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Scientists have demonstrated that quantum computing has presented novel approaches to address computational challenges, each varying in complexity. Adapting problem-solving strategies is crucial to harness the full potential of quantum computing. Nonetheless, there are defined boundaries to the capabilities of quantum computing. This paper concentrates on aggregating prior research efforts dedicated to solving intricate classical computational problems through quantum computing. The objective is to systematically compile an exhaustive inventory of these solutions and categorize a collection of demanding open problems that await further exploration. Through statistical analysis, we help the researchers with their further investigations., Comment: arXiv admin note: text overlap with arXiv:1612.07366 by other authors

Given a set of points labeled with $k$ labels, we introduce the heat map sorting problem as reordering and merging the points and dimensions while preserving the clusters (labels). A cluster is preserved if it remains connected, i.e., if it is not split into several clusters and no two clusters are merged. We prove the problem is NP-hard and we give a fixed-parameter algorithm with a constant number of rounds in the massively parallel computation model, where each machine has a sublinear memory and the total memory of the machines is linear. We give an approximation algorithm for a NP-hard special case of the problem. We empirically compare our algorithm with k-means and density-based clustering (DBSCAN) using a dimensionality reduction via locality-sensitive hashing on several directed and undirected graphs of email and computer networks.

In a metric space, a set of point sets of roughly the same size and an integer $k\geq 1$ are given as the input and the goal of data-distributed $k$-center is to find a subset of size $k$ of the input points as the set of centers to minimize the maximum distance from the input points to their closest centers. Metric $k$-center is known to be NP-hard which carries to the data-distributed setting. We give a $2$-approximation algorithm of $k$-center for sublinear $k$ in the data-distributed setting, which is tight. This algorithm works in several models, including the massively parallel computation model (MPC).

Sublinear time complexity is required by the massively parallel computation (MPC) model. Breaking dynamic programs into a set of sparse dynamic programs that can be divided, solved, and merged in sublinear time. The rectangle escape problem (REP) is defined as follows: For $n$ axis-aligned rectangles inside an axis-aligned bounding box $B$, extend each rectangle in only one of the four directions: up, down, left, or right until it reaches $B$ and the density $k$ is minimized, where $k$ is the maximum number of extensions of rectangles to the boundary that pass through a point inside bounding box $B$. REP is NP-hard for $k>1$. If the rectangles are points of a grid (or unit squares of a grid), the problem is called the square escape problem (SEP) and it is still NP-hard. We give a $2$-approximation algorithm for SEP with $k\geq2$ with time complexity $O(n^{3/2}k^2)$. This improves the time complexity of existing algorithms which are at least quadratic. Also, the approximation ratio of our algorithm for $k\geq 3$ is $3/2$ which is tight. We also give a $8$-approximation algorithm for REP with time complexity $O(n\log n+nk)$ and give a MPC version of this algorithm for $k=O(1)$ which is the first parallel algorithm for this problem.

A sparse graph that preserves an approximation of the shortest paths between all pairs of points in a plane is called a geometric spanner. Using range trees of sublinear size, we design an algorithm in massively parallel computation (MPC) model for constructing a geometric spanner known as Yao-graph. This improves the total time and the total memory of existing algorithms for geometric spanners from subquadratic to near-linear.

Delaunay Triangulation(DT) is one of the important geometric problems that is used in various branches of knowledge such as computer vision, terrain modeling, spatial clustering and networking. Kinetic data structures have become very important in computational geometry for dealing with moving objects. However, when dealing with moving points, maintaining a dynamically changing Delaunay triangulation can be challenging. So, In this case, we have to update triangulation repeatedly. If points move so far, it is better to rebuild the triangulation. One approach to handle moving points is to use an incremental algorithm. For the case that points move slowly, we can give a faster algorithm than rebuilding it. Furthermore, sequential algorithms can be computationally expensive for large datasets. So, one way to compute as fast as possible is parallelism. In this paper, we propose a parallel algorithm for moving points. we propose an algorithm that divides datasets into equal partitions and give every partition to one block. Each block satisfay the Delaunay constraints after each time step and uses delete and insert algorithms to do this. We show this algorithm works faster than serial algorithms.

