $G$ is any set $F \subseteq E$ such that $d_F(v) \in B(v)$ for each vertex $v$, where $d_F(v)$ denotes the number of edges of $F$ incident to $v$. The general factor problem asks the existence of a $B$-factor in a given graph. A set $B(v)$ is said to have a {\em gap of length} $p$ if there exists a natural number $k \in B(v)$ such that $k+1, \ldots, k+p \notin B(v)$ and $k+p+1 \in B(v)$. Without any restrictions the general factor problem is NP-complete. However, if no set $B(v)$ contains a gap of length greater than $1$, then the problem can be solved in polynomial time and Cornuejols \cite{Cor} presented an algorithm for finding a $B$-factor, if it exists. In this paper we consider a weighted version of the general factor problem, in which each edge has a nonnegative weight and we are interested in finding a $B$-factor of maximum (or minimum) weight. In particular, this version comprises the minimum/maximum cardinality variant of the general factor problem, where we want to find a $B$-factor having a minimum/maximum number of edges. We present an algorithm for the maximum/minimum weight $B$-factor for the case when no set $B(v)$ contains a gap of length greater than $1$. This also yields the first polynomial time algorithm for the maximum/minimum cardinality $B$-factor for this case

Systems of Linear Equations over F2\mathbb{F}_2 and Problems Parameterized Above Average

In the problem Max Lin, we are given a system $Az=b$ of $m$ linear equations with $n$ variables over $\mathbb{F}_2$ in which each equation is assigned a positive weight and we wish to find an assignment of values to the variables that maximizes the excess, which is the total weight of satisfied equations minus the total weight of falsified equations. Using an algebraic approach, we obtain a lower bound for the maximum excess. Max Lin Above Average (Max Lin AA) is a parameterized version of Max Lin introduced by Mahajan et al. (Proc. IWPEC'06 and J. Comput. Syst. Sci. 75, 2009). In Max Lin AA all weights are integral and we are to decide whether the maximum excess is at least $k$, where $k$ is the parameter. It is not hard to see that we may assume that no two equations in $Az=b$ have the same left-hand side and $n={\rm rank A}$. Using our maximum excess results, we prove that, under these assumptions, Max Lin AA is fixed-parameter tractable for a wide special case: $m\le 2^{p(n)}$ for an arbitrary fixed function $p(n)=o(n)$. Max $r$-Lin AA is a special case of Max Lin AA, where each equation has at most $r$ variables. In Max Exact $r$-SAT AA we are given a multiset of $m$ clauses on $n$ variables such that each clause has $r$ variables and asked whether there is a truth assignment to the $n$ variables that satisfies at least $(1-2^{-r})m + k2^{-r}$ clauses. Using our maximum excess results, we prove that for each fixed $r\ge 2$, Max $r$-Lin AA and Max Exact $r$-SAT AA can be solved in time $2^{O(k \log k)}+m^{O(1)}.$ This improves $2^{O(k^2)}+m^{O(1)}$-time algorithms for the two problems obtained by Gutin et al. (IWPEC 2009) and Alon et al. (SODA 2010), respectively

Deviations from the mean field predictions for the phase behaviour of random copolymers melts

We investigate the phase behaviour of random copolymers melts via large scale Monte Carlo simulations. We observe macrophase separation into A and B--rich phases as predicted by mean field theory only for systems with a very large correlation lambda of blocks along the polymer chains, far away from the Lifshitz point. For smaller values of lambda, we find that a locally segregated, disordered microemulsion--like structure gradually forms as the temperature decreases. As we increase the number of blocks in the polymers, the region of macrophase separation further shrinks. The results of our Monte Carlo simulation are in agreement with a Ginzburg criterium, which suggests that mean field theory becomes worse as the number of blocks in polymers increases.Comment: 6 pages, 4 figures, Late

Folding of small proteins: A matter of geometry?

We review some of our recent results obtained within the scope of simple lattice models and Monte Carlo simulations that illustrate the role of native geometry in the folding kinetics of two state folders.Comment: To appear in Molecular Physic

Conformations of Randomly Linked Polymers

We consider polymers in which M randomly selected pairs of monomers are restricted to be in contact. Analytical arguments and numerical simulations show that an ideal (Gaussian) chain of N monomers remains expanded as long as M<<N; its mean squared end to end distance growing as r^2 ~ M/N. A possible collapse transition (to a region of order unity) is related to percolation in a one dimensional model with long--ranged connections. A directed version of the model is also solved exactly. Based on these results, we conjecture that the typical size of a self-avoiding polymer is reduced by the links to R > (N/M)^(nu). The number of links needed to collapse a polymer in three dimensions thus scales as N^(phi), with (phi) > 0.43.Comment: 6 pages, 3 Postscript figures, LaTe