APPROXIMATION ALGORITHM

We consider a scheduling problem in which a bounded number of jobs can be processed simultaneously by a single machine. The input is a set of n jobs J = {J 1 , . . . , J n }. Each job, J j , is associated with an interval [s j , c j ] along which it should be processed. Also given is the parallelism parameter g ≥ 1, which is the maximal number of jobs that can be processed simultaneously by a single machine. Each machine operates along a contiguous time interval, called its busy interval, which contains all the intervals corresponding to the jobs it processes. The goal is to assign the jobs to machines such that the total busy time of the machines is minimized.

For an edge-weighted connected undirected graph, the minimum k-way cut problem is to find a subset of edges of minimum total weight whose removal separates the graph into k connected components. The problem is NP-hard when k is part of the input and W[1]-hard when k is taken as a parameter. A simple algorithm for approximating a minimum k-way cut

For a given graph G over n vertices, let OPT G denote the size of an optimal solution in G of a particular minimization problem (e.g., the size of a minimum vertex cover). A randomized algorithm will be called an α-approximation algorithm with an additive error for this minimization problem, if for any given additive error parameter > 0 it computes a value OPT such that, with probability at least 2/3, it holds that OPT G ≤ OPT ≤ α · OPT G + n .

Existing works on variational bayesian (VB) treatment for factor analysis (FA) model such as . Variational inference for Bayesian mixture of factor analysers. In Advances in neural information proceeding systems. Cambridge, MA: MIT Press; Nielsen, F. B. . Variational approach to factor analysis and related models. Master's thesis, The Institute of Informatics and Mathematical Modelling, Technical University of Denmark.] are found theoretically and empirically to suffer two problems: x penalize the model more heavily than BIC and y perform unsatisfactorily in low noise cases as redundant factors can not be effectively suppressed. A novel VB treatment is proposed in this paper to resolve the two problems and a simulation study is conducted to testify its improved performance over existing treatments.

Given a metric d defined on a set V of points (a metric space), we define the ball B(v, r) centered at v ∈ V and having radius r ≥ 0 to be the set {q ∈ V |d(v, q) ≤ r}. In this work, we consider the problem of computing a minimum cost k-cover for a given set P ⊆ V of n points, where k > 0 is some given integer which is also part of the input. For κ ≥ 0, a κ-cover for subset Q ⊆ P is a set of at most κ balls, each centered at a point in P , whose union covers (contains) Q. The cost of a set D of balls, denoted cost(D), is the sum of the radii of those balls.

For an edge-weighted connected undirected graph, the minimum k-way cut problem is to find a subset of edges of minimum total weight whose removal separates the graph into k connected components. The problem is NP-hard when k is part of the input and W[1]-hard when k is taken as a parameter. A simple algorithm for approximating a minimum k-way cut

The Steiner problem on networks asks for a shortest subgraph spanning a given subset of distinguished vertices. We give a !-approximation algorithm for the special case in which the underlying network is complete and all edge lengths are either 1 or 2. We also relate the Steiner problem to a complexity class recently defined by Papadimitriou and Yannakakis by showing that this special case is MAX SNP-hard, which may be evidence that the Steiner problem on networks has no polynomial-time approximation scheme.

A strategy for adaptive control and energetic optimization of aerobic fermentors was implemented, with both air flow and agitation speed as manipulated variables. This strategy is separable in its components: control, optimization, estimation. We optimized parameter’s estimation (from the usual KLa correlation) using sinusoidal excitation of air flow and agitation speed. We have implemented parameter’s estimation trough recursive least squares algorithm with forgetting factor. We carried separate essays on control, optimization and estimation algorithms. We carried our essays using an original computational simulation environment, with noise and delay generating facilities for data sampling and filtering.

Our results show the convergence and robustness of the estimation algorithm used, improved with use of both forgetting factor and KLa dead-band facilities. Control algorithm used in our work compares favorably with PID using the integrated area criteria for deviation between oxygen molarity and critical molarity (set point). Optimization algorithm clearly reduces energetic consumption, respecting critical molarity. Integration of control, optimization and adaptive algorithms was implemented, but future work is needed for stability. Methods were defined and implemented for stability improvement. We have implemented data acquisition and computer manipulation of air flow and agitation speed for actual fermentors.

