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Let $P$ be a set of $n$ points in $\mathbb{R}^d$. In the projective clustering problem, given $k, q$ and norm $\rho \in [1,\infty]$, we have to compute a set $\mathcal{F}$ of $k$ $q$-dimensional flats such that $(\sum_{p\in P}d(p, \mathcal{F})^\rho)^{1/\rho}$ is minimized; here $d(p, \mathcal{F})$ represents the (Euclidean) distance of $p$ to the closest flat in $\mathcal{F}$. We let $f_k^q(P,\rho)$ denote the minimal value and interpret $f_k^q(P,\infty)$ to be $\max_{r\in P}d(r, \mathcal{F})$. When $\rho=1,2$ and $\infty$ and $q=0$, the problem corresponds to the $k$-median, $k$-mean and the $k$-center clustering problems respectively. For every $0 < \epsilon < 1$, $S\subset P$ and $\rho \ge 1$, we show that the orthogonal projection of $P$ onto a randomly chosen flat of dimension $O(((q+1)^2\log(1/\epsilon)/\epsilon^3) \log n)$ will $\epsilon$-approximate $f_1^q(S,\rho)$. This result combines the concepts of geometric coresets and subspace embeddings based on the Johnson-Lindenstrauss Lemma. As a consequence, an orthogonal projection of $P$ to an $O(((q+1)^2 \log ((q+1)/\epsilon)/\epsilon^3) \log n)$ dimensional randomly chosen subspace $\epsilon$-approximates projective clusterings for every $k$ and $\rho$ simultaneously. Note that the dimension of this subspace is independent of the number of clusters~$k$. Using this dimension reduction result, we obtain new approximation and streaming algorithms for projective clustering problems. For example, given a stream of $n$ points, we show how to compute an $\epsilon$-approximate projective clustering for every $k$ and $\rho$ simultaneously using only $O((n+d)((q+1)^2\log ((q+1)/\epsilon))/\epsilon^3 \log n)$ space. Compared to standard streaming algorithms with $\Omega(kd)$ space requirement, our approach is a significant improvement when the number of input points and their dimensions are of the same order of magnitude.

We consider two knapsack problems. The time-Invariant Incremental Knapsack problem (IIK) is a generalization of Maximum Knapsack to a discrete multi-period setting. At each time the capacity increases and items can be added, but not removed from the knapsack. The goal is to maximize the sum of profits over all times. While IIK is strongly NP-Hard, we design a PTAS for it and some of its generalizations. The Minimum Knapsack problem (Min-K) aims at minimizing a linear function over the $0/1$ points that satisfy a single linear constraint. Despite the existence of an FPTAS, it is an open question whether one can obtain a poly-size linear formulation with constant integrality gap for Min-K. This motivated recent work on disjunctive formulations having integrality gap of at most $1+\epsilon$ for a fixed objective function. We give such a formulation of size polynomial in $n$ and subexponential in $\frac{1}{\epsilon}$.

In a standard $f$-connectivity network design problem, we are given an undirected graph $G=(V,E)$, a cut-requirement function $f:2^V \rightarrow {\mathbb{N}}$, and non-negative costs $c(e)$ for all $e \in E$. We are then asked to find a minimum-cost vector $x \in {\mathbb{N}}^E$ such that $x(\delta(S)) \geq f(S)$ for all $S \subseteq V$. We focus on the class of such problems where $f$ is a proper function. This encodes many well-studied NP-hard problems such as the generalized survivable network design problem. In this paper we present the first strongly polynomial time FPTAS for solving the LP relaxation of the standard IP formulation of the $f$-connectivity problem with general proper functions $f$. Implementing Jain's algorithm, this yields a strongly polynomial time $(2+\epsilon)$-approximation for the generalized survivable network design problem (where we consider rounding up of rationals an arithmetic operation).

In this paper we investigate quasi-stationary distributions {\mu}_N of stochastic approximation algorithms with constant step size which can be viewed as random perturbations of a time-continuous dynamical system. Inspired by ecological models these processes have a closed absorbing set corresponding to extinction. Under some large deviation assumptions and the existence of an interior attractor for the ODE, we show that the weak* limit points of the QSD {\mu}_N are invariant measures for the ODE with support in the interior attractors.

The Joint Replenishment Problem (JRP) is a fundamental optimization problem in supply-chain management, concerned with optimizing the flow of goods over time from a supplier to retailers. Over time, in response to demands at the retailers, the supplier sends shipments, via a warehouse, to the retailers. The objective is to schedule shipments to minimize the sum of shipping costs and retailers' waiting costs. We study the approximability of JRP with deadlines, where instead of waiting costs the retailers impose strict deadlines. We study the integrality gap of the standard linear-program (LP) relaxation, giving a lower bound of 1.207, and an upper bound and approximation ratio of 1.574. The best previous upper bound and approximation ratio was 1.667; no lower bound was previously published. For the special case when all demand periods are of equal length we give an upper bound of 1.5, a lower bound of 1.2, and show APX-hardness.