Consider two entities with constant but not necessarily equal velocities, moving on two given piece-wise linear trajectories inside a simple polygon $P$. The Trajectory Range Visibility problem deals with determining the sub-trajectories on which two entities become visible to each other. A more straightforward decision version of this problem is called Trajectory Visibility, where the trajectories are line segments. The decision version specifies whether the entities can see one another. This version was studied by P. Eades et al. in 2020, where they supposed given constant velocities for the entities. However, the approach presented in this paper supports non-constant complexity trajectories. Furthermore, we report every pair of constant velocities with which the entities can see each other. In particular, for every constant velocity of a moving entity, we specify: $(1)$ All visible parts of the other entity's trajectory. $(2)$ All possible constant velocities of the other entity to become visible. Regarding line-segment trajectories, we present $\mathcal{O}(n \log n)$ running time algorithm which obtains all pairs of sub-trajectories on which the moving entities become visible to one another, where $n$ is the complexity of $P$. Regarding the general case, we provide an algorithm with $\mathcal{O}(n \log n + m(\log m + \log n))$ running time, where $m$ indicates the complexity of both trajectories. We offer $\mathcal{O}(\log n)$ query time for line segment trajectories and $\mathcal{O}(\log m + k)$ for the non-constant complexity ones s.t. $k$ is the number of velocity ranges reported in the output. Interestingly, our results require only $\mathcal{O}(n + m)$ space for non-constant complexity trajectories.

Given a simple polygon $\cal P$, in the Art Gallery problem the goal is to find the minimum number of guards needed to cover the entire $\cal P$, where a guard is a point and can see another point $q$ when $\overline{pq}$ does not cross the edges of $\cal P$. This paper studies a variant of the Art Gallery problem in which guards are restricted to lie on a dense grid inside $\cal P$. In the general problem, guards can be anywhere inside or on the boundary of $\cal P$. The general problem is called the \emph{point} guarding problem. It was proved that the point guarding problem is APX-complete, meaning that we cannot do better than a constant-factor approximation algorithm unless $P = NP$. A huge amount of research is committed to the studies of combinatorial and algorithmic aspects of this problem, and as of this time, we could not find a constant factor approximation for simple polygons. The last best-known approximation factor for point guarding a simple polygon was $\mathcal{O}(\log (|OPT|))$ introduced by E. Bonnet and T. Miltzow in 2020, where $|OPT|$ is the size of the optimal solution. Here, we propose an algorithm with a constant approximation factor for the point guarding problem where the location of guards is restricted to a grid. The running time of the proposed algorithm depends on the number of cells of the grid. The approximation factor is constant regardless of the grid we use, the running time could be super-polynomial if the grid size becomes exponential.

The classic cake-cutting problem provides a model for addressing the fair and efficient allocation of a divisible, heterogeneous resource among agents with distinct preferences. Focusing on a standard formulation of cake cutting, in which each agent must receive a contiguous piece of the cake in an offline setting, this work instead focuses on online allocating non-contiguous pieces of cake among agents and establishes algorithmic results for fairness measures. In this regard, we made use of classic adversarial multi-armed bandits to achieve sub-linear Fairness and Revenue Regret at the same time. Adversarial bandits are powerful tools to model the adversarial reinforcement learning environments, that provide strong upper-bounds for regret of learning with just observing one action's reward in each step by applying smart trade-off between exploration and exploitation. This work studies the power of the famous EXP_3 algorithm that is based on exponential wight{-}importance updating probability distribution through time horizon.