The maximum 2-satisÿability problem (MAX-2-SAT) is: given a Boolean formula in 2-CNF, ÿnd a truth assignment that satisÿes the maximum possible number of its clauses. MAX-2-SAT is MAX-SNP-complete. Recently, this problem received much attention in the contexts of (polynomial-time) approximation algorithms and (exponential-time) exact algorithms. In this paper, we present an exact algorithm solving MAX-2-SAT in time poly(L) · 2 K=5 , where K is the number of clauses and L is their total length. In fact, the running time is only poly(L)·2 K 2 =5 , where K2 is the number of clauses containing two literals. This bound implies the bound poly(L) · 2 L=10 . Our results signiÿcantly improve previous bounds: poly(L) · 2 K=2:88 (J. Algorithms 36 62-88) and poly(L) · 2 K=3:44 .))

This paper considers variants of the one-dimensional bin packing (and stock cutting) problem in which both the ordering and orientation of items in a container influences the validity and quality of a solution. Two new real-world problems of this type are introduced, the first that involves the creation of wooden trapezoidal-shaped trusses for use in the roofing industry, the second that requires the cutting and scoring of rectangular pieces of cardboard in the construction of boxes. To tackle these problems, two variants of a local search-based approximation algorithm are proposed, the first that attempts to determine item ordering and orientation via simple heuristics, the second that employs more accurate but costly branchand-bound procedures. We investigate the inevitable trade-off between speed and accuracy that occurs with these variants and highlight the circumstances under which each scheme is advantageous.

The Multiple Subset Sum Problem (MSSP) is the variant of bin packing in which the number of bins is given and one would like to maximize the overall weight of the items packed in the bins. The problem is also a special case of the multiple knapsack problem in which all knapsacks have the same capacity and the item profits and weights coincide. Recently, polynomial time approximation schemes have been proposed for MSSP and its generalizations, see A. Caprara, H. Kellerer, and U. schemes are only of theoretical interest, since they require either the solution of huge integer linear programs, or the enumeration of a huge number of possible solutions, for any reasonable value of required accuracy. In this paper, we present a polynomial-time 3/4-approximation algorithm which runs fast also in practice. Its running time is linear in the number of items and quadratic in the number of bins. The "core" of the algorithm is a procedure to pack triples of "large" items into the bins. As a byproduct of our analysis, we get the approximation guarantee for a natural greedy heuristic for the 3-Partitioning Problem.

While preparing these proceedings, we received news of Timo Raita's tragic death, after a severe illness. He came to the Department of Computer Science, University of Turku, in 1980, received his PhD in 1988, and acted first as a lecturer, then as a professor of computer science. Timo was a member of the organizing committee of this workshop, which remained one of his last official duties.

Research in VLSI placement, an NP-hard problem, has branched in two different directions. The first one employs iterative heuristics with many tunable parameters to produce a near-optimal solution but without theoretical guarantee on its quality. The other one considers ...

A full-rank under-determined linear system of equations Ax = b has in general infinitely many possible solutions. In recent years there is a growing interest in the sparsest solution of this equation-the one with the fewest non-zero entries, measured by x 0 . Such solutions find applications in signal and image processing, where the topic is typically referred to as "sparse representation". Considering the columns of A as atoms of a dictionary, it is assumed that a given signal b is a linear composition of few such atoms. Recent work established that if the desired solution x is sparse enough, uniqueness of such a result is guaranteed. Also, pursuit algorithms, approximation solvers for the above problem, are guaranteed to succeed in finding this solution.

In this paper, we present new approximation results for the offline problem of single machine scheduling with sequenceindependent set-ups and item availability, where the jobs to be scheduled are independent (i.e., have no precedence constraints) and have a common release time.

For an edge-weighted connected undirected graph, the minimum k-way cut problem is to find a subset of edges of minimum total weight whose removal separates the graph into k connected components. The problem is NP-hard when k is part of the input and W[1]-hard when k is taken as a parameter. A simple algorithm for approximating a minimum k-way cut

In this paper, we consider the problem of maximizing the lifetime of a target-covering sensor network in which each sensor can adjust its sensing range. The network model consists of a large number of sensors with adjustable sensing ranges being deployed to monitor a set of targets. Since more than one sensor can cover a target, in order to be energy efficient, one can activate successive subsets of sensors that cover all targets. This paper addresses the problem of maximizing the total lifetime of such an activation schedule.

