Geometric representation of graphs in low dimension

An axis-parallel k–dimensional box is a Cartesian product R1 × R2 × · · · × Rk where Ri (for 1 ≤ i ≤ k) is a closed interval of the form [ai, bi] on the real line. For a graph G, its boxicity box(G) is the minimum dimension k, such that G is representable as the intersection graph of (axis–parallel) boxes in k–dimensional space. The concept of boxicity finds applications in various areas such as ecology, operation research etc. A number of NP-hard problems are either polynomial time solvable or have much better approximation ratio on low boxicity graphs. For example, the max-clique problem is polynomial time solvable on bounded boxicity graphs and the maximum independent set problem has log n approximation ratio for boxicity 2 graphs. In most cases, the first step usually is computing a low dimensional box representation of the given graph. Deciding whether the boxicity of a graph is at most 2 itself is NP-hard. We give an efficient randomized algorithm to construct a box representation of any graph G on n vertices in 1.5( ∆ + 2) ln n dimensions, where ∆ is the maximum degree of G. We also show that box(G) ≤ (∆+2) lnn for any graph G. Our bound is tight up to a factor of ln n. The only previously known general upper bound for boxicity was given by Roberts, namely box(G) ≤ n/2. Our result gives an exponentially better upper bound for bounded degree graphs. We also show that our randomized algorithm can be derandomized to get a polynomial time deterministic algorithm. Though our general upper bound is in terms of maximum degree ∆, we show that for almost all graphs on n vertices, its boxicity is upper bound by c · (dav + 1) lnn where dav is the average degree and c is a small constant. Also, we show that for any graph G, box(G) ≤ √ 8ndav ln n, which is tight up to a factor of b √ ln n for a constant b

Analysis of HIV- type 1 protease and reverse transcriptase in Brazilian children failing highly active antiretroviral therapy (HAART) Análise da protease e transcriptase reversa do HIV-1 em crianças com falha terapêutica em uso de terapia anti-retroviral altamente eficaz (HAART)

The aim of this study was to evaluate the genotypic resistance profiles of HIV-1 in children failing highly active antiretroviral therapy (HAART). Forty-one children (median age = 67 months) receiving HAART were submitted to genotypic testing when virological failure was detected. cDNA was extracted from PBMCs and amplified by nested PCR for the reverse transcriptase and protease regions of the pol gene. Drug resistance genotypes were determined from DNA sequencing. According to the genotypic analysis, 12/36 (33.3%) and 6/36 (16.6%) children showed resistance and possible resistance, respectively, to ZDV; 5/36 (14%) and 4/36 (11.1%), respectively, showed resistance and possible resistance to ddI; 4/36 (11.1%) showed resistance to 3TC and D4T; and 3/36 (8.3%) showed resistance to Abacavir. A high percentage (54%) of children exhibited mutations conferring resistance to NNRTI class drugs. Respective rates of resistance and possible resistance to PIs were: RTV (12.2%, 7.3%); APV (2.4%, 12.1%); SQV(0%, 12.1%); IDV (14.6%, 4.9%), NFV (22%, 4.9%), LPV/RTV (2.4%, 12.1%). Overall, 37/41 (90%) children exhibited virus with mutations related to drug resistance, while 9% exhibited resistance to all three antiretroviral drug classes.<br>O objetivo deste estudo foi avaliar o perfil de resistência genotípica do HIV-1 em crianças com falha terapêutica ao tratamento anti-retroviral (HAART). Quarenta e uma crianças (idade mediana = 67 meses) em uso de HAART foram submetidas ao teste de genotipagem no momento da detecção de falha ao tratamento. Foi realizada extração de cDNA de células periféricas mononucleares e amplificação do mesmo (regiões da transcriptase reversa e protease do gene pol) através de PCR-nested. O perfil genotípico foi determinado através do seqüenciamnto de nucleotídeos. De acordo com a análise genotípica, 12/36 (33,3%) e 6/36 (16,6%) crianças apresentaram, respectivamente, resistência e possível resistência ao AZT; 5/36 (14%) e 4/36 (11,1%), respectivamente, eram resistentes e possivelmente resistentes ao ddI; 4/36 %11,1%) apresentaram resistência ao 3TC e D4T, e 3/36 (8,3%) eram resistentes ao ABC. Uma alta porcentagem de crianças (54%) apresentou mutações relacionadas à resistência aos inibidores da trancriptase reversa não-análogos de nucleosídeos. As taxas de resistência e possível resistência aos inibidores da protease foram, respectivamente: RTV (12,2%; 7,3%); APV (2,4%; 12,1%); SQV (0%; 12,1%); IDV (14,6%; 4,9%); NFV (22%; 4,9%); LPV/RTV (2,4%; 12,1%). No total, 37/41 (90%) crianças apresentaram vírus com mutações relacionadas à resistência a alguma droga, sendo que 9% delas tinham vírus resistentes às três classes de drogas anti-retrovirais disponíveis