Approximation Algorithms for Minimum-Load k-Facility Location

We consider a facility-location problem that abstracts settings where the cost of serving the clients assigned to a facility is incurred by the facility. Formally, we consider the minimum-load k-facility location (MLkFL) problem, which is defined as follows. We have a set F of facilities, a set C of clients, and an integer k > 0. Assigning client j to a facility f incurs a connection cost d(f, j). The goal is to open a set F\u27 of k facilities, and assign each client j to a facility f(j) in F\u27 so as to minimize maximum, over all facilities in F\u27, of the sum of distances of clients j assigned to F\u27 to F\u27. We call this sum the load of facility f. This problem was studied under the name of min-max star cover in [6, 2], who (among other results) gave bicriteria approximation algorithms for MLkFL for when F = C. MLkFL is rather poorly understood, and only an O(k)-approximation is currently known for MLkFL, even for line metrics. Our main result is the first polynomial time approximation scheme (PTAS) for MLkFL on line metrics (note that no non-trivial true approximation of any kind was known for this metric). Complementing this, we prove that MLkFL is strongly NP-hard on line metrics. We also devise a quasi-PTAS for MLkFL on tree metrics. MLkFL turns out to be surprisingly challenging even on line metrics, and resilient to attack by the variety of techniques that have been successfully applied to facility-location problems. For instance, we show that: (a) even a configuration-style LP-relaxation has a bad integrality gap; and (b) a multi-swap k-median style local-search heuristic has a bad locality gap. Thus, we need to devise various novel techniques to attack MLkFL. Our PTAS for line metrics consists of two main ingredients. First, we prove that there always exists a near-optimal solution possessing some nice structural properties. A novel aspect of this proof is that we first move to a mixed-integer LP (MILP) encoding the problem, and argue that a MILP-solution minimizing a certain potential function possesses the desired structure, and then use a rounding algorithm for the generalized-assignment problem to "transfer" this structure to the rounded integer solution. Complementing this, we show that these structural properties enable one to find such a structured solution via dynamic programming

The possibility of access to the kidneys from posterior axillary line in supine position for percutaneous nephrolithotomy

Please cite this article as: Tabibi A, Kashi AH, Mirjalili SAM, Mahmoudnejad N, Kashani P, Salavatipour B, Soltani MH. The possibility of access to the kidneys from posterior axillary line in supine position for percutaneous nephrolithotomy. Novel Biomed 2013;1(2):43-47.Objectives: To evaluate the possibility of access to the kidneys from posterior axillary line (PAL) in supine position for percutaneous nephrolithotomy.Materials and Methods: 102 consecutive patients who were candidated for abdominal CT scan, enrolled in this study. In cases of impossible access, the point on the posterior surface of body which permitted safe access was determined and the percent of movement toward body midline (relative to PAL) was calculated (M.PER).Results: Percutaneous access was simulated from upper and middle calyces of the kidney in 13% and 75% of cases, respectively. Access to the lower region was possible in 90% of right and 79% of left lower calyces, respectively (p=0.03). In cases with impossible access from PAL, the M.PER for a safe access was 46-47% for upper region and 34- 38% for middle and lower calyces of the kidney (P = 0.0001).Conclusions: Access to upper calyces from PAL was limited in some cases regarding to the presence of solid organs. Presence of colon made access impossible in the lower right and left calyces in about 10% and 20% of cases, respectively. In upper region, more deviation toward midline was necessary to establish a safe access compared with middle and lower calyces