The $k$-center problem is to choose a subset of size $k$ from a set of $n$ points such that the maximum distance from each point to its nearest center is minimized. Let $Q=\{Q_1,\ldots,Q_n\}$ be a set of polygons or segments in the region-based uncertainty model, in which each $Q_i$ is an uncertain point, where the exact locations of the points in $Q_i$ are unknown. The geometric objects segments and polygons can be models of a point set. We define the uncertain version of the $k$-center problem as a generalization in which the objective is to find $k$ points from $Q$ to cover the remaining regions of $Q$ with minimum or maximum radius of the cluster to cover at least one or all exact instances of each $Q_i$, respectively. We modify the region-based model to allow multiple points to be chosen from a region and call the resulting model the aggregated uncertainty model. All these problems contain the point version as a special case, so they are all NP-hard with a lower bound 1.822. We give approximation algorithms for uncertain $k$-center of a set of segments and polygons. We also have implemented some of our algorithms on a data-set to show our theoretical performance guarantees can be achieved in practice., Comment: Accepted in Fundamenta Informaticae

Given a simple polygon $\cal P$, in the Art Gallery problem, the goal is to find the minimum number of guards needed to cover the entire $\cal P$, where a guard is a point and can see another point $q$ when $\overline{pq}$ does not cross the edges of $\cal P$. This paper studies a variant of the Art Gallery problem in which the boundaries of $\cal P$ are replaced by single specular-reflection edges, allowing the view rays to reflect once per collision with an edge. This property allows the guards to see through the reflections, thereby viewing a larger portion of the polygon. For this problem, the position of the guards in $\cal P$ can be determined with our proposed $\mathcal{O}(\log n)$-approximation algorithm. Besides presenting an algorithm with the mentioned approximation factor, we will see that reflection can decrease the number of guards in practice. The proposed algorithm converts the generalized problem to the Set Cover problem.

A polygonal curve $P$ with $n$ vertices is $c$-packed, if the sum of the lengths of the parts of the edges of the curve that are inside any disk of radius $r$ is at most $cr$, for any $r>0$. Similarly, the concept of $c$-packedness can be defined for any scaling of a given shape. Assuming $L$ is the diameter of $P$ and $\delta$ is the minimum distance between points on disjoint edges of $P$, we show the approximation factor of the existing $O(\frac{\log (L/\delta)}{\epsilon}n^3)$ time algorithm is $1+\epsilon$-approximation algorithm. The massively parallel versions of these algorithms run in $O(\log (L/\delta))$ rounds. We improve the existing $O((\frac{n}{\epsilon^3})^{\frac 4 3}\polylog \frac n \epsilon)$ time $(6+\epsilon)$-approximation algorithm by providing a $(4+\epsilon)$-approximation $O(n(\log^2 n)(\log^2 \frac{1}{\epsilon})+\frac{n}{\epsilon})$ time algorithm, and the existing $O(n^2)$ time $2$-approximation algorithm improving the existing $O(n^2\log n)$ time $2$-approximation algorithm. Our exact $c$-packedness algorithm takes $O(n^5)$ time, which is the first exact algorithm for disks. We show using $\alpha$-fat shapes instead of disks adds a factor $\alpha^2$ to the approximation. We also give a data-structure for computing the curve-length inside query disks. It has $O(n^6\log n)$ construction time, uses $O(n^6)$ space, and has query time $O(\log n+k)$, where $k$ is the number of intersected segments with the query shape. We also give a massively parallel algorithm for relative $c$-packedness with $O(1)$ rounds.

This paper studies a variant of the Art Gallery problem in which the ``walls" can be replaced by \emph{reflecting edges}, which allows the guards to see further and thereby see a larger portion of the gallery. Given a simple polygon $\cal P$, first, we consider one guard as a point viewer, and we intend to use reflection to add a certain amount of area to the visibility polygon of the guard. We study visibility with specular and diffuse reflections where the specular type of reflection is the mirror-like reflection, and in the diffuse type of reflection, the angle between the incident and reflected ray may assume all possible values between $0$ and $\pi$. Lee and Aggarwal already proved that several versions of the general Art Gallery problem are $NP$-hard. We show that several cases of adding an area to the visible area of a given point guard are $NP$-hard, too. Second, we assume all edges are reflectors, and we intend to decrease the minimum number of guards required to cover the whole gallery. Chao Xu proved that even considering $r$ specular reflections, one may need $\lfloor \frac{n}{3} \rfloor$ guards to cover the polygon. Let $r$ be the maximum number of reflections of a guard's visibility ray. In this work, we prove that considering $r$ \emph{diffuse} reflections, the minimum number of \emph{vertex or boundary} guards required to cover a given simple polygon $\cal P$ decreases to { $\bf \lceil \frac{\alpha}{1+ \lfloor \frac{r}{8} \rfloor} \rceil$}, where $\alpha$ indicates the minimum number of guards required to cover the polygon without reflection. We also generalize the $\mathcal{O}(\log n)$-approximation ratio algorithm of the vertex guarding problem to work in the presence of reflection., Comment: 32 pages, 10 figures