Time-varying coverage, namely sweep coverage is a recent development in the area of wireless sensor networks, where a small number of mobile sensors sweep or monitor comparatively large number of locations periodically. In this article we study barrier sweep coverage with mobile sensors where the barrier is considered as a finite length continuous curve on a plane. The coverage at every point on the curve is time-variant. We propose an optimal solution for sweep coverage of a finite length continuous curve. Usually energy source of a mobile sensor is battery with limited power, so energy restricted sweep coverage is a challenging problem for long running applications. We propose an energy restricted sweep coverage problem where every mobile sensors must visit an energy source frequently to recharge or replace its battery. We propose a $\frac{13}{3}$-approximation algorithm for this problem. The proposed algorithm for multiple curves achieves the best possible approximation factor 2 for a special case. We propose a 5-approximation algorithm for the general problem. As an application of the barrier sweep coverage problem for a set of line segments, we formulate a data gathering problem. In this problem a set of mobile sensors is arbitrarily monitoring the line segments one for each. A set of data mules periodically collects the monitoring data from the set of mobile sensors. We prove that finding the minimum number of data mules to collect data periodically from every mobile sensor is NP-hard and propose a 3-approximation algorithm to solve it.

We consider optimal route planning when the objective function is a general nonlinear and non-monotonic function. Such an objective models user behavior more accurately, for example, when a user is risk-averse, or the utility function needs to capture a penalty for early arrival. It is known that as nonlinearity arises, the problem becomes NP-hard and little is known about computing optimal solutions when in addition there is no monotonicity guarantee. We show that an approximately optimal non-simple path can be efficiently computed under some natural constraints. In particular, we provide a fully polynomial approximation scheme under hop constraints. Our approximation algorithm can extend to run in pseudo-polynomial time under a more general linear constraint that sometimes is useful. As a by-product, we show that our algorithm can be applied to the problem of finding a path that is most likely to be on time for a given deadline.

We consider approximation of diameter of a set $S$ of $n$ points in dimension $m$. E$\tilde{g}$ecio$\tilde{g}$lu and Kalantari \cite{kal} have shown that given any $p \in S$, by computing its farthest in $S$, say $q$, and in turn the farthest point of $q$, say $q'$, we have ${\rm diam}(S) \leq \sqrt{3} d(q,q')$. Furthermore, iteratively replacing $p$ with an appropriately selected point on the line segment $pq$, in at most $t \leq n$ additional iterations, the constant bound factor is improved to $c_*=\sqrt{5-2\sqrt{3}} \approx 1.24$. Here we prove when $m=2$, $t=1$. This suggests in practice a few iterations may produce good solutions in any dimension. Here we also propose a randomized version and present large scale computational results with these algorithm for arbitrary $m$. The algorithms outperform many existing algorithms. On sets of data as large as $1,000,000$ points, the proposed algorithms compute solutions to within an absolute error of $10^{-4}$.

We consider approximation of diameter of a set $S$ of $n$ points in dimension $m$. E$\tilde{g}$ecio$\tilde{g}$lu and Kalantari \cite{kal} have shown that given any $p \in S$, by computing its farthest in $S$, say $q$, and in turn the farthest point of $q$, say $q'$, we have ${\rm diam}(S) \leq \sqrt{3} d(q,q')$. Furthermore, iteratively replacing $p$ with an appropriately selected point on the line segment $pq$, in at most $t \leq n$ additional iterations, the constant bound factor is improved to $c_*=\sqrt{5-2\sqrt{3}} \approx 1.24$. Here we prove when $m=2$, $t=1$. This suggests in practice a few iterations may produce good solutions in any dimension. Here we also propose a randomized version and present large scale computational results with these algorithm for arbitrary $m$. The algorithms outperform many existing algorithms. On sets of data as large as $1,000,000$ points, the proposed algorithms compute solutions to within an absolute error of $10^{-4}$.

In the past few years powerful generalizations to the Euclidean k-means problem have been made, such as Bregman clustering [7], co-clustering (i.e., simultaneous clustering of rows and columns of an input matrix) [9,18], and tensor clustering [8,34]. Like k-means, these more general problems also suffer from the NP-hardness of the associated optimization. Researchers have developed approximation algorithms of varying degrees of sophistication for k-means, k-medians, and more recently also for Bregman clustering [2]. However, there seem to be no approximation algorithms for Bregman co- and tensor clustering. In this paper we derive the first (to our knowledge) guaranteed methods for these increasingly important clustering settings. Going beyond Bregman divergences, we also prove an approximation factor for tensor clustering with arbitrary separable metrics. Through extensive experiments we evaluate the characteristics of our method, and show that it also has practical impact.

We provide polynomial-time approximately optimal Bayesian mechanisms for makespan minimization on unrelated machines as well as for max-min fair allocations of indivisible goods, with approximation factors of $2$ and $\min\{m-k+1, \tilde{O}(\sqrt{k})\}$ respectively, matching the approximation ratios of best known polynomial-time \emph{algorithms} (for max-min fairness, the latter claim is true for certain ratios of the number of goods $m$ to people $k$). Our mechanisms are obtained by establishing a polynomial-time approximation-sensitive reduction from the problem of designing approximately optimal {\em mechanisms} for some arbitrary objective ${\cal O}$ to that of designing bi-criterion approximation {\em algorithms} for the same objective ${\cal O}$ plus a linear allocation cost term. Our reduction is itself enabled by extending the celebrated "equivalence of separation and optimization"[GLSS81,KP80] to also accommodate bi-criterion approximations. Moreover, to apply the reduction to the specific problems of makespan and max-min fairness we develop polynomial-time bi-criterion approximation algorithms for makespan minimization with costs and max-min fairness with costs, adapting the algorithms of [ST93], [BD05] and [AS07] to the type of bi-criterion approximation that is required by the reduction.

