Approximation Algorithms for Scheduling with Reservations

We study the problem of non-preemptively scheduling n independent sequential jobs on a system of m identical parallel machines in the presence of reservations, where m is constant. This setting is practically relevant because for various reasons, some machines may not be available during specified time intervals. The objective is to minimize the makespan C _max, which is the maximum completion time.     

Approximation Algorithms for Requirement Cut on Graphs

In this paper, we unify several graph partitioning problems including multicut, multiway cut, and k-cut, into a single problem. The input to the requirement cut problem is an undirected edge-weighted graph G=(V,E), and g groups of vertices X ₁,…,X _g ⊆V, with each group X _i having a requirement r _i between 0 and |X _i |. The goal is to find a minimum cost set of edges whose removal separates each group X _i into at least r _i disconnected components. We give an O(log n⋅log (gR)) approximation algorithm for the requirement cut problem, where n is the total number of vertices, g is the number of groups, and R is the maximum requirement. We also show that the integrality gap of a natural LP relaxation for this problem is bounded by O(log n⋅log (gR)). On trees, we obtain an improved guarantee of O(log (gR)). There is an Ω(log g) hardness of approximation for the requirement cut problem, even on trees.     

Approximation Algorithms for Reconstructing the Duplication History of Tandem Repeats

Computing the duplication history of a tandem repeated region is an important problem in computational biology (Fitch in Genetics 86:623–644, 1977; Jaitly et al. in J. Comput. Syst. Sci. 65:494–507, 2002; Tang et al. in J. Comput. Biol. 9:429–446, 2002). In this paper, we design a polynomial-time approximation scheme (PTAS) for the case where the size of the duplication block is 1. Our PTAS is faster than the previously best PTAS in Jaitly et al. (J. Comput. Syst. Sci. 65:494–507, 2002). For example, to achieve a ratio of 1.5, our PTAS takes O(n ⁵) time while the PTAS in Jaitly et al. (J. Comput. Syst. Sci. 65:494–507, 2002) takes O(n ¹¹) time. We also design a ratio-6 polynomial-time approximation algorithm for the case where the size of each duplication block is at most 2. This is the first polynomial-time approximation algorithm with a guaranteed ratio for this case. Part of work was done during a Z.-Z. Chen visit at City University of Hong Kong.     

Fractional Path Coloring in Bounded Degree Trees with Applications

This paper studies the natural linear programming relaxation of the path coloring problem. We prove constructively that finding an optimal fractional path coloring is Fixed Parameter Tractable (FPT), with the degree of the tree as parameter: the fractional coloring of paths in a bounded degree trees can be done in a time which is linear in the size of the tree, quadratic in the load of the set of paths, while exponential in the degree of the tree. We give an algorithm based on the generation of an efficient polynomial size linear program. Our algorithm is able to explore in polynomial time the exponential number of different fractional colorings, thanks to the notion of trace of a coloring that we introduce. We further give an upper bound on the cost of such a coloring in binary trees and extend this algorithm to bounded degree graphs with bounded treewidth. Finally, we also show some relationships between the integral and fractional problems, and derive a 1+5/3e≈1.61—approximation algorithm for the path coloring problem in bounded degree trees, improving on existing results. This classic combinatorial problem finds applications in the minimization of the number of wavelengths in wavelength division multiplexing (wdm) optical networks.     

Fixed-Parameter Algorithms for Cluster Vertex Deletion

We initiate the first systematic study of the NP-hard Cluster Vertex Deletion (CVD) problem (unweighted and weighted) in terms of fixed-parameter algorithmics. In the unweighted case, one searches for a minimum number of vertex deletions to transform a graph into a collection of disjoint cliques. The parameter is the number of vertex deletions. We present efficient fixed-parameter algorithms for CVD applying the fairly new iterative compression technique. Moreover, we study the variant of CVD where the maximum number of cliques to be generated is prespecified. Here, we exploit connections to fixed-parameter algorithms for (weighted) Vertex Cover.     

Faster Combinatorial Algorithms for Determinant and Pfaffian

Computation of a determinant is a very classical problem. The related concept is a Pfaffian of a matrix defined for skew-symmetric matrices. The classical algorithm for computing the determinant is Gaussian elimination. It needs O(n ³) additions, subtractions, multiplications and divisions. The algorithms of Mahajan and Vinay compute determinant and Pfaffian in a completely non-classical and combinatorial way, by reducing these problems to summation of paths in some acyclic graphs. The attractive feature of these algorithms is that they are division-free. We present a novel algebraic view of these algorithms: a relation to a pseudo-polynomial dynamic-programming algorithm for the knapsack problem. The main phase of Mahajan-Vinay algorithm can be interpreted as a computation of an algebraic version of the knapsack problem, which is an alternative to the graph-theoretic approach used in the original algorithm. Our main results show how to implement Mahajan-Vinay algorithms without divisions, in time $\tilde{O}(n^{3.03})$ using the fast matrix multiplication algorithm.     

