Algorithms for finding distance-edge-colorings of graphs

For a bounded integer ℓ, we wish to color all edges of a graph G so that any two edges within distance ℓ have different colors. Such a coloring is called a distance-edge-coloring or an ℓ-edge-coloring of G. The distance-edge-coloring problem is to compute the minimum number of colors required for a distance-edge-coloring of a given graph G. A partial k-tree is a graph with tree-width bounded by a fixed constant k. We first present a polynomial-time exact algorithm to solve the problem for partial k-trees, and then give a polynomial-time 2-approximation algorithm for planar graphs.     

Inverse optimization for linearly constrained convex separable programming problems

In this paper, we study inverse optimization for linearly constrained convex separable programming problems that have wide applications in industrial and managerial areas. For a given feasible point of a convex separable program, the inverse optimization is to determine whether the feasible point can be made optimal by adjusting the parameter values in the problem, and when the answer is positive, find the parameter values that have the smallest adjustments. A sufficient and necessary condition is given for a feasible point to be able to become optimal by adjusting parameter values. Inverse optimization formulations are presented with ℓ₁ and ℓ₂ norms. These inverse optimization problems are either linear programming when ℓ₁ norm is used in the formulation, or convex quadratic separable programming when ℓ₂ norm is used.     

Asymptotic Behaviour of Best lp-Approximations from Affi ne Subspaces

In this paper we consider the problem of best approximation in ℓ_pⁿ, 1<p∞. If h_p, 1<p<∞, denotes the best ℓ_p-approximation of the element h ⁿ from a proper affine subspace K of ⁿ, hK, then lim_p→∞h_p=h_∞^*, where h_∞^* is a best uniform approximation of h from K, the so-called strict uniform approximation. Our aim is to prove that for all r there are α_j ⁿ, 1jr, such that

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, with γ_p^{(r) n} and γ_p^(r)= (p^−r−1).     

Approximations for node-weighted Steiner tree in unit disk graphs

Given a node-weighted connected graph and a subset of terminals, the problem node-weighted Steiner tree (NWST) seeks a lightest tree connecting a given set of terminals in a node-weighted graph. While NWST in general graphs are as hard as Set Cover, NWST restricted to unit-disk graphs (UDGs) admits constant-approximations. Recently, Zou et al. (Lecture notes in computer science, vol 5165, COCOA, 2008, pp 278–285) showed that any μ-approximation algorithm for the classical edge-weighted Steiner tree problem can be used to produce 2.5 μ-approximation algorithm for NWST restricted to UDGs. With the best known approximation bound 1.55 for the classical Steiner tree problem, they obtained an approximation bound 3.875 for NWST restricted to UDGs. In this paper, we present three approximation algorithms for NWST restricted to UDGs, the k-Restricted Relative Greedy Algorithm whose approximation bound converges to 1 + ln 5 ≈ 2.61 as k → ∞, the 3-Restricted Greedy Algorithm with approximation bound 4\frac13{4\frac{1}{3}} , and the k-Restricted Variable Metric Algorithm whose approximation bound converges to 3.9334 as k → ∞.     

An improved monotone algorithm for scheduling related machines with precedence constraints

We answer an open question posed by Krumke et al. (2008) [6] by showing how to turn the algorithm of Chekuri and Bender for scheduling related machines with precedence constraints into an O(logm)-approximation algorithm that is monotone in expectation. This significantly improves on the previously best known monotone approximation algorithms for this problem, from Krumke et al. [6] and Thielen and Krumke (2008) [8], which have an approximation guarantee of O(m^2/3).     

Approximation Algorithms for Precedence-Constrained Scheduling Problems on Parallel Machines that Run at Different Speeds

We present new approximation algorithms for the problem of scheduling precedence-constrained jobs on parallel machines that are uniformly related. That is, there arenjobs andmmachines; each jobjrequiresp_junits of processing, and is to be processed on one machine without interruption; if it is assigned to machinei, which runs at a given speeds_i, it takesp_j/s_itime units. There also is a partial order on the jobs, wherej kimplies that jobkmay not start processing until jobjhas been completed. We consider two objective functions:C_max = max_j C_j, whereC_jdenotes the completion time of jobj, and ∑_jw_jC_j, wherew_jis a weight that is given for each jobj. For the first objective, the best previously known result is an -approximation algorithm, which was shown by Jaffe more than 15 years ago. We give anO(log m)-approximation algorithm. We also show how to extend this result to obtain anO(log m)-approximation algorithm for the second objective, albeit with a somewhat larger constant. These results also extend to settings in which each jobjhas a release dater_jbefore which the job may not begin processing. In addition, we obtain stronger performance guarantees if there are a limited number of distinct speeds. Our results are based on a new linear programming-based technique for estimating the speed at which each job should be run, and a variant of the list scheduling algorithm of Graham that can exploit this additional information.     