Approximation algorithms for classical constraint satisfaction problems are one of the main research areas in theoretical computer science. Here we define a natural approximation version of the QMA-complete local Hamiltonian problem and initiate its study. We present two main results. The first shows that a non-trivial approximation ratio can be obtained in the class NP using product states. The second result (which builds on the first one), gives a polynomial time (classical) algorithm providing a similar approximation ratio for dense instances of the problem. The latter result is based on an adaptation of the "exhaustive sampling method" by Arora et al. [J. Comp. Sys. Sci. 58, p.193 (1999)] to the quantum setting, and might be of independent interest.

A critical problem in the emerging high-throughput genotyping protocols is to minimize the number of polymerase chain reaction (PCR) primers required to amplify the single nucleotide polymorphism loci of interest. In this paper we study PCR primer set selection with amplification length and uniqueness constraints from both theoretical and practical perspectives. We give a greedy algorithm that achieves a logarithmic approximation factor for the problem of minimizing the number of primers subject to a given upperbound on the length of PCR amplification products. We also give, using randomized rounding, the first non-trivial approximation algorithm for a version of the problem that requires unique amplification of each amplification target. Empirical results on randomly generated testcases as well as testcases extracted from the from the National Center for Biotechnology Information's genomic databases show that our algorithms are highly scalable and produce better results compared to previous heuristics.

Raghavendra (STOC 2008) gave an elegant and surprising result: if Khot's Unique Games Conjecture (STOC 2002) is true, then for every constraint satisfaction problem (CSP), the best approximation ratio is attained by a certain simple semidefinite programming and a rounding scheme for it. In this paper, we show that similar results hold for constant-time approximation algorithms in the bounded-degree model. Specifically, we present the followings: (i) For every CSP, we construct an oracle that serves an access, in constant time, to a nearly optimal solution to a basic LP relaxation of the CSP. (ii) Using the oracle, we give a constant-time rounding scheme that achieves an approximation ratio coincident with the integrality gap of the basic LP. (iii) Finally, we give a generic conversion from integrality gaps of basic LPs to hardness results. All of those results are \textit{unconditional}. Therefore, for every bounded-degree CSP, we give the best constant-time approximation algorithm among all. A CSP instance is called $\epsilon$-far from satisfiability if we must remove at least an $\epsilon$-fraction of constraints to make it satisfiable. A CSP is called testable if there is a constant-time algorithm that distinguishes satisfiable instances from $\epsilon$-far instances with probability at least $2/3$. Using the results above, we also derive, under a technical assumption, an equivalent condition under which a CSP is testable in the bounded-degree model.

The link scheduling in wireless multi-hop networks is addressed. Different from most of work that adopt the protocol interference model which merely take consideration of packet collisions, our proposed algorithms use the physical interference model to reflect the aggregated signal to interference and noise ratio (SINR), which is a more accurate abstraction of the real scenario. We first propose a centralized scheduling method based on the Integer Linear Programming (ILP) and resolve it by an approximate solution based on the randomized rounding method. The probability bound of getting a guaranteed approximate factor is given. We then extend the centralized algorithm to a distributed solution, which is favorable in wireless networks. It is proven that with the distributed scheduling method, all links can transmit without interference, and the approximate ratio of the algorithm is also given.

We determine the jet vertex for Mueller-Navelet jets and forward jets in the small-cone approximation for two particular choices of jet algoritms: the kt algorithm and the cone algorithm. These choices are motivated by the extensive use of such algorithms in the phenomenology of jets. The differences with the original calculations of the small-cone jet vertex by Ivanov and Papa, which is found to be equivalent to a formerly algorithm proposed by Furman, are shown at both analytic and numerical level, and turn out to be sizeable. A detailed numerical study of the error introduced by the small-cone approximation is also presented, for various observables of phenomenological interest. For values of the jet "radius" R=0.5, the use of the small-cone approximation amounts to an error of about 5% at the level of cross section, while it reduces to less than 2% for ratios of distributions such as those involved in the measure of the azimuthal decorrelation of dijets.

Network applications, such as multimedia streaming and video conferencing, impose growing requirements over Quality of Service (QoS), including bandwidth, delay, jitter, etc. Meanwhile, networks are expected to be load-balanced, energy-efficient, and resilient to some degree of failures. It is observed that the above requirements could be better met with multiple disjoint QoS paths than a single one. Let $G=(V,\, E)$ be a digraph with nonnegative integral cost and delay on every edge, $s,\, t\in V$ be two specified vertices, and $D\in\mathbb{Z}_{0}^{+}$ be a delay bound (or some other constraint), the \emph{$k$ Disjoint Restricted Shortest Path} ($k$\emph{RSP})\emph{ Problem} is computing $k$ disjoint paths between $s$ and $t$ with total cost minimized and total delay bounded by $D$. Few efficient algorithms have been developed because of the hardness of the problem. In this paper, we propose efficient algorithms with provable performance guarantees for the $k$RSP problem. We first present a pseudo-polynomial-time approximation algorithm with a bifactor approximation ratio of $(1,\,2)$, then improve the algorithm to polynomial time with a bifactor ratio of $(1+\epsilon,\,2+\epsilon)$ for any fixed $\epsilon>0$, which is better than the current best approximation ratio $(O(1+\gamma),\, O(1+\frac{1}{\gamma})\})$ for any fixed $\gamma>0$ \cite{orda2004efficient}. To the best of our knowledge, this is the first constant-factor algorithm that almost strictly obeys the constraint for the $k$RSP problem.