We consider the graph balancing problem of providing orientations to edges in an undirected multi-graph to minimize the maximum load. We first obtain an FPTAS when the multi-graph is restricted to a tree. We also obtain some additional results for other restricted cases by showing equivalencies with related combinatorial problems.

We consider a multi-agent scheduling problem on a single machine in which each agent is responsible for his own set of jobs and wishes to minimize the total weighted completion time of his own set of jobs. It is known that the unweighted problem with two agents is NP-hard in the ordinary sense. For this case, we can reduce our problem to a Multi-Objective Shortest-Path (MOSP) problem and this reduction leads to several results including Fully Polynomial Time Approximation Schemes (FPTAS). We also provide an efficient approximation algorithm with a reasonably good worst-case ratio.

Partitioning a permutation into a minimum number of monotone subsequences is N Phard. We extend this complexity result to minimum partitioning into k-modal subsequences; here unimodal is the special case k = 1. Based on a network flow interpretation we formulate both, the monotone and the k-modal version, as mixed integer programs. This is the first proposal to obtain provably optimal partitions of permutations. LP rounding gives a 2-approximation for minimum monotone partitions and a (k+1)-approximation for minimum (upper) k-modal partitions. For the online problem, in which the permutation becomes known to an algorithm sequentially, we derive a logarithmic lower bound on the competitive ratio for minimum monotone partitions, and we analyze two (bin packing) online algorithms. These immediately apply to online cocoloring of permutation graphs.

Due to their axiomatic foundation and their favorable computational properties convex risk measures are becoming a powerful tool in financial risk management. In this paper we will review the fundamental structural concepts of convex risk measures within the framework of convex analysis. Then we will exploit it for deriving strong duality relations in a generic portfolio optimization context. In particular, the duality relationship can be used for designing new, efficient approximation algorithms based on Nesterov's smoothing techniques for non-smooth convex optimization. Furthermore, the presented concepts enable us to formalize the notion of flexibility as the (marginal) risk absorption capacity of a technology or (available) resources.

Most game programs have a large number of parameters that are crucial for their performance. While tuning these parameters by hand is rather difficult, successful applications of automatic optimisation algorithms in game programs are known only for parameters that belong to certain components (e.g. evaluation-function parameters). The SPSA (Simultaneous Perturbation Stochastic Approximation) algorithm is an attractive choice for optimising any kind of parameters of a game program, both for its generality and its simplicity. It's disadvantage is that it can be very slow. In this article we propose several methods to speed up SPSA, in particular, the combination with RPROP, using common random numbers, antithetic variables and averaging. We test the resulting algorithm for tuning various types of parameters in two domains, poker and LOA. From the experimental study, we conclude that using SPSA is a viable approach for optimisation in game programs, especially if no good alternative exists for the types of parameters considered.

Society relies heavily on its networked physical infrastructure and information systems. Accurately assessing the vulnerability of these systems against disruptive events is vital for planning and risk management. Existing approaches to vulnerability assessments of large-scale systems mainly focus on investigating inhomogeneous properties of the underlying graph elements. These measures and the associated heuristic solutions are limited in evaluating the vulnerability of large-scale network topologies. Furthermore, these approaches often fail to provide performance guarantees of the proposed solutions. In this paper, we propose a vulnerability measure, pairwise connectivity, and use it to formulate network vulnerability assessment as a graph-theoretical optimization problem, referred to as -disruptor. The objective is to identify the minimum set of critical network elements, namely nodes and edges, whose removal results in a specific degradation of the network global pairwise connectivity. We prove the NP-completeness and inapproximability of this problem and propose an (log log log ) pseudo-approximation algorithm to computing the set of critical nodes and an (log 1 5 ) pseudo-approximation algorithm for computing the set of critical edges. The results of an extensive simulation-based experiment show the feasibility of our proposed vulnerability assessment framework and the efficiency of the proposed approximation algorithms in comparison to other approaches.