Preemptive scheduling with rejection

 We consider the problem of preemptively scheduling a set of n jobs on m (identical, uniformly related, or unrelated) parallel machines. The scheduler may reject a subset of the jobs and thereby incur job-dependent penalties for each rejected job, and he must construct a schedule for the remaining jobs so as to optimize the preemptive makespan on the m machines plus the sum of the penalties of the jobs rejected. We provide a complete classification of these scheduling problems with respect to complexity and approximability. Our main results are on the variant with an arbitrary number of unrelated machines. This variant is APX-hard, and we design a 1.58-approximation algorithm for it. All other considered variants are weakly -hard, and we provide fully polynomial time approximation schemes for them. Finally, we argue that our results for unrelated machines can be carried over to the corresponding preemptive open shop scheduling problem with rejection. Received: October 30, 2000 / Accepted: September 26, 2001 Published online: September 5, 2002 Key words. scheduling – preemption – approximation algorithm – worst case ratio – computational complexity – in-approximability Supported in part by the EU Thematic Network APPOL, Approximation and Online Algorithms, IST-1999-14084 Supported by the START program Y43-MAT of the Austrian Ministry of Science.     

Complexity of Weighted Approximation over

We study approximation of univariate functions defined over the reals. We assume that the rth derivative of a function is bounded in a weighted L_p norm with a weight ψ. Approximation algorithms use the values of a function and its derivatives up to order r−1. The worst case error of an algorithm is defined in a weighted L_q norm with a weight ρ. We study the worst case (information) complexity of the weighted approximation problem, which is equal to the minimal number of function and derivative evaluations needed to obtain error . We provide necessary and sufficient conditions in terms of the weights ψ and ρ, as well as the parameters r, p, and q for the weighted approximation problem to have finite complexity. We also provide conditions which guarantee that the complexity of weighted approximation is of the same order as the complexity of the classical approximation problem over a finite interval. Such necessary and sufficient conditions are also provided for a weighted integration problem since its complexity is equivalent to the complexity of the weighted approximation problem for q=1.     

An Efficient Approximation Algorithm for Minimizing Makespan on Uniformly Related Machines

We give a new and efficient approximation algorithm for scheduling precedence-constrained jobs on machines with different speeds. The problem is as follows. We are given n jobs to be scheduled on a set of m machines. Jobs have processing times and machines have speeds. It takes p_j/s_i units of time for machine i with speed s_i to process job j with processing requirement p_j. Precedence constraints between jobs are given in the form of a partial order. If j k, processing of job k cannot start until job j's execution is completed. The objective is to find a non-preemptive schedule to minimize the makespan of the schedule. Chudak and Shmoys (1997, “Proceedings of the Eighth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA),” pp. 581–590) gave an algorithm with an approximation ratio of O(log m), significantly improving the earlier ratio of due to Jaffe (1980, Theoret. Comput. Sci.26, 1–17). Their algorithm is based on solving a linear programming relaxation. Building on some of their ideas, we present a combinatorial algorithm that achieves a similar approximation ratio but runs in O(n³) time. Our algorithm is based on a new and simple lower bound which we believe is of independent interest.     

On the chromatic number of random graphs

In this paper we consider the classical Erdős–Rényi model of random graphs G_n,p. We show that for p=p(n)n^−3/4−δ, for any fixed δ>0, the chromatic number χ(G_n,p) is a.a.s. ℓ, ℓ+1, or ℓ+2, where ℓ is the maximum integer satisfying 2(ℓ−1)log(ℓ−1)p(n−1).     

Polynomial time approximation schemes for general multiprocessor job shop scheduling

We study preemptive and non-preemptive versions of the general multiprocessor job shop scheduling problem: Given a set of n tasks each consisting of at most μ ordered operations that can be processed on different (possibly all) subsets of m machines with different processing times, compute a schedule (preemptive or non-preemptive, depending on the model) with minimum makespan where operations belonging to the same task have to be scheduled according to the specified order. We propose algorithms for both preemptive and non-preemptive variants of this problem that compute approximate solutions of any positive ε accuracy and run in O(n) time for any fixed values of m, μ, and ε. These results include (as special cases) many recent developments on polynomial time approximation schemes for scheduling jobs on unrelated machines, multiprocessor tasks, and classical open, flow and job shops.     

An approximation algorithm for sorting by reversals and transpositions

Genome rearrangement algorithms are powerful tools to analyze gene orders in molecular evolution. Analysis of genomes evolving by reversals and transpositions leads to a combinatorial problem of sorting by reversals and transpositions, the problem of finding a shortest sequence of reversals and transpositions that sorts one genome into the other. In this paper we present a 2k-approximation algorithm for sorting by reversals and transpositions for unsigned permutations where k is the approximation ratio of the algorithm used for cycle decomposition. For the best known value of k our approximation ratio becomes 2.8386+δ for any δ>0. We also derive a lower bound on reversal and transposition distance of an unsigned permutation.     

