Sharif Digital Repository

We revisit the problem of finding k paths with a minimum number of shared edges between two vertices of a graph. An edge is called shared if it is used in more than one of the k paths. We provide a ⌊k/2⌋-approximation algorithm for this problem, improving the best previous approximation factor of k - 1. We also provide the first approximation algorithm for the problem with a sublinear approximation factor of O(n3/4), where n is the number of vertices in the input graph. For sparse graphs, such as bounded-degree and planar graphs, we show that the approximation factor of our algorithm can be improved to O(√n). While the problem is NP-hard, and even hard to approximate to within an O(log n)... 

In this paper we study envy free pricing problem in general graphs where there is not a seller in every graph's nodes. We assume unique establishment cost for initiating a store in each node and we wish to find an optimal set of nodes in which we would make the maximum profit by initiating stores in them. Our model is motivated from the observation that a same product has different prices in different locations and there is also an establishing cost for initiating any store. We consider both of these issues in our model: first where should we establish the stores, and second at what price should we sell our items in them to gain maximum possible profit. We prove that in a case of constant... 

We study a classical problem in communication and wireless networks called Finding White Space Regions. In this problem, we are given a set of antennas (points) some of which are noisy (black) and the rest are working fine (white). The goal is to find a set of convex hulls with maximum total area that cover all white points and exclude all black points. In other words, these convex hulls make it safe for white antennas to communicate with each other without any interference with black antennas. We study the problem on three different settings (based on overlapping between different convex hulls) and find hardness results and good approximation algorithms  

In this paper, we consider the following problem: given an undirected graph G=(V,E) and an integer k, find I⊆V2 with |I|≤k in such a way that G'=(V,E∪I) has the maximum number of triangles (a cycle of length 3). We first prove that this problem is NP-hard and then give an approximation algorithm for it  

For a set of n disjoint line segments S in R2, the visibility testing problem (VTP) is to test whether the query point p sees a query segment s∈S. For this configuration, the visibility counting problem (VCP) is to preprocess S such that the number of visible segments in S from any query point p can be computed quickly. In this paper, we solve VTP in expected logarithmic query time using quadratic preprocessing time and space. Moreover, we propose a (1+δ)-approximation algorithm for VCP using at most quadratic preprocessing time and space. The query time of this method is Oε (1/δ 2√n) where Oε (f(n))=O(f(n)nε) and ε>0 is an arbitrary constant number  

We consider a class of optimization problems called movement minimization on euclidean plane. Given a set of m nodes on the plane, the aim is to achieve some specific property by minimum movement of the nodes. We consider two specific properties, namely the connectivity (Con) and realization of a given topology (Topol). By minimum movement, we mean either the sum of all movements (Sum) or the maximum movement (Max). We obtain several approximation algorithms and some hardness results for these four problems. We obtain an O(m) -factor approximation for ConMax and ConSum and extend some known result on graphical grounds and obtain inapproximability results on the geometrical grounds. For the... 

Motivated by the bus escape routing problem in printed circuit boards, we study the following rectangle escape problem: given a set S of n axis-aligned rectangles inside an axis-aligned rectangular region R, extend each rectangle in S toward one of the four borders of R so that the maximum density over the region R is minimized. The density of each point p∈R is defined as the number of extended rectangles containing p. We show that the problem is hard to approximate to within a factor better than 3/2 in general. When the optimal density is sufficiently large, we provide a randomized algorithm that achieves an approximation factor of 1+ε with high probability improving over the current best... 

We study the problem of fair allocation for indivisible goods. We use the maxmin share paradigm introduced by Budish [16] as a measure for fairness. Kurokawa, Procaccia, and Wang [36] were the first to investigate this fundamental problem in the additive setting. They show that a maxmin guarantee (1-MMS allocation) is not always possible even when the number of agents is limited to 3. While the existence of an approximation solution (e.g. a 1/2-MMS allocation) is quite straightforward, improving the guarantee becomes subtler for larger constants. Kurokawa et al. [36] provide a proof for the existence of a 2/3-MMS allocation and leave the question open for better guarantees. Our main... 