The problem of grooming is central in studies of optical networks. In graph-theoretic terms, this can be viewed as assigning colors to the lightpaths so that at most g of them (g being the grooming factor) can share one edge. The cost of a coloring is the number of optical switches (ADMs); each lightpath uses two ADM’s, one at each endpoint, and in case g lightpaths of the same wavelength enter through the same edge to one node, they can all use the same ADM (thus saving g – 1 ADMs). The goal is to minimize the total number of ADMs. This problem was shown to be NP-complete for g = 1 and for a general g. Exact solutions are known for some specific cases, and approximation algorithms for certain topologies exist for g = 1. We present an approximation algorithm for this problem. For every value of g the running time of the algorithm is polynomial in the input size, and its approximation ratio for a wide variety of network topologies – including the ring topology – is shown to be 2 ln g + o(ln g). This is the first approximation algorithm for the grooming problem with a general grooming factor g.

We consider the m-machine ordered flow shop scheduling problem with machines subject to maintenance and with the makespan as objective. It is assumed that the maintenances are scheduled in advance and that the jobs are resumable. We consider permutation schedules and show that the problem is strongly NP-hard; it remains NP-hard in the ordinary sense even in the case of a single maintenance. We show that if the first (last) machine is the slowest and if maintenances occur only on the first (last) machine, then sequencing the jobs in the LPT (SPT) order yields an optimal schedule for the m-machine problem. As a special case of the ordered flow shop, we focus on the proportionate flow shop where the processing times of any given job on all the machines are identical. We prove that the proportionate flow shop problem with two maintenance periods is NP-hard, while the problem with a single maintenance period can be solved in polynomial time. Furthermore, we show that the optimal algorithm for the single maintenance case is a 3 2 -approximation algorithm for the two maintenance case. In our conclusion we discuss also the computational complexity of other objective functions.

The maximum cut problem is known to be an important NPcomplete problem with many applications. In this paper, we investigate this problem (which we call the nonnal maximum cut problem) and a variant of it (which we call the connected maximum cut problem). Our main results are as follows. 1) We show that any n-vertex e-edge graph admits a cut with at least the fraction 1/2 + 1/2n of its edges, thus improving the ratio 1/2 + 2/e known before. 2) We show that it is NPcomplete to decide if a given graph has a normal maximum cut with at least a fraction (1/2 + 6) of its edges, where the positive constant E can be taken smaller than any value of our choice. 3) We exhibit, however, an approximation algorithm for the normal maximum cut problem on any graph that rnns in O((e log e + n log n ) / p + log p . log n ) parallel time using p (1 5 p 5 e + n ) processors, that gnarantees a ratio of at least L1/2 + l / 2 n J I given a matching of size e / n in G. 4) We take up the connected maximum cut problem and show that, unlike the normal maximum cut problem, this problem admits an infinity of instances where the fraction of the edges in the connected maximum cut is arbitrarily close to zero. 5) We then show that the connected maximum cut problem is NP-complete even for planar graphs, in clear contrast to the normal maximum cut problem which is solvable in polynomial time on planar graphs.

We present a short overview on polynomial approximation of NP-hard problems. We present the main approximability classes together with examples of problems belonging to them. We also describe the general concept of approximability preserving reductions together with a discussion about their capacities and their limits. Finally, we present a quick description of what it is commonly called inapproximability results. Such results provide limits on the approximability of the problems tackled.

We study a variation of the vertex cover problem where it is required that the graph induced by the vertex cover is connected. We prove that this problem is polynomial in chordal graphs, has a PTAS in planar graphs, is APX-hard in bipartite graphs and is 5/3-approximable in any class of graphs where the vertex cover problem is polynomial (in particular in bipartite graphs).

We study the problem of packing element-disjoint Steiner trees in graphs. We are given a graph and a designated subset of terminal nodes, and the goal is to find a maximum cardinality set of elementdisjoint trees such that each tree contains every terminal node. An element means a non-terminal node or an edge. (Thus, each non-terminal node and each edge must be in at most one of the trees.) We show that the problem is APX-hard when there are only three terminal nodes, thus answering an open question. Our main focus is on the special case when the graph is planar. We show that the problem of finding two element-disjoint Steiner trees in a planar graph is NP-hard. Similarly, the problem of finding two edge-disjoint Steiner trees in a planar graph is NP-hard. We design an algorithm for planar graphs that achieves an approximation guarantee close to 2. In fact, given a planar graph that is k elementconnected on the terminals (k is an upper bound on the number of element-disjoint Steiner trees), the algorithm returns k 2 − 1 element-disjoint Steiner trees. Using this algorithm, we get an approximation algorithm for the edge-disjoint version of the problem on planar graphs that improves on the previous approximation guarantees. We also show that the natural LP relaxation of the planar problem has an integrality ratio approaching 2.