In this paper, we prove that the Max-Morse Matching Problem is approximable, thus resolving an open problem posed by Joswig and Pfetsch. We describe two different approximation algorithms for the Max-Morse Matching Problem. For $D$-dimensional simplicial complexes, we obtain a $\frac{(D+1)}{(D^2+D+1)}$-factor approximation ratio using a simple edge reorientation algorithm that removes cycles. Our second result is an algorithm that provides a $\frac{2}{D}$-factor approximation for simplicial manifolds by processing the simplices in increasing order of dimension. One application of these algorithms is towards efficient homology computation of simplicial complexes. Experiments using a prototype implementation on several datasets indicate that the algorithm computes near optimal results.

We present approximation algorithms with O(n^3) processing time for the minimum vertex and edge guard problems in simple polygons. It is improved from previous O(n^4) time algorithms of Ghosh. For simple polygon, there are O(n^3) visibility regions, thus any approximation algorithm for the set covering problem with approximation ratio of log(n) can be used for the approximation of n vertex and edge guard problems with O(n^3) visibility sequence. We prove that the visibility of all points in simple polygons is guaranteed by covering O(n^2) sinks from vertices and edges : It comes to O(n^3) time bound.

We study the approximability of instances of the minimum entropy set cover problem, parameterized by the average frequency of a random element in the covering sets. We analyze an algorithm combining a greedy approach with another one biased towards large sets. The algorithm is controled by the percentage of elements to which we apply the biased approach. The optimal parameter choice has a phase transition around average density $e$ and leads to improved approximation guarantees when average element frequency is less than $e$.

This article concerns the following question arising in computational evolutionary biology. For a given subclass of phylogenetic networks, what is the maximum value of 0

In this paper modified variants of the sparse Fourier transform algorithms from [14] are presented which improve on the approximation error bounds of the original algorithms. In addition, simple methods for extending the improved sparse Fourier transforms to higher dimensional settings are developed. As a consequence, approximate Fourier transforms are obtained which will identify a near-optimal k-term Fourier series for any given input function, $f : [0, 2 pi] -> C, in O(k^2 \cdot D^4)$ time (neglecting logarithmic factors). Faster randomized Fourier algorithm variants with runtime complexities that scale linearly in the sparsity parameter k are also presented.

This paper has been merged into 1110.4604.

We consider the {\em mobile facility location} (\mfl) problem. We are given a set of facilities and clients located in a common metric space. The goal is to move each facility from its initial location to a destination and assign each client to the destination of some facility so as to minimize the sum of the movement-costs of the facilities and the client-assignment costs. This abstracts facility-location settings where one has the flexibility of moving facilities from their current locations to other destinations so as to serve clients more efficiently by reducing their assignment costs. We give the first {\em local-search based} approximation algorithm for this problem and achieve the best-known approximation guarantee. Our main result is $(3+\epsilon)$-approximation for this problem for any constant $\epsilon>0$ using local search. The previous best guarantee was an 8-approximation algorithm based on LP-rounding. Our guarantee {\em matches} the best-known approximation guarantee for the $k$-median problem. Since there is an approximation-preserving reduction from the $k$-median problem to \mfl, any improvement of our result would imply an analogous improvement for the $k$-median problem. Furthermore, {\em our analysis is tight} (up to $o(1)$ factors) since the tight example for the local-search based 3-approximation algorithm for $k$-median can be easily adapted to show that our local-search algorithm has a tight approximation ratio of 3. One of the chief novelties of the analysis is that in order to generate a suitable collection of local-search moves whose resulting inequalities yield the desired bound on the cost of a local-optimum, we define a tree-like structure that (loosely speaking) functions as a "recursion tree", using which we spawn off local-search moves by exploring this tree to a constant depth.

We are given a set of jobs, each one specified by its release date, its deadline and its processing volume (work), and a single (or a set of) speed-scalable processor(s). We adopt the standard model in speed-scaling in which if a processor runs at speed s then the energy consumption is s^{\alpha} per time unit, where \alpha>1. Our goal is to find a schedule respecting the release dates and the deadlines of the jobs so that the total energy consumption is minimized. While most previous works have studied the preemptive case of the problem, where a job may be interrupted and resumed later, we focus on the non-preemptive case where once a job starts its execution, it has to continue until its completion without any interruption. We propose improved approximation algorithms for particular instances of the multiprocessor non-preemptive speed-scaling problem.

Routing and scheduling problems are fundamental problems in combinatorial optimization, and also have many applications. Most variations of these problems are NP-Hard, so we need to use heuristics to solve these problems on large instances, which are fast and yet come close to the optimal value. In this thesis, we study the design and analysis of approximation algorithms for such problems. We focus on two important class of problems. The first is the Unsplittable Flow Problem and some of its variants and the second is the Resource Allocation for Job Scheduling Problem and some of its variants. The first is a packing problem, whereas the second is a covering problem.