Accessing to required data on the internet is wide via search engines in the last two decades owing to the huge amount of available data and the high rate of new data is generating daily. Accordingly, search engines are encouraged to make the most valuable existing data on the web searchable. Knowing how to handle a large amount of data in each step of a search engines' procedure from crawling to indexing and ranking is just one of the challenges that a professional search engine should solve. Moreover, it should also have the best practices in handling users' traffics, state-of-the-art natural language processing tools, and should also address many other challenges on the edge of science and technology. As a result, evaluating these systems is too challenging due to the level of internal complexity they have, and is crucial for finding the improvement path of the existing system. Therefore, an evaluation procedure is a normal subsystem of a search engine that has the role of building its roadmap. Recently, several countries have developed national search engine programs to build an infrastructure to provide special services based on their needs on the available data of their language on the web. This research is conducted accordingly to enlighten the advancement path of two Iranian national search engines: Yooz and Parsijoo in comparison with two international ones, Google and Bing. Unlike related work, it is a semi-automatic method to evaluate the search engines at the first pace. Eventually, we obtained some interesting results which based on them the component-based improvement roadmap of national search engines could be illustrated concretely.

An arrangement of $n$ curves in the plane is given. The query is a point $q$ and the goal is to find the face of the arrangement that contains $q$. A data-structure for point-location, preprocesses the curves into a data structure of polynomial size in $n$, such that the queries can be answered in time polylogarithmic in $n$. We design a data structure for solving the point location problem queries in $O(\log C(n)+\log S(n))$ time using $O(T(n)+S(n)\log(S(n)))$ preprocessing time, if a polygonal subdivision of total size $S(n)$, with cell complexity at most $C(n)$ can be computed in time $T(n)$, such that the order of the parts of the curves inside each cell has a monotone order with respect to at least one segment of the boundary of the cell. We call such a partitioning a curve-monotone polygonal subdivision.

A set of piecewise linear functions, called polylines, $P_1,\ldots,P_L$ each with at most $n$ vertices can be simplified into a polyline $M$ with $k$ vertices, such that the Fr\'echet distances $\epsilon_1,\ldots,\epsilon_L$ to each of these polylines are minimized under the $L_p$ distance. We call $M$ for $L_p$ with $p\geq 1$ a $p$-mean curve ($p$-MC). We discuss $p\geq 1$, for which $L_p$ distance satisfies the triangle inequality and $p$-mean has not been discussed before for most values $p$. Computing the $p$-mean polyline is NP-hard for $L=\Omega(1)$ and some values of $p$, so we discuss approximation algorithms. We give a $O(n^2\log k)$ time exact algorithm for $L=2$ and $p\geq 1$. Also, we reduce the Fr\'echet distance to the discrete Fr\'echet distance which adds a factor $2$ to both $k$ and $\epsilon$. Then we use our exact algorithm to find a $3$-approximation for $L>2$ in $\operatorname{poly}(n,L)$ time. Our method is based on a generalization of the free-space diagram (FSD) for Fr\'echet distance and composable core-sets for approximate summaries.