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We consider the problems of finding the maximum number of vertex-disjoint triangles (VTP) and edge-disjoint triangles (ETP) in a simple graph. Both problems are NP-hard. The algorithm with the best approximation ratio known so far for these problems has ratio 3/2 + ε, a result that follows from a more general algorithm for set packing obtained by Hurkens and Schrijver [On the size of systems of sets every t of which have an SDR, with an application to the worst-case ratio of heuristics for packing problems, SIAM J. Discrete Math. 2(1) (1989) 68-72]. We present improvements on the approximation ratio for restricted cases of VTP and ETP that are known to be APX-hard: we give an approximation algorithm for VTP on graphs with maximum degree 4 with ratio slightly less than 1.2, and for ETP on graphs with maximum degree 5 with ratio 4/3. We also present an exact linear-time algorithm for VTP on the class of indifference graphs.

The paper describes and studies an iterative algorithm for finding small values of a set of linear forms over vectors of integers. The algorithm uses a linear recurrence relation to generate a vector sequence, the basic idea being to choose the integral coefficients in the recurrence relation in such a way that the linear forms take small values, subject to the requirement that the integers should not become too large. The problem of choosing good coefficients for the recurrence relation is thus related to the problem of finding a good approximation of a given vector by a vector in a certain one-parameter family of lattices; the novel feature of our approach is that practical formulae for the coefficients are obtained by considering the limit as the parameter tends to zero. The paper discusses two rounding procedures to solve the underlying inhomogeneous Diophantine approximation problem: the first, which we call "naive rounding" leads to a multidimensional continued fraction algorithm with suboptimal asymptotic convergence properties; in particular, when it is applied to the familiar problem of simultaneous rational approximation, the algorithm reduces to the classical Jacobi-Perron algorithm. The second rounding procedure is Babai's nearest-plane procedure. We compare the two rounding procedures numerically; our experiments suggest that the multidimensional continued fraction corresponding to nearest-plane rounding converges at an optimal asymptotic rate.

This paper considers the problem of scheduling a two-stage flowshop that consists of a common critical machine in stage one and two independent dedicated machines in stage two. All the jobs require processing first on the common critical machine. Each job after completing its critical operation in stage one will proceed to the dedicated machine of its type for further processing in stage two. The objective is to minimize the weighted sum of stage-two machine completion times. We show that the problem is strongly NP-hard, and develop an O(n 3 ) polynomial time algorithm to solve the special case where the sequences of both types of jobs are given. We also design an approximation algorithm with a tight performance ratio of 4 3 for the general case.

We present a novel technique for approximating finite-impulse-response (FIR) filters with infinite-impulse-response (IIR) structures through extending the vector fitting (VF) algorithm, used extensively for continuous-time frequency-domain rational approximation, to its discrete-time counterpart called VFz. VFz directly computes the candidate filter poles and iteratively relocates them for progressively better approximation. Each VFz iteration consists of the solutions of an overdetermined linear equation and an eigenvalue problem, with real-domain arithmetic to accommodate complex poles. Pole flipping and maximum pole radius constraint guarantee stability and robustness against finite-precision implementation. Comparison against existing algorithms confirms that VFz generally exhibits fast convergence and produces highly accurate IIR approximants.

Maximum Independent Set (MIS) and its relative Maximum Weight Independent Set (MWIS) are well-known problems in combinatorial optimization; they are NP-hard even in the geometric setting of unit disk graphs. In this paper, we study the Maximum Area Independent Set (MAIS) problem, a natural restricted version of MWIS in disk intersection graphs where the weight equals the disk area. We obtain: (i) Quantitative bounds on the maximum total area of an independent set relative to the union area; (ii) Practical constant-ratio approximation algorithms for finding an independent set with a large total area relative to the union area.

Using flow and matching algorithms to solve the problem of finding disjoint paths through a given node, and with a technique of Chekuri and Khanna, we give an O(√ n) approximation for the edge-disjoint paths problem in undirected graphs, directed acyclic graphs and directed graphs with edge capacity at least 2.