We study the problem of approximating the quality of a disperser. A bipartite graph $G$ on $([N],[M])$ is a $(\rho N,(1-\delta)M)$-disperser if for any subset $S\subseteq [N]$ of size $\rho N$, the neighbor set $\Gamma(S)$ contains at least $(1-\delta)M$ distinct vertices. Our main results are strong integrality gaps in the Lasserre hierarchy and an approximation algorithm for dispersers. \begin{enumerate} \item For any $\alpha>0$, $\delta>0$, and a random bipartite graph $G$ with left degree $D=O(\log N)$, we prove that the Lasserre hierarchy cannot distinguish whether $G$ is an $(N^{\alpha},(1-\delta)M)$-disperser or not an $(N^{1-\alpha},\delta M)$-disperser. \item For any $\rho>0$, we prove that there exist infinitely many constants $d$ such that the Lasserre hierarchy cannot distinguish whether a random bipartite graph $G$ with right degree $d$ is a $(\rho N, (1-(1-\rho)^d)M)$-disperser or not a $(\rho N, (1-\Omega(\frac{1-\rho}{\rho d + 1-\rho}))M)$-disperser. We also provide an efficient algorithm to find a subset of size exact $\rho N$ that has an approximation ratio matching the integrality gap within an extra loss of $\frac{\min\{\frac{\rho}{1-\rho},\frac{1-\rho}{\rho}\}}{\log d}$. \end{enumerate} Our method gives an integrality gap in the Lasserre hierarchy for bipartite expanders with left degree~$D$. $G$ on $([N],[M])$ is a $(\rho N,a)$-expander if for any subset $S\subseteq [N]$ of size $\rho N$, the neighbor set $\Gamma(S)$ contains at least $a \cdot \rho N$ distinct vertices. We prove that for any constant $\epsilon>0$, there exist constants $\epsilon'

Consider the Maximum Weight Independent Set problem for rectangles: given a family of weighted axis-parallel rectangles in the plane, find a maximum-weight subset of non-overlapping rectangles. The problem is notoriously hard both in the approximation and in the parameterized setting. The best known polynomial-time approximation algorithms achieve super-constant approximation ratios [Chalermsook and Chuzhoy, SODA 2009; Chan and Har-Peled, Discrete & Comp. Geometry 2012], even though there is a $(1+\epsilon)$-approximation running in quasi-polynomial time [Adamaszek and Wiese, FOCS 2013; Chuzhoy and Ene, FOCS 2016]. When parameterized by the target size of the solution, the problem is $\mathsf{W}[1]$-hard even in the unweighted setting [Marx, FOCS 2007]. To achieve tractability, we study the following shrinking model: one is allowed to shrink each input rectangle by a multiplicative factor $1-\delta$ for some fixed $\delta>0$, but the performance is still compared against the optimal solution for the original, non-shrunk instance. We prove that in this regime, the problem admits an EPTAS with running time $f(\epsilon,\delta)\cdot n^{\mathcal{O}(1)}$, and an FPT algorithm with running time $f(k,\delta)\cdot n^{\mathcal{O}(1)}$, in the setting where a maximum-weight solution of size at most $k$ is to be computed. This improves and significantly simplifies a PTAS given earlier for this problem [Adamaszek et al., APPROX 2015], and provides the first parameterized results for the shrinking model. Furthermore, we explore kernelization in the shrinking model, by giving efficient kernelization procedures for several variants of the problem when the input rectangles are squares.

We consider the distributed version of the Multiple Knapsack Problem (MKP), where $m$ items are to be distributed amongst $n$ processors, each with a knapsack. We propose different distributed approximation algorithms with a tradeoff between time and message complexities. The algorithms are based on the greedy approach of assigning the best item to the knapsack with the largest capacity. These algorithms obtain a solution with a bound of $\frac{1}{n+1}$ times the optimum solution, with either $\mathcal{O}\left(m\log n\right)$ time and $\mathcal{O}\left(m n\right)$ messages, or $\mathcal{O}\left(m\right)$ time and $\mathcal{O}\left(mn^{2}\right)$ messages.

We consider the stochastic $k$-TSP problem where rewards at vertices are random and the objective is to minimize the expected length of a tour that collects reward $k$. We present an adaptive $O(\log k)$-approximation algorithm, and a non-adaptive $O(\log^2k)$-approximation algorithm. We also show that the adaptivity gap of this problem is between $e$ and $O(\log^2k)$.

Given two rooted phylogenetic trees on the same set of taxa X, the Maximum Agreement Forest problem (MAF) asks to find a forest that is, in a certain sense, common to both trees and has a minimum number of components. The Maximum Acyclic Agreement Forest problem (MAAF) has the additional restriction that the components of the forest cannot have conflicting ancestral relations in the input trees. There has been considerable interest in the special cases of these problems in which the input trees are required to be binary. However, in practice, phylogenetic trees are rarely binary, due to uncertainty about the precise order of speciation events. Here, we show that the general, nonbinary version of MAF has a polynomial-time 4-approximation and a fixed-parameter tractable (exact) algorithm that runs in O(4^k poly(n)) time, where n = |X| and k is the number of components of the agreement forest minus one. Moreover, we show that a c-approximation algorithm for nonbinary MAF and a d-approximation algorithm for the classical problem Directed Feedback Vertex Set (DFVS) can be combined to yield a d(c+3)-approximation for nonbinary MAAF. The algorithms for MAF have been implemented and made publicly available.

The Final Proceedings for Algorithms for Approximation IV (A4A4), 16 July 2001-20 July 2001, a multidisciplinary conference addressing many areas of interest to the Air Force. Of primary interest are the potential applications to Modeling and Simulation. Specifically, the topics to be covered include in the following four major areas: Algorithms, Efficiency, Software, and Applications. Each major topic is divided into subtopics as follows: Algorithms- Approximation of Functions, Data Fitting, Geometric and Surface Modelling, Splines, Wavelets, Radial Basis Functions, Support Vector Machines, Norms and Metrics, Errors in Data, Uncertainty Estimation; Efficiency- Numerical Analysis, Parallel Processing; Software- Standards, Libraries, New Routines, World Wide Web; Applications- Metrology (Science of Measurement), Data Fusion, Neural Networks and Intelligent Systems, Spherical Data and Geodetics, Medical Data.