A set of points $P$ in a metric space and a constant integer $k$ are given. The $k$-center problem finds $k$ points as centers among $P$, such that the maximum distance of any point of $P$ to their closest centers $(r)$ is minimized. Doubling metrics are metric spaces in which for any $r$, a ball of radius $r$ can be covered using a constant number of balls of radius $r/2$. Fixed dimensional Euclidean spaces are doubling metrics. The lower bound on the approximation factor of $k$-center is $1.822$ in Euclidean spaces, however, $(1+\epsilon)$-approximation algorithms with exponential dependency on $\frac{1}{\epsilon}$ and $k$ exist. For a given set of sets $P_1,\ldots,P_L$, a composable coreset independently computes subsets $C_1\subset P_1, \ldots, C_L\subset P_L$, such that $\cup_{i=1}^L C_i$ contains an approximation of a measure of the set $\cup_{i=1}^L P_i$. We introduce a $(1+\epsilon)$-approximation composable coreset for $k$-center, which in doubling metrics has size sublinear in $|P|$. This results in a $(2+\epsilon)$-approximation algorithm for $k$-center in MapReduce with a constant number of rounds in doubling metrics for any $\epsilon>0$ and sublinear communications, which is based on parametric pruning. We prove the exponential nature of the trade-off between the number of centers $(k)$ and the radius $(r)$, and give a composable coreset for a related problem called dual clustering. Also, we give a new version of the parametric pruning algorithm with $O(\frac{nk}{\epsilon})$ running time, $O(n)$ space and $2+\epsilon$ approximation factor for metric $k$-center., Comment: The preliminary version of this paper has appeared in Proceedings of the 30th Canadian Conference on Computational Geometry, (CCCG 2018)

In Spatial Voting Theory, distortion is a measure of how good the winner is. It is proved that no deterministic voting mechanism can guarantee a distortion better than $3$, even for simple metrics such as a line. In this study, we wish to answer the following question: how does the distortion value change if we allow less motivated agents to abstain from the election? We consider an election with two candidates and suggest an abstention model, which is a more general form of the abstention model proposed by Kirchgassner. We define the concepts of the expected winner and the expected distortion to evaluate the distortion of an election in our model. Our results fully characterize the distortion value and provide a rather complete picture of the model., Comment: Revised version of the paper appeared in AAAI-19

We consider extending the visibility polygon of a given point $q$, inside a simple polygon $P$ by converting some edges of $P$ to mirrors. We will show that several variations of the problem of finding mirror-edges to add precisely $k$ units of area to $VP(q)$ are NP-complete. The optimal cases are NP-hard. We are unaware of any result on adding an exact number to a polygon, or covering an area with an exact surface. We deal with both single and multiple reflecting mirrors for both specular or diffuse types of reflections.

One of the important yet insufficiently studied subjects in fair allocation is the externality effect among agents. For a resource allocation problem, externalities imply that a bundle allocated to an agent may affect the utilities of other agents. In this paper, we conduct a study of fair allocation of indivisible goods when the externalities are not negligible. We present a simple and natural model, namely \emph{network externalities}, to capture the externalities. To evaluate fairness in the network externalities model, we generalize the idea behind the notion of maximin-share ($\MMS$) to achieve a new criterion, namely, \emph{extended-maximin-share} ($\EMMS$). Next, we consider two problems concerning our model. First, we discuss the computational aspects of finding the value of $\EMMS$ for every agent. For this, we introduce a generalized form of partitioning problem that includes many famous partitioning problems such as maximin, minimax, and leximin partitioning problems. We show that a $1/2$-approximation algorithm exists for this partitioning problem. Next, we investigate on finding approximately optimal $\EMMS$ allocations. That is, allocations that guarantee every agent a utility of at least a fraction of his extended-maximin-share. We show that under a natural assumption that the agents are $\alpha$-self-reliant, an $\alpha/2$-$\EMMS$ allocation always exists. The combination of this with the former result yields a polynomial-time $\alpha/4$-$\EMMS$ allocation algorithm.