We introduce a new combinatorial optimization problem in this article, called the minimum common integer partition (MCIP) problem, which was inspired by computational biology applications including ortholog assignment and DNA fingerprint assembly. A partition of a positive integer n is a multiset of positive integers that add up to exactly n , and an integer partition of a multiset S of integers is defined as the multiset union of partitions of integers in S . Given a sequence of multisets S 1 , S 2 , …, S k of integers, where k ≥ 2, we say that a multiset is a common integer partition if it is an integer partition of every multiset S i , 1 ≤ i ≤ k . The MCIP problem is thus defined as to find a common integer partition of S 1 , S 2 , …, S k with the minimum cardinality, denoted as MCIP( S 1 , S 2 , …, S k ). It is easy to see that the MCIP problem is NP-hard, since it generalizes the well-known subset sum problem. We can in fact show that it is APX-hard. We will also present a 5/4-...

In online learning, each training example is processed separately and then discarded. Environments that require online learning are often non-stationary and their underlying distributions may change over time (concept drift). Even though ensembles of learning machines have been used for handling concept drift, there has been no deep study of why they can be helpful for dealing with drifts and which of their features can contribute for that. The thesis mainly investigates how ensemble diversity affects accuracy in online learning in the presence of concept drift and how to use diversity in order to improve accuracy in changing environments. This is Firstly, many thanks to my supervisor, Xin Yao, whose advice on conducting research has been excellent. I would also like to thank him for his professionalism, generosity and understanding. He has shown to be a respectful person not only for his achievements, but also for his personality. Many thanks also to the thesis group members (Peter Tiño and John Bullinaria) for their helpful discussions. The current and previous members of the Data Mining Reading Group

Computational fluid dynamics (CFD) is a powerful tool to extent knowledge of biomechanical processes in cardiovascular implants. To provide a standardized method the U.S. Food and Drug Administration (FDA) initialized a CFD round robin study. One of the developed benchmark standard models is a generic nozzle geometry, consisting of a cylindrical throat with a conical collector and sudden expansion on either side. Several fluid mechanical data obtained from international institutes by means of CFD and particle image velocimetry (PIV) measurements under different flow regimes (Re = 500, 2000, 3500, 5000 and 6500) are freely available. This database includes only steady state simulations. In this study we performed pulsatile CFD simulations to consider the physiological environment of the coronary vessels. Furthermore, the nozzle geometry was scaled down to coronary dimension (D inlet = 12 mm to 3 mm) while retaining the average Reynolds number Re = 500 constant. The pulsatile character is described by a Womersley number of Wo = 2.065. Our CFD code was previously validated by using FDA´s data for steady state inflow conditions. It could be shown that time averaged wall shear stress and shear stress values agree well with steady state results. We conclude that steady state simulations are valid for hemodynamic analyses if only time averaged values are needed. This could save computational costs of future hemodynamic investigations. In addition, this study expands FDA´s benchmark case by pulsatile inlet condition for further code validation. This could be necessary for the development of new numerical methods as well as for validation of CFD codes used in the approval process of medical devices.

In multi-objective convex optimization it is necessary to compute an infinite set of nondominated points. We propose a method for approximating the nondominated set of a multi-objective nonlinear programming problem, where the objective functions and the feasible set are convex. This method is an extension of Benson's outer approximation algorithm for multi-objective linear programming problems. We prove that this method provides a set of weakly ε-nondominated points. For the case that the objectives and constraints are differentiable, we describe an efficient way to carry out the main step of the algorithm, the construction of a hyperplane separating an exterior point from the feasible set in objective space. We provide examples that show that this cannot always be done in the same way in the case of non-differentiable objectives or constraints.

We study the problem of computing the similarity between two piecewise-linear bivariate functions defined over a common domain, where the surfaces they define in 3D-polyhedral terrains-can be transformed vertically by a linear transformation of the third coordinate (scaling and translation). We present a randomized algorithm that minimizes the maximum vertical distance between the graphs of the two functions, over all linear transformations of one of the terrains, in O(n 4/3 polylog n) expected time, where n is the total number of vertices in the graphs of the two functions. We also study the computation of similarity between two univariate or bivariate functions by minimizing the area or volume between their graphs. For univariate functions we give a (1 + ε)-approximation algorithm for minimizing the area that runs in O(n/ √ ε) time, for any fixed ε > 0. The (1 + ε)approximation algorithm for the bivariate version, where volume is minimized, runs in O(n/ε 2 ) time, for any fixed ε > 0, provided the two functions are defined over the same triangulation of their domain.