The forty-one papers in this volume have been arranged into three primary sections: I Development of Algorithms, II Applications, and Catalogue of Algorithms. The first two sections have been subdivided into eight groups: (1) Spline approximation, (2) Polynomial and piecewise polynomial approximation, (3) Interpolation, (4) Smoothing and constraint methods, (5) Complex approximation, (6) Computer-aided design and geometric modelling, (7) Applications in numerical analysis, and (8) Applications in other disciplines. Such a division into sections, while giving the book a useful structure, is somewhat arbitrary, and we apologize to any authors who may feel that their work has been incorrectly categorized. Several papers could have been placed in up to three groups (especially spline approximation, piecewise polynomial approximation, and computer-aided design). Moreover the CAD group, which we have placed in the Applications section could perfectly well have been placed in Section I. Although there is no group headed nonlinear approximation , there are several 'nonlinear algorithms' (in Section II in particular), and of course the complex algorithms (in group 5 and elsewhere) could have come under this heading

We will survey recent work in the design of approximation algorithms for several discrete stochastic optimization problems, with a particular focus on 2-stage problems with recourse. In each of the problems we discuss, we are given a probability distribution over inputs, and the aim is to find a feasible solution that minimizes the expected cost of the solution found (with respect to the input distribution); an approximation algorithm finds a solution that is guaranteed to be nearly optimal. Among the specific problems that we shall discuss are stochastic generalizations of the traditional deterministic facility location problem, a simple single-machine scheduling problem, and the traveling salesman problem.These results build on techniques initially developed in the context of deterministic approximation, including rounding approaches, primal-dual algorithms, as well as a simple random sampling technique. Furthermore, although the focus of this stream of work was for discrete optimization problems, new insights for solving 2-stage stochastic linear programming problems were gained along the way. ©2008 Microsoft Corporation. All rights reserved.

Book Source: Digital Library of India Item 2015.193454 dc.contributor.author: Samarjit Chakraborty dc.date.accessioned: 2015-07-08T02:45:22Z dc.date.available: 2015-07-08T02:45:22Z dc.date.digitalpublicationdate: 2005-09-27 dc.identifier.barcode: 1990010091784 dc.identifier.origpath: /rawdataupload/upload/0091/784 dc.identifier.copyno: 1 dc.identifier.uri: http://www.new.dli.ernet.in/handle/2015/193454 dc.description.scannerno: 14 dc.description.scanningcentre: IIIT, Allahabad dc.description.main: 1 dc.description.tagged: 0 dc.description.totalpages: 82 dc.format.mimetype: application/pdf dc.language.iso: English dc.publisher: Indian Institute Of Technology Kanpur dc.rights: Out_of_copyright dc.source.library: Indian Institute Of Technology Kanpur dc.subject.classification: Technology dc.subject.classification: Engineering. Technology In General dc.subject.classification: Computer Science & Engineering dc.title: Approximation Algorithms For 3-d Common Substructure Identification In Drug Amp Protein Molecules

Many data dissemination and publish-subscribe systems that guarantee the privacy and authenticity of the participants rely on symmetric key cryptography. An important problem in such a system is to maintain the shared group key as the group membership changes. We consider the problem of determining a key hierarchy that minimizes the average communication cost of an update, given update frequencies of the group members and an edge-weighted undirected graph that captures routing costs. We first present a polynomial-time approximation scheme for minimizing the average number of multicast messages needed for an update. We next show that when routing costs are considered, the problem is NP-hard even when the underlying routing network is a tree network or even when every group member has the same update frequency. Our main result is a polynomial time constant-factor approximation algorithm for the general case where the routing network is an arbitrary weighted graph and group members have nonuniform update frequencies.

Given a convex polygon $P$ with $n$ vertices, the two-center problem is to find two congruent closed disks of minimum radius such that they completely cover $P$. We propose an algorithm for this problem in the streaming setup, where the input stream is the vertices of the polygon in clockwise order. It produces a radius $r$ satisfying $r\leq2r_{opt}$ using $O(1)$ space, where $r_{opt}$ is the optimum solution. Next, we show that in non-streaming setup, we can improve the approximation factor by $r\leq 1.84 r_{opt}$, maintaining the time complexity of the algorithm to $O(n)$, and using $O(1)$ extra space in addition to the space required for storing the input.

We introduce the {\it Bipartite Multi-cut} problem. This is a generalization of the {\it st-Min-cut} problem, is similar to the {\it Multi-cut} problem (except for more stringent requirements) and also turns out to be an immediate generalization of the {\it Min UnCut} problem. We prove that this problem is {\bf NP}-hard and then present LP and SDP based approximation algorithms. While the LP algorithm is based on the Garg-Vazirani-Yannakakis algorithm for {\it Multi-cut}, the SDP algorithm uses the {\it Structure Theorem} of $\ell_2^2$ Metrics.

We will survey recent work in the design of approximation algorithms for several discrete stochastic optimization problems, with a particular focus on 2-stage problems with recourse. In each of the problems we discuss, we are given a probability distribution over inputs, and the aim is to find a feasible solution that minimizes the expected cost of the solution found (with respect to the input distribution); an approximation algorithm finds a solution that is guaranteed to be nearly optimal. Among the specific problems that we shall discuss are stochastic generalizations of the traditional deterministic facility location problem, a simple single-machine scheduling problem, and the traveling salesman problem.These results build on techniques initially developed in the context of deterministic approximation, including rounding approaches, primal-dual algorithms, as well as a simple random sampling technique. Furthermore, although the focus of this stream of work was for discrete optimization problems, new insights for solving 2-stage stochastic linear programming problems were gained along the way. ©2008 Microsoft Corporation. All rights reserved.