Given a set of points in the plane, the unit clustering problem asks for finding a minimum-size set of unit disks that cover the whole input set. We study the unit clustering problem in a distributed setting, where input data is partitioned among several machines. We present a (3 + ϵ)-approximation algorithm for the problem in the Euclidean plane, and a (4 + ϵ)-approximation algorithm for the problem under general Lp metric (p1). We also study the capacitated version of the problem, where each cluster has a limited capacity for covering the points. We present a distributed algorithm for the capacitated version of the problem that achieves an approximation factor of 4+" in the L2 plane, and a... 

We focus on two metric clusterings namely r-gather and (r, ?)-gather. The objective of r-gather is to minimize the radius of clustering, such that each cluster has at least r points. (r, ?)-gather is a version of r-gather with the extra condition that at most n? points can be left unclustered (outliers). MapReduce is a model used for processing big data. In each round, it distributes data to multiple servers, then simultaneously processes each server's data. We prove a lower bound 2 on the approximation factor of metric r-gather in the MapReduce model, even if an optimal algorithm for r-gather exists. Then, we give a (4+ δ)-approximation algorithm for r-gather in MapReduce which runs in O(... 

Given a set of points in the plane, the unit clustering problem asks for finding a minimum-size set of unit disks that cover the whole input set. We study the unit clustering problem in a distributed setting, where input data is partitioned among several machines. We present a (3 + ϵ)-approximation algorithm for the problem in the Euclidean plane, and a (4 + ϵ)-approximation algorithm for the problem under general Lp metric (p1). We also study the capacitated version of the problem, where each cluster has a limited capacity for covering the points. We present a distributed algorithm for the capacitated version of the problem that achieves an approximation factor of 4+" in the L2 plane, and a... 

We focus on two metric clusterings namely r-gather and (r, ?)-gather. The objective of r-gather is to minimize the radius of clustering, such that each cluster has at least r points. (r, ?)-gather is a version of r-gather with the extra condition that at most n? points can be left unclustered (outliers). MapReduce is a model used for processing big data. In each round, it distributes data to multiple servers, then simultaneously processes each server's data. We prove a lower bound 2 on the approximation factor of metric r-gather in the MapReduce model, even if an optimal algorithm for r-gather exists. Then, we give a (4+ δ)-approximation algorithm for r-gather in MapReduce which runs in O(... 

Edit distance and longest common subsequence are among the most fundamental problems in combinatorial optimization. Recent developments have proven strong lower bounds against subquadratic time solutions for both problems. Moreover, the best approximation factors for subquadratic time solutions have been limited to 3 for edit distance and super constant for longest common subsequence. Improved approximation algorithms for these problems1 are some of the biggest open questions in combinatorial optimization. In this work, we present improved algorithms for both edit distance and longest common subsequence. The running times are truly subquadratic, though we obtain 1 + o(1) approximate... 

Edit distance is one of the most fundamental problems in combinatorial optimization to measure the similarity between strings. Ulam distance is a special case of edit distance where no character is allowed to appear more than once in a string. Recent developments have been very fruitful for obtaining fast and parallel algorithms for both edit distance and Ulam distance. In this work, we present an almost optimal MPC (massively parallel computation) algorithm for Ulam distance and improve MPC algorithms for edit distance. Our algorithm for Ulam distance is almost optimal in the sense that (1) the approximation factor of our algorithm is $1+epsilon$1+ϵ, (2) the round complexity of our... 

Motivated by the bus escape routing problem in printed circuit boards (PCBs), we study the following optimization problem: given a set of rectangles attached to the boundary of a rectangular region, find a subset of nonoverlapping rectangles with maximum total weight. We present an efficient algorithm that solves this problem optimally in O(n4) time, where n is the number of rectangles in the input instance. This improves over the best previous O(n6) -time algorithm available for the problem. We also present two efficient approximation algorithms for the problem that find near-optimal solutions with guaranteed approximation factors. The first algorithm finds a 2-approximate solution in O(n2)...