Constant-Factor Approximation for TSP with Disks

We revisit the traveling salesman problem with neighborhoods (TSPN) and present the first constant-ratio approximation for disks in the plane: Given a set of $n$ disks in the plane, a TSP tour whose length is at most $O(1)$ times the optimal can be computed in time that is polynomial in $n$. Our result is the first constant-ratio approximation for a class of planar convex bodies of arbitrary size and arbitrary intersections. In order to achieve a $O(1)$-approximation, we reduce the traveling salesman problem with disks, up to constant factors, to a minimum weight hitting set problem in a geometric hypergraph. The connection between TSPN and hitting sets in geometric hypergraphs, established here, is likely to have future applications.Comment: 14 pages, 3 figure

Expression of the apoptosis repressor with caspase recruitment domain (ARC) in liver metastasis of colorectal cancer and its correlation with clinical data and other prognostic and predictive proteins

Liver metastasis in colorectal cancer is still common and the primary treatment is chemotherapy. Until now there is no routinely used test in the clinical practice to predict effectiveness of conventional chemotherapy. Therefore, biomarkers with predictive value also for conventional chemotherapy would be of considerable benefit in the treatment planning. Apoptotic signaling is one of the most important processes in the measurement of chemotherapeutic effectiveness. In apoptotic machinery various pathways and proteins are involved (i.e. mismatch repair proteins, p53 etc.). One of the regulatory proteins is ARC, which can inhibit not only the extrinsic but also the intrinsic apoptotic signaling. In this study we investigated the expression levels of ARC in colorectal liver metastasis and compared them with the expression of mismatch repair proteins and p53. Furthermore, we investigated ARC expression level depending on sex, age, tumor grade, mucin production, tumor size and number of liver metastasis. ARC expression level in colorectal cancer liver metastasis was independent from clinical data (i.e. age, gender, tumor size, tumor number or mucin production) but strongly correlated with MSH2 and MSH6 expression, which further supported the evidence for the regulatory role of MSH2 and MSH6 in apoptosis: i.e. in case of sufficient MSH2 and MSH6 expression significantly higher ARC level is required to suppress the apoptosis. A regulatory interaction between ARC and p53 has been described, but we found no correlation between p53 expression levels and ARC levels. In the second phase we analysed three proteins (ERCC1, RRM1 and TUBB3) in colorectal cancer liver metastasis. We used tissue microarray slides with hundred and one liver metastasis; stained for ERCC1, RRM1 and TUBB3 and established scoring systems (fitted for tissue microarray) for each protein. In statistical analysis we compared the expression of ERCC1, RRM1 and TUBB3 to mismatch proteins (MLH1, MSH2, MSH6 and PMS2), p53 and to apoptosis repressor protein (ARC). Statistically significant correlations were found between ERCC1, TUBB3 and MLH1, MSH2 and RRM1 and MSH2, MSH6. Noteworthy, our analysis declares strong significant correlation between cytoplasmic ARC expression and RRM1, TUBB3, implying additional role of TUBB3 and RRM1 not only in therapy resistance but also in the apoptotic machinery. Our data strengthens the importance of ERCC1, TUBB3 and RRM1 in prediction of chemotherapy effectiveness and suggest new functional connections in DNA repair, microtubule network and apoptotic signaling (i.e. ARC protein). CDX2 is well-established as a diagnostic marker for colorectal cancer, but less is known about its regulation, especially about its possible interactions with DNA repair proteins, APC and β-catenin in non-transcriptional manner. In this study we analysed the protein expression of CDX2 depending on the expression of DNA repair proteins (mismatch repair proteins, MGMT and ERCC1) and crucial member of Wnt signalling. CDX2 loss of expression was found in 38.5% of our cases of colorectal cancer liver metastasis. We found statistically significant association between CDX2 and each of the investigated mismatch repair protein: MLH1, MSH2, MSH6, and PMS2. Furthermore, loss of MGMT and ERCC1 was also associated with CDX2 loss . In addition, CDX2 and ERCC1 were inversely associated with metastatic tumor size. Sustained CDX2 expression was associated with higher expression of cytoplasmic/membranous β-catenin and with nuclear APC expression. In conclusion, CDX2 expression loss is not a rare event in liver metastasis of colorectal cancer and our results suggest that CDX2 is involved in mechanisms resulting in loss of DNA repair protein expression (i.e. methylation) and may be a part of this mechanism; however, its exact function in this context remains to be investigated further. We showed the importance and need of predictive biomarkers in metastasized colorectal cancer and pointed out the relevance not only of single predictive markers but also of their interactions with other known and newly explored relations between different signaling pathways. In conclusion, we can state that further studies are needed to define the exact role of ARC in apoptotic signaling and thus its role in chemoresistence and survival of tumor cells. In this study, we were able to describe the expression manner of ARC protein and could demonstrate an important link between the nuclear and cytoplasmic expression of ARC to MMR proteins and ERCC1, TUBB3 and RRM1 in colorectal cancer liver metastasis, which has not been shown before

Analysis of the quality variances of asphalt production by Monte Carlo Simulation

The first part of the article discusses the control level and characteristics analysis of the production of an asphalt mixture produced by a normal asphalt mix plant. By examining the stability of production, the analysis identifies the so called process capability indices as well as the density and distribution functions of the tested parameters. With the help of these, a prediction can be made on the Operating Compliance Level (OCL) of the mixture plant provided the existing processes will remain unchanged in the future. In the second part of the article certain stochastic methods i.e. the Monte Carlo Simulation techniques are used to examine the reliability of the production process in order to provide a review on how the dynamic modulus of the asphalt mixture is influenced by the variation of the tested parameters

New bounds on the average distance from the Fermat-Weber center of a planar convex body

The Fermat-Weber center of a planar body $Q$ is a point in the plane from which the average distance to the points in $Q$ is minimal. We first show that for any convex body $Q$ in the plane, the average distance from the Fermat-Weber center of $Q$ to the points of $Q$ is larger than ${1/6} \cdot \Delta(Q)$, where $\Delta(Q)$ is the diameter of $Q$. This proves a conjecture of Carmi, Har-Peled and Katz. From the other direction, we prove that the same average distance is at most $\frac{2(4-\sqrt3)}{13} \cdot \Delta(Q) < 0.3490 \cdot \Delta(Q)$. The new bound substantially improves the previous bound of $\frac{2}{3 \sqrt3} \cdot \Delta(Q) \approx 0.3849 \cdot \Delta(Q)$ due to Abu-Affash and Katz, and brings us closer to the conjectured value of ${1/3} \cdot \Delta(Q)$. We also confirm the upper bound conjecture for centrally symmetric planar convex bodies.Comment: 13 pages, 2 figures. An earlier version (now obsolete): A. Dumitrescu and Cs. D. T\'oth: New bounds on the average distance from the Fermat-Weber center of a planar convex body, in Proceedings of the 20th International Symposium on Algorithms and Computation (ISAAC 2009), 2009, LNCS 5878, Springer, pp. 132-14