We consider the problem of finding a lowest cost dominating set in a given disk graph containing $n$ disks. The problem has been extensively studied on subclasses of disk graphs, yet the best known approximation for disk graphs has remained $O(\log n)$ -- a bound that is asymptotically no better than the general case. We improve the status quo in two ways: for the unweighted case, we show how to obtain a PTAS using the framework recently proposed (independently)by Mustafa and Ray [SoCG 09] and by Chan and Har-Peled [SoCG 09]; for the weighted case where each input disk has an associated rational weight with the objective of finding a minimum cost dominating set, we give a randomized algorithm that obtains a dominating set whose weight is within a factor $2^{O(\log^* n)}$ of a minimum cost solution, with high probability -- the technique follows the framework proposed recently by Varadarajan [STOC 10].

The radius and diameter are fundamental graph parameters. They are defined as the minimum and maximum of the eccentricities in a graph, respectively, where the eccentricity of a vertex is the largest distance from the vertex to another node. In directed graphs, there are several versions of these problems. For instance, one may choose to define the eccentricity of a node in terms of the largest distance into the node, out of the node, the sum of the two directions (i.e. roundtrip) and so on. All versions of diameter and radius can be solved via solving all-pairs shortest paths (APSP), followed by a fast postprocessing step. Solving APSP, however, on $n$-node graphs requires $\Omega(n^2)$ time even in sparse graphs, as one needs to output $n^2$ distances. Motivated by known and new negative results on the impossibility of computing these measures exactly in general graphs in truly subquadratic time, under plausible assumptions, we search for \emph{approximation} and \emph{fixed parameter subquadratic} algorithms, and for reasons why they do not exist. Our results include: - Truly subquadratic approximation algorithms for most of the versions of Diameter and Radius with \emph{optimal} approximation guarantees (given truly subquadratic time), under plausible assumptions. In particular, there is a $2$-approximation algorithm for directed Radius with one-way distances that runs in $\tilde{O}(m\sqrt{n})$ time, while a $(2-\delta)$-approximation algorithm in $O(n^{2-\epsilon})$ time is unlikely. - On graphs with treewidth $k$, we can solve the problems in $2^{O(k\log{k})}n^{1+o(1)}$ time. We show that these algorithms are near optimal since even a $(3/2-\delta)$-approximation algorithm that runs in time $2^{o(k)}n^{2-\epsilon}$ would refute the plausible assumptions.

We parallelize several previously proposed algorithms for the minimum routing cost spanning tree problem and some related problems.

Several problems that are NP-hard on general graphs are efficiently solvable on graphs with bounded treewidth. Efforts have been made to generalize treewidth and the related notion of pathwidth to digraphs. Directed treewidth, DAG-width and Kelly-width are some such notions which generalize treewidth, whereas directed pathwidth generalizes pathwidth. Each of these digraph width measures have an associated decomposition structure. In this paper, we present approximation algorithms for all these digraph width parameters. In particular, we give an O(sqrt{logn})-approximation algorithm for directed treewidth, and an O({\log}^{3/2}{n})-approximation algorithm for directed pathwidth, DAG-width and Kelly-width. Our algorithms construct the corresponding decompositions whose widths are within the above mentioned approximation factors.

In this paper we study algorithms to find a Gaussian approximation to a target measure defined on a Hilbert space of functions; the target measure itself is defined via its density with respect to a reference Gaussian measure. We employ the Kullback-Leibler divergence as a distance and find the best Gaussian approximation by minimizing this distance. It then follows that the approximate Gaussian must be equivalent to the Gaussian reference measure, defining a natural function space setting for the underlying calculus of variations problem. We introduce a computational algorithm which is well-adapted to the required minimization, seeking to find the mean as a function, and parameterizing the covariance in two different ways: through low rank perturbations of the reference covariance; and through Schr\"odinger potential perturbations of the inverse reference covariance. Two applications are shown: to a nonlinear inverse problem in elliptic PDEs, and to a conditioned diffusion process. We also show how the Gaussian approximations we obtain may be used to produce improved pCN-MCMC methods which are not only well-adapted to the high-dimensional setting, but also behave well with respect to small observational noise (resp. small temperatures) in the inverse problem (resp. conditioned diffusion).

Motivated by applications in grid computing and project management, we study multiprocessor scheduling in scenarios where there is uncertainty in the successful execution of jobs when assigned to processors. We consider the problem of multiprocessor scheduling under uncertainty, in which we are given n unit-time jobs and m machines, a directed acyclic graph C giving the dependencies among the jobs, and for every job j and machine i, the probability p_{ij} of the successful completion of job j when scheduled on machine i in any given particular step. The goal of the problem is to find a schedule that minimizes the expected makespan, that is, the expected completion time of all the jobs. The problem of multiprocessor scheduling under uncertainty was introduced by Malewicz and was shown to be NP-hard even when all the jobs are independent. In this paper, we present polynomial-time approximation algorithms for the problem, for special cases of the dag C. We obtain an O(log(n))-approximation for the case of independent jobs, an O(log(m)log(n)log(n+m)/loglog(n+m))-approximation when C is a collection of disjoint chains, an O(log(m)log^2(n))-approximation when C is a collection of directed out- or in-trees, and an O(log(m)log^2(n)log(n+m)/loglog(n+m))-approximation when C is a directed forest.