Parallel algorithms for finding polynomial Roots on OTIS-torus

We present two parallel algorithms for finding all the roots of an N-degree polynomial equation on an efficient model of Optoelectronic Transpose Interconnection System (OTIS), called OTIS-2D torus. The parallel algorithms are based on the iterative schemes of Durand–Kerner and Ehrlich methods. We show that the algorithm for the Durand–Kerner method requires (N ^0.75+0.5N ^0.25−1) electronic moves + 2(N ^0.5−1) OTIS moves using N processors. The parallel algorithm for Ehrlich method is shown to run in (N ^0.75+0.5N ^0.25−1) electronic moves + 2(N ^0.5−1) OTIS moves with the same number of processors. The algorithms have lower AT cost than the algorithms presented in Jana (Parallel Comput 32:301–312, 2006). The scalability of the algorithms is also discussed.     

Better and Simpler Approximation Algorithms for?the?Stable Marriage Problem

We first consider the problem of finding a maximum size stable matching if incomplete lists and ties are both allowed, but ties are on one side only. For this problem we give a simple, linear time 3/2-approximation algorithm, improving on the best known approximation factor 5/3 of Irving and Manlove (J. Comb. Optim., doi:10.1007/s10878-007-9133-x, 2007). Next, we show how this extends to the Hospitals/Residents problem with the same ratio if the residents have strict orders. We also give a simple linear time algorithm for the general problem with approximation factor 5/3, improving the best known 15/8-approximation algorithm of Iwama, Miyazaki and Yamauchi (SODA ??07: Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pp.?288?C297, 2007). For the cases considered in this paper it is NP-hard to approximate within a factor of 21/19 by the result of Halldórsson et?al. (ACM Transactions on Algorithms 3(3):30, 2007). Our algorithms not only give better approximation ratios than the cited ones, but are much simpler and run significantly faster. Also we may drop a restriction used in (J. Comb. Optim., doi:10.1007/s10878-007-9133-x, 2007) and the analysis is substantially more moderate. Preliminary versions of this paper appeared in (Király, Egres Technical Report TR-2008-04, www.cs.elte.hu/egres/, 2008; Király in Proceedings of MATCH-UP 2008: Matching Under Preferences??Algorithms and Complexity, Satellite Workshop of ICALP, July 6, 2008, Reykjavík, Iceland, pp.?36?C45, 2008; Király in ESA 2008, Lecture Notes in Computer Science, vol.?5193, pp.?623?C634, 2008). For the related results obtained thenceforth see Sect.?5.     

Simple and Improved Parameterized Algorithms for Multiterminal Cuts

Given a graph G=(V,E) with n vertices and m edges, and a subset T of k vertices called terminals, the Edge (respectively, Vertex) Multiterminal Cut problem is to find a set of at most l edges (non-terminal vertices), whose removal from G separates each terminal from all the others. These two problems are NP-hard for k≥3 but well-known to be polynomial-time solvable for k=2 by the flow technique. In this paper, based on a notion farthest minimum isolating cut, we design several simple and improved algorithms for Multiterminal Cut. We show that Edge Multiterminal Cut can be solved in O(2 ^l kT(n,m)) time and Vertex Multiterminal Cut can be solved in O(k ^l T(n,m)) time, where T(n,m)=O(min?(n ^2/3,m ^1/2)m) is the running time of finding a minimum (s,t) cut in an unweighted graph. Furthermore, the running time bounds of our algorithms can be further reduced for small values of k: Edge 3-Terminal Cut can be solved in O(1.415 ^l T(n,m)) time, and Vertex {3,4,5,6}-Terminal Cuts can be solved in O(2.059 ^l T(n,m)), O(2.772 ^l T(n,m)), O(3.349 ^l T(n,m)) and O(3.857 ^l T(n,m)) time respectively. Our results on Multiterminal Cut can also be used to obtain faster algorithms for Multicut: $O((\min(\sqrt{2k},l)+1)^{2k}2^{l}T(n,m))Given a graph G=(V,E) with n vertices and m edges, and a subset T of k vertices called terminals, the Edge (respectively, Vertex) Multiterminal Cut problem is to find a set of at most l edges (non-terminal vertices), whose removal from G separates each terminal from all the others. These two problems are NP-hard for k≥3 but well-known to be polynomial-time solvable for k=2 by the flow technique. In this paper, based on a notion farthest minimum isolating cut, we design several simple and improved algorithms for Multiterminal Cut. We show that Edge Multiterminal Cut can be solved in O(2 ^l kT(n,m)) time and Vertex Multiterminal Cut can be solved in O(k ^l T(n,m)) time, where T(n,m)=O(min (n ^2/3,m ^1/2)m) is the running time of finding a minimum (s,t) cut in an unweighted graph. Furthermore, the running time bounds of our algorithms can be further reduced for small values of k: Edge 3-Terminal Cut can be solved in O(1.415 ^l T(n,m)) time, and Vertex {3,4,5,6}-Terminal Cuts can be solved in O(2.059 ^l T(n,m)), O(2.772 ^l T(n,m)), O(3.349 ^l T(n,m)) and O(3.857 ^l T(n,m)) time respectively. Our results on Multiterminal Cut can also be used to obtain faster algorithms for Multicut: O((min(?{2k},l)+1)^2k2^lT(n,m))O((\min(\sqrt{2k},l)+1)^{2k}2^{l}T(n,m)) -time algorithm for Edge Multicut and O((2k) ^k+l/2 T(n,m))-time algorithm for Vertex Multicut.     