The edit distance between two strings is defined as the smallest number of insertions, deletions, and substitutions that need to be made to transform one of the strings to another one. Approximating edit distance in subquadratic time is "one of the biggest unsolved problems in the field of combinatorial pattern matching". Our main result is a quantum constant approximation algorithm for computing the edit distance in truly subquadratic time. More precisely, we give an $O(n^{1.858})$ quantum algorithm that approximates the edit distance within a factor of $7$. We further extend this result to an $O(n^{1.781})$ quantum algorithm that approximates the edit distance within a larger constant factor. Our solutions are based on a framework for approximating edit distance in parallel settings. This framework requires as black box an algorithm that computes the distances of several smaller strings all at once. For a quantum algorithm, we reduce the black box to \textit{metric estimation} and provide efficient algorithms for approximating it. We further show that this framework enables us to approximate edit distance in distributed settings. To this end, we provide a MapReduce algorithm to approximate edit distance within a factor of $3$, with sublinearly many machines and sublinear memory. Also, our algorithm runs in a logarithmic number of rounds., Comment: A preliminary version of this paper was presented at SODA 2018

We consider extending visibility polygon $(VP)$ of a given point $q$ $(VP(q))$, inside a simple polygon $\P$ by converting some edges of $\P$ to mirrors. We will show that several variations of the problem of finding mirror-edges to add at least $k$ units of area to $VP(q)$ are NP-complete, or NP-hard. Which $k$ is a given value. We deal with both single and multiple reflecting mirrors, and also specular or diffuse types of reflections. In specular reflection, a single incoming direction is reflected into a single outgoing direction. In this paper diffuse reflection is regarded as reflecting lights at all possible angles from a given surface. The paper deals with finding mirror-edges to add \emph{at least} $k$ units of area to $VP(q)$. In the case of specular type of reflections we only consider single reflections, and the multiple case is still open. Specular case of the problem is more tricky. We construct a simple polygon for every given instance of a 3-SAT problem. There are some specific spikes which are visible only by some particular mirror-edges. Consequently, to have minimum number of mirror-edges it is required to choose only one of these mirrors to see a particular spike. There is a reduction polygon which contains a clause-gadget corresponding to every clause, and a variable-gadget corresponding to every variable. 3-SAT formula has $n$ variables and $m$ clauses, so the minimum number of mirrors required to add an area of at least $k$ to $V P(q)$ is $l = 3m+n+1$ if and only if the 3-SAT formula is satisfiable. This reduction works in these two cases: adding at least $k$ vertex of $\P$ to $VP(q)$, and expanding $VP(q)$ at least $k$ units of area., Comment: The writing is not well and I should rewrite the paper

We study the problem of fair allocation for indivisible goods. We use the the maxmin share paradigm introduced by Budish as a measure for fairness. Procaccia and Wang (EC'14) were first to investigate this fundamental problem in the additive setting. In contrast to what real-world experiments suggest, 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. Procaccia provide a proof for existence of a $2/3$-MMS allocation and leave the question open for better guarantees. Our main contribution is an answer to the above question. We improve the result of [Procaccia and Wang] to a $3/4$ factor in the additive setting. The main idea for our $3/4$-MMS allocation method is clustering the agents. To this end, we introduce three notions and techniques, namely reducibility, matching allocation, and cycle-envy-freeness, and prove the approximation guarantee of our algorithm via non-trivial applications of these techniques. Our analysis involves coloring and double counting arguments that might be of independent interest. One major shortcoming of the current studies on fair allocation is the additivity assumption on the valuations. We alleviate this by extending our results to the case of submodular, fractionally subadditive, and subadditive settings. More precisely, we give constant approximation guarantees for submodular and XOS agents, and a logarithmic approximation for the case of subadditive agents. Furthermore, we complement our results by providing close upper bounds for each class of valuation functions. Finally, we present algorithms to find such allocations for additive, submodular, and XOS settings in polynomial time.