Given an undirected graph on a node set $V$ and positive integers $k$ and $m$, a $k$-connected $m$-dominating set ($(k,m)$-CDS) is defined as a subset $S$ of $V$ such that each node in $V \setminus S$ has at least $m$ neighbors in $S$, and a $k$-connected subgraph is induced by $S$. The weighted $(k,m)$-CDS problem is to find a minimum weight $(k,m)$-CDS in a given node-weighted graph. The problem is called the unweighted $(k,m)$-CDS problem if the objective is to minimize the cardinality of a $(k,m)$-CDS. These problems have been actively studied for unit disk graphs, motivated by the application of constructing a virtual backbone in a wireless ad hoc network. However, constant-approximation algorithms are known only for $k \leq 3$ in the unweighted $(k,m)$-CDS problem, and for $(k,m)=(1,1)$ in the weighted $(k,m)$-CDS problem. In this paper, we consider the case in which $m \geq k$, and we present a simple $O(5^k k!)$-approximation algorithm for the unweighted $(k,m)$-CDS problem, and a primal-dual $O(k^2 \log k)$-approximation algorithm for the weighted $(k,m)$-CDS problem. Both algorithms achieve constant approximation factors when $k$ is a fixed constant.

We study the prize-collecting version of the Node-weighted Steiner Tree problem (NWPCST) restricted to planar graphs. We give a new primal-dual Lagrangian-multiplier-preserving (LMP) 3-approximation algorithm for planar NWPCST. We then show a ($2.88 + \epsilon$)-approximation which establishes a new best approximation guarantee for planar NWPCST. This is done by combining our LMP algorithm with a threshold rounding technique and utilizing the 2.4-approximation of Berman and Yaroslavtsev for the version without penalties. We also give a primal-dual 4-approximation algorithm for the more general forest version using techniques introduced by Hajiaghay and Jain.

The power dominating set (PDS) problem is the following extension of the well-known dominating set problem: find a smallest-size set of nodes $S$ that power dominates all the nodes, where a node $v$ is power dominated if (1) $v$ is in $S$ or $v$ has a neighbor in $S$, or (2) $v$ has a neighbor $w$ such that $w$ and all of its neighbors except $v$ are power dominated. We show a hardness of approximation threshold of $2^{\log^{1-\epsilon}{n}}$ in contrast to the logarithmic hardness for the dominating set problem. We give an $O(\sqrt{n})$ approximation algorithm for planar graphs, and show that our methods cannot improve on this approximation guarantee. Finally, we initiate the study of PDS on directed graphs, and show the same hardness threshold of $2^{\log^{1-\epsilon}{n}}$ for directed \emph{acyclic} graphs. Also we show that the directed PDS problem can be solved optimally in linear time if the underlying undirected graph has bounded tree-width.

We consider the Generalized Bin Covering (GBC) problem: We are given $m$ bin types, where each bin of type $i$ has profit $p_i$ and demand $d_i$. Furthermore, there are $n$ items, where item $j$ has size $s_j$. A bin of type $i$ is covered if the set of items assigned to it has total size at least the demand $d_i$. In that case, the profit of $p_i$ is earned and the objective is to maximize the total profit. To the best of our knowledge, only the cases $p_i = d_i = 1$ (Bin Covering) and $p_i = d_i$ (Variable-Sized Bin Covering (VSBC)) have been treated before. We study two models of bin supply: In the unit supply model, we have exactly one bin of each type, i.\,e., we have individual bins. By contrast, in the infinite supply model, we have arbitrarily many bins of each type. Clearly, the unit supply model is a generalization of the infinite supply model. To the best of our knowledge the unit supply model has not been studied yet. Our results for the unit supply model hold not only asymptotically, but for all instances. This contrasts most of the previous work on \prob{Bin Covering}. We prove that there is a combinatorial 5-approximation algorithm for GBC with unit supply, which has running time $\bigO{nm\sqrt{m+n}}$. Furthermore, for VSBC we show that the natural and fast Next Fit Decreasing ($\NFD$) algorithm is a 9/4-approximation in the unit supply model. The bound is tight for the algorithm and close to being best-possible. We show that there is an AFPTAS for VSBC in the \emph{infinite} supply model.

Book Source: Digital Library of India Item 2015.239354 dc.contributor.author: Samarjit Chakraborty dc.date.accessioned: 2015-07-10T20:05:26Z dc.date.available: 2015-07-10T20:05:26Z dc.date.digitalpublicationdate: 2005-09-27 dc.identifier.barcode: 5990010125673 dc.identifier.origpath: /rawdataupload1/upload/0125/449 dc.identifier.copyno: 1 dc.identifier.uri: http://www.new.dli.ernet.in/handle/2015/239354 dc.description.scannerno: 14 dc.description.scanningcentre: IIIT, Allahabad dc.description.main: 1 dc.description.tagged: 0 dc.description.totalpages: 81 dc.format.mimetype: application/pdf dc.language.iso: English dc.publisher: Indian Institute Of Technology Kanpur dc.rights: Out_of_copyright dc.source.library: Indian Institute Of Technology Kanpur dc.subject.classification: Technology dc.subject.classification: Engineering. Technology In General dc.subject.classification: Computer Science & Engineering dc.title: Approximation Algorithms For 3-d Common Substructure Identification In Drug Amp Protein Molecules