Multicuts in unweighted graphs and digraphs with bounded degree and bounded tree-width

The Multicut problem can be defined as: given a graph G and a collection of pairs of distinct vertices {s_i,t_i} of G, find a minimum set of edges of G whose removal disconnects each s_i from the corresponding t_i. Multicut is known to be NP-hard and Max SNP-hard even when the input graph is restricted to being a tree. The main result of the paper is a polynomial-time approximation scheme (PTAS) for Multicut in unweighted graphs with bounded degree and bounded tree-width. That is, for any ε>0, we present a polynomial-time (1+ε)-approximation algorithm. In the particular case when the input is a bounded-degree tree, we have a linear-time implementation of the algorithm. We also provide some hardness results: we prove that Multicut is still NP-hard for binary trees and that it is Max SNP-hard if we drop any of the three conditions (unweighted, bounded-degree, bounded tree-width). Finally we show that some of these results extend to the vertex version of Multicut and to a directed version of Multicut.     

Minimum Color Sum of Bipartite Graphs

The problem ofminimum color sumof a graph is to color the vertices of the graph such that the sum (average) of all assigned colors is minimum. Recently it was shown that in general graphs this problem cannot be approximated withinn^{1 − ε}, for any ε > 0, unlessNP = ZPP(Bar-Noyet al., Information and Computation140(1998), 183–202). In the same paper, a 9/8-approximation algorithm was presented for bipartite graphs. The hardness question for this problem on bipartite graphs was left open. In this paper we show that the minimum color sum problem for bipartite graphs admits no polynomial approximation scheme, unlessP = NP. The proof is byL-reducing the problem of finding the maximum independent set in a graph whose maximum degree is four to this problem. This result indicates clearly that the minimum color sum problem is much harder than the traditional coloring problem, which is trivially solvable in bipartite graphs. As for the approximation ratio, we make a further step toward finding the precise threshold. We present a polynomial 10/9-approximation algorithm. Our algorithm uses a flow procedure in addition to the maximum independent set procedure used in previous solutions.     

On the existence of the best discrete approximation in norm by reciprocals of real polynomials

For the given data (w_i,x_i,y_i), i=1,…,M, we consider the problem of existence of the best discrete approximation in l_p norm (1≤p<∞) by reciprocals of real polynomials. For this problem, the existence of best approximations is not always guaranteed. In this paper, we give a condition on data which is necessary and sufficient for the existence of the best approximation in l_p norm. This condition is theoretical in nature. We apply it to obtain several other existence theorems very useful in practice. Some illustrative examples are also included.     

Approximation algorithms for scheduling unrelated parallel machines

We consider the following scheduling problem. There arem parallel machines andn independent jobs. Each job is to be assigned to one of the machines. The processing of jobj on machinei requires timep _ij . The objective is to find a schedule that minimizes the makespan.Our main result is a polynomial algorithm which constructs a schedule that is guaranteed to be no longer than twice the optimum. We also present a polynomial approximation scheme for the case that the number of machines is fixed. Both approximation results are corollaries of a theorem about the relationship of a class of integer programming problems and their linear programming relaxations. In particular, we give a polynomial method to round the fractional extreme points of the linear program to integral points that nearly satisfy the constraints.In contrast to our main result, we prove that no polynomial algorithm can achieve a worst-case ratio less than 3/2 unlessP = NP. We finally obtain a complexity classification for all special cases with a fixed number of processing times.A preliminary version of this paper appeared in theProceedings of the 28th Annual IEEE Symposium on the Foundations of Computer Science (Computer Society Press of the IEEE, Washington, D.C., 1987) pp. 217–224.     

Admission control with advance reservations in simple networks

In the admission control problem we are given a network and a set of connection requests, each of which is associated with a path, a time interval, a bandwidth requirement, and a weight. A feasible schedule is a set of connection requests such that at any given time, the total bandwidth requirement on every link in the network is at most 1. Our goal is to find a feasible schedule with maximum total weight.We consider the admission control problem in two simple topologies: the line and the tree. We present a 12c-approximation algorithm for the line topology, where c is the maximum number of requests on a link at some time instance. This result implies a 12c-approximation algorithm for the rectangle packing problem, where c is the maximum number of rectangles that cover simultaneously a point in the plane. We also present an O(logt)-approximation algorithm for the tree topology, where t is the size of the tree. We consider the loss minimization version of the admission control problem in which the goal is to minimize the weight of unscheduled requests. We present a c-approximation algorithm for loss minimization problem in the tree topology. This result is based on an approximation algorithm for a generalization of set cover, in which each element has a covering requirement, and each set has a covering potential. The approximation ratio of this algorithm is Δ, where Δ is the maximum number of sets that contain the same element.     

Average CaseL∞-Approximation in the Presence of Gaussian Noise

We consider the average caseL_∞-approximation of functions fromC^r([0, 1]) with respect to ther-fold Wiener measure. An approximation is based onnfunction evaluations in the presence of Gaussian noise with varianceσ²>0. We show that the n th minimal average error is of ordern^{−(2r+1)/(4r+4)} ln^1/2 n, and that it can be attained either by the piecewise polynomial approximation using repetitive observations, or by the smoothing spline approximation using non-repetitive observations. This completes the already known results forL_q-approximation withq<∞ andσ0, and forL_∞-approximation withσ=0.