We study the minimum diameter problem for a set of inexact points. By inexact, we mean that the precise location of the points is not known. Instead, the location of each point is restricted to a contineus region ($\impre$ model) or a finite set of points ($\indec$ model). Given a set of inexact points in one of $\impre$ or $\indec$ models, we wish to provide a lower-bound on the diameter of the real points. In the first part of the paper, we focus on $\indec$ model. We present an $O(2^{\frac{1}{\epsilon^d}} \cdot \epsilon^{-2d} \cdot n^3 )$ time approximation algorithm of factor $(1+\epsilon)$ for finding minimum diameter of a set of points in $d$ dimensions. This improves the previously proposed algorithms for this problem substantially. Next, we consider the problem in $\impre$ model. In $d$-dimensional space, we propose a polynomial time $\sqrt{d}$-approximation algorithm. In addition, for $d=2$, we define the notion of $\alpha$-separability and use our algorithm for $\indec$ model to obtain $(1+\epsilon)$-approximation algorithm for a set of $\alpha$-separable regions in time $O(2^{\frac{1}{\epsilon^2}}\allowbreak . \frac{n^3}{\epsilon^{10} .\sin(\alpha/2)^3} )$.

We study fair allocation of indivisible goods to agents with unequal entitlements. Fair allocation has been the subject of many studies in both divisible and indivisible settings. Our emphasis is on the case where the goods are indivisible and agents have unequal entitlements. This problem is a generalization of the work by Procaccia and Wang wherein the agents are assumed to be symmetric with respect to their entitlements. Although Procaccia and Wang show an almost fair (constant approximation) allocation exists in their setting, our main result is in sharp contrast to their observation. We show that, in some cases with $n$ agents, no allocation can guarantee better than $1/n$ approximation of a fair allocation when the entitlements are not necessarily equal. Furthermore, we devise a simple algorithm that ensures a $1/n$ approximation guarantee. Our second result is for a restricted version of the problem where the valuation of every agent for each good is bounded by the total value he wishes to receive in a fair allocation. Although this assumption might seem w.l.o.g, we show it enables us to find a $1/2$ approximation fair allocation via a greedy algorithm. Finally, we run some experiments on real-world data and show that, in practice, a fair allocation is likely to exist. We also support our experiments by showing positive results for two stochastic variants of the problem, namely stochastic agents and stochastic items.

In a multi-robot system, a number of autonomous robots would sense, communicate, and decide to move within a given domain to achieve a common goal. In this paper, we consider a new variant of the pursuit-evasion problem in which the robots (pursuers) each move back and forth along an orthogonal line segment inside a simple orthogonal polygon $P$. A point $p$ can be covered by a sliding robot that moves along a line segment s, if there exists a point $q\in s$ such that $\overline{pq}$ is a line segment perpendicular to $s$. In the pursuit-evasion problem, a polygonal region is given and a robot called a pursuer tries to find some mobile targets called evaders. The goal of this problem is to design a motion strategy for the pursuer such that it can detect all the evaders. We assume that $P$ includes unpredictable, moving evaders that have unbounded speed. We propose a motion-planning algorithm for a group of sliding robots, assuming that they move along the pre-located line segments with a constant speed to detect all the evaders with unbounded speed., Comment: 11 pages, 6 figures

Given a set $S$ of $n$ disjoint line segments in $\mathbb{R}^{2}$, the visibility counting problem (VCP) is to preprocess $S$ such that the number of segments in $S$ visible from any query point $p$ can be computed quickly. This problem can trivially be solved in logarithmic query time using $O(n^{4})$ preprocessing time and space. Gudmundsson and Morin proposed a 2-approximation algorithm for this problem with a tradeoff between the space and the query time. They answer any query in $O_{\epsilon}(n^{1-\alpha})$ with $O_{\epsilon}(n^{2+2\alpha})$ of preprocessing time and space, where $\alpha$ is a constant $0\leq \alpha\leq 1$, $\epsilon > 0$ is another constant that can be made arbitrarily small, and $O_{\epsilon}(f(n))=O(f(n)n^{\epsilon})$. In this paper, we propose a randomized approximation algorithm for VCP with a tradeoff between the space and the query time. We will show that for an arbitrary constants $0\leq \beta\leq \frac{2}{3}$ and $0<\delta <1$, the expected preprocessing time, the expected space, and the query time of our algorithm are $O(n^{4-3\beta}\log n)$, $O(n^{4-3\beta})$, and $O(\frac{1}{\delta^3}n^{\beta}\log n)$, respectively. The algorithm computes the number of visible segments from $p$, or $m_p$, exactly if $m_p\leq \frac{1}{\delta^3}n^{\beta}\log n$. Otherwise, it computes a $(1+\delta)$-approximation $m'_p$ with the probability of at least $1-\frac{1}{\log n}$, where $m_p\leq m'_p\leq (1+\delta)m_p$.