Competitive Analysis of Scheduling Algorithms for Aggregated Links

We study an online job scheduling problem arising in networks with aggregated links. The goal is to schedule n jobs, divided into k disjoint chains, on m identical machines, without preemption, so that the jobs within each chain complete in the order of release times and the maximum flow time is minimized. We present a deterministic online algorithm with competitive ratio , and show a matching lower bound, even for randomized algorithms. The performance bound for we derive in the paper is, in fact, more subtle than a standard competitive ratio bound, and it shows that in overload conditions (when many jobs are released in a short amount of time), ’s performance is close to the optimum. We also show how to compute an offline solution efficiently for k=1, and that minimizing the maximum flow time for k,m≥2 is -hard. As by-products of our method, we obtain two offline polynomial-time algorithms for minimizing makespan: an optimal algorithm for k=1, and a 2-approximation algorithm for any k. W. Jawor and M. Chrobak supported by NSF grants OISE-0340752 and CCR-0208856. Work of C. Dürr conducted while being affiliated with the Laboratoire de Recherche en Informatique, Université Paris-Sud, 91405 Orsay. Supported by the CNRS/NSF grant 17171 and ANR Alpage.     

Improved Algorithms for Maximum Agreement and Compatible Supertrees

Consider a set of labels L and a set of unordered trees \(\mathcal{T}=\{\mathcal{T}^{(1)},\mathcal{T}^{(2)},\ldots ,\allowbreak \mathcal{T}^{(k)}\}\) where each tree \(\mathcal{T}^{(i)}\) is distinctly leaf-labeled by some subset of L. One fundamental problem is to find the biggest tree (denoted as supertree) to represent \(\mathcal{T}\) which minimizes the disagreements with the trees in \(\mathcal{T}\) under certain criteria. In this paper, we focus on two particular supertree problems, namely, the maximum agreement supertree problem (MASP) and the maximum compatible supertree problem (MCSP). These two problems are known to be NP-hard for k≥3. This paper gives improved algorithms for both MASP and MCSP. In particular, our results imply the first polynomial time algorithms for both MASP and MCSP when both k and the maximum degree D of the input trees are constant.     

“Algorithms for Degree—Raising of Splines”中的问题及其…

研究表明，Ｃｏｈｅｎ，Ｌｙｃｈｅ和Ｓｃｈｕｍａｋｅｒ于１９８５年在ＡＣＭＴｒａｎｓｃｃｔｉｏｎｓｏｎＧｒａｐｈｉｃｓ杂志上所发表的论文“ＡｌｇｏｒｔｉｔｈｍｓｆｏｒＤｅｇｒｅｅ－ＲａｓｉｓｉｎｇｏｆＳｐｌｉｎｅｓ”中存在着严重问题，文中给出解决Ｂ样条曲线升阶的经典理论中的这些问题的三个算法，同时也给出了计算实例。     

An Approximation Algorithm for Binary Searching in Trees

We consider the problem of computing efficient strategies for searching in trees. As a generalization of the classical binary search for ordered lists, suppose one wishes to find a (unknown) specific node of a tree by asking queries to its arcs, where each query indicates the endpoint closer to the desired node. Given the likelihood of each node being the one searched, the objective is to compute a search strategy that minimizes the expected number of queries. Practical applications of this problem include file system synchronization and software testing. Here we present a linear time algorithm which is the first constant factor approximation for this problem. This represents a significant improvement over previous O(log n)-approximation.     