We consider the problem of walking in an unknown street, for a robot that has a minimal sensing capability. The robot is equipped with a sensor that only detects the discontinuities in depth information (gaps) and can locate the target point as enters in its visibility region. First, we propose an online deterministic search strategy that generates an optimal search path for the simple robot to reach the target t, starting from s. In contrast with previously known research, the path is designed without memorizing any portion of the scene has seen so far. Then, we present a randomized search strategy, based on the deterministic strategy. We prove that the expected distance traveled by the robot is at most a 5.33 times longer than the shortest path to reach the target.

In this paper we consider the problem of computing the weak visibility polygon of any query line segment $pq$ (or $WVP(pq)$) inside a given polygon $P$. Our first non-trivial algorithm runs in simple polygons and needs $O(n^3 \log n)$ time and $O(n^3)$ space in the preprocessing phase to report $WVP(pq)$ of any query line segment $pq$ in time $O(\log n + |WVP(pq)|)$. We also give an algorithm to compute the weak visibility polygon of a query line segment in a non-simple polygon with $h$ pairwise-disjoint polygonal obstacles with a total of $n$ vertices. Our algorithm needs $O(n^2 \log n)$ time and $O(n^2)$ space in the preprocessing phase and computes $WVP(pq)$ in query time of $O(n\hbar \log n + k)$, in which $\hbar$ is an output sensitive parameter of at most $\min(h,k)$, and $k = O(n^2h^2)$ is the output size. This is the best query-time result on this problem so far., Comment: 10 pages. arXiv admin note: substantial text overlap with arXiv:1309.7803

This paper considers the problem of computing the weak visibility polygon (WVP) of any query line segment pq (or WVP(pq)) inside a given simple polygon P. We present an algorithm that preprocesses P and creates a data structure from which WVP(pq) is efficiently reported in an output sensitive manner. Our algorithm needs O(n^2 log n) time and O(n^2) space in the preprocessing phase to report WVP(pq) of any query line segment pq in time O(log^2 n + |WVP(pq)|). We improve the preprocessing time and space of current results for this problem at the expense of more query time.

The spread of influence in social networks is studied in two main categories: the progressive model and the non-progressive model (see e.g. the seminal work of Kempe, Kleinberg, and Tardos in KDD 2003). While the progressive models are suitable for modeling the spread of influence in monopolistic settings, non-progressive are more appropriate for modeling non-monopolistic settings, e.g., modeling diffusion of two competing technologies over a social network. Despite the extensive work on the progressive model, non-progressive models have not been studied well. In this paper, we study the spread of influence in the non-progressive model under the strict majority threshold: given a graph $G$ with a set of initially infected nodes, each node gets infected at time $\tau$ iff a majority of its neighbors are infected at time $\tau-1$. Our goal in the \textit{MinPTS} problem is to find a minimum-cardinality initial set of infected nodes that would eventually converge to a steady state where all nodes of $G$ are infected. We prove that while the MinPTS is NP-hard for a restricted family of graphs, it admits an improved constant-factor approximation algorithm for power-law graphs. We do so by proving lower and upper bounds in terms of the minimum and maximum degree of nodes in the graph. The upper bound is achieved in turn by applying a natural greedy algorithm. Our experimental evaluation of the greedy algorithm also shows its superior performance compared to other algorithms for a set of real-world graphs as well as the random power-law graphs. Finally, we study the convergence properties of these algorithms and show that the non-progressive model converges in at most $O(|E(G)|)$ steps.