Linear Time Algorithms for Generalizations of the Longest Common Substring Problem

In its simplest form, the longest common substring problem is to find a longest substring common to two or multiple strings. Using (generalized) suffix trees, this problem can be solved in linear time and space. A first generalization is the k -common substring problem: Given m strings of total length n, for all k with 2≤k≤m simultaneously find a longest substring common to at least k of the strings. It is known that the k-common substring problem can also be solved in O(n) time (Hui in Proc. 3rd Annual Symposium on Combinatorial Pattern Matching, volume 644 of Lecture Notes in Computer Science, pp. 230–243, Springer, Berlin, 1992). A further generalization is the k -common repeated substring problem: Given m strings T ⁽¹⁾,T ⁽²⁾,…,T ^(m) of total length n and m positive integers x ₁,…,x _m , for all k with 1≤k≤m simultaneously find a longest string ω for which there are at least k strings \(T^{(i_{1})},T^{(i_{2})},\ldots,T^{(i_{k})}\) (1≤i ₁<i ₂<???<i _k ≤m) such that ω occurs at least \(x_{i_{j}}\) times in \(T^{(i_{j})}\) for each j with 1≤j≤k. (For x ₁=???=x _m =1, we have the k-common substring problem.) In this paper, we present the first O(n) time algorithm for the k-common repeated substring problem. Our solution is based on a new linear time algorithm for the k-common substring problem.     

Feasible Operations on Proofs: The Logic of Proofs for Bounded Arithmetic

This paper presents a semantics for the logic of proofs in which all the operations on proofs are realized by feasibly computable functions. More precisely, we will show that the completeness of for the semantics of proofs of Peano Arithmetic extends to the semantics of proofs in Buss’ bounded arithmetic . In view of applications in epistemology of in particular and justification logics in general this result shows that explicit knowledge in the propositional framework can be made computationally feasible. This research supported by CUNY Community College Collaborative Incentive Research Grant 91639-0001 “Mathematical Foundations of Knowledge Representation”.     

Adaptive Finite Element Approximation for a Constrained Optimal Control Problem via Multi-meshes

In this paper, we study adaptive finite element approximation schemes for a constrained optimal control problem. We derive the equivalent a posteriori error estimators for both the state and the control approximation, which particularly suit an adaptive multi-mesh finite element scheme. The error estimators are then implemented and tested with promising numerical results.     

Faster Fixed-Parameter Tractable Algorithms for Matching and Packing Problems

We obtain faster algorithms for problems such as r-dimensional matching and r-set packing when the size k of the solution is considered a parameter. We first establish a general framework for finding and exploiting small problem kernels (of size polynomial in k). This technique lets us combine Alon, Yuster and Zwick’s color-coding technique with dynamic programming to obtain faster fixed-parameter algorithms for these problems. Our algorithms run in time O(n+2 ^O(k)), an improvement over previous algorithms for some of these problems running in time O(n+k ^O(k)). The flexibility of our approach allows tuning of algorithms to obtain smaller constants in the exponent. Research initiated at the International Workshop on Fixed Parameter Tractability in Computational Geometry and Games, Bellairs Research Institute of McGill University, Holetown, Barbados, Feb. 7–13, 2004, organized by S. Whitesides. D.M. Thilikos supported by the EU within the 6th Framework Programme under contract 001907 (DELIS) and by the Spanish CICYT project TIC-2002-04498-C05-03 (TRACER).     

Branch and Recharge: Exact Algorithms for Generalized Domination

In this paper we present branching algorithms for infinite classes of problems.     

A Better Constant-Factor Approximation for Selected-Internal Steiner Minimum Tree

The selected-internal Steiner minimum tree problem is a generalization of original Steiner minimum tree problem. Given a weighted complete graph G=(V,E) with weight function c, and two subsets R ^′ ⊊ R⊆V with |R−R ^′|≥2, selected-internal Steiner minimum tree problem is to find a minimum subtree T of G interconnecting R such that any leaf of T does not belong to R ^′. In this paper, suppose c is metric, we obtain a (1+ρ)-approximation algorithm for this problem, where ρ is the best-known approximation ratio for the Steiner minimum tree problem.     

Fully Polynomial Approximation Schemes for a Symmetric Quadratic Knapsack Problem and its Scheduling Applications

We design a fully polynomial-time approximation scheme (FPTAS) for a knapsack problem to minimize a symmetric quadratic function. We demonstrate how the designed FPTAS can be adopted for several single machine scheduling problems to minimize the sum of the weighted completion times. The applications presented in this paper include problems with a single machine non-availability interval (for both the non-resumable and the resumable scenarios) and a problem of planning a single machine maintenance period; the latter problem is closely related to a single machine scheduling problem with two competing agents. The running time of each presented FPTAS is strongly polynomial.