The Directed Orienteering Problem

This paper studies vehicle routing problems on asymmetric metrics. Our starting point is the directed k-TSP problem: given an asymmetric metric (V,d), a root r∈V and a target k≤|V|, compute the minimum length tour that contains r and at least k other vertices. We present a polynomial time O(\fraclog² nloglogn·logk)O(\frac{\log^{2} n}{\log\log n}\cdot\log k)-approximation algorithm for this problem. We use this algorithm for directed k-TSP to obtain an O(\fraclog² nloglogn)O(\frac{\log^{2} n}{\log\log n})-approximation algorithm for the directed orienteering problem. This answers positively, the question of poly-logarithmic approximability of directed orienteering, an open problem from Blum et al. (SIAM J. Comput. 37(2):653–670, 2007). The previously best known results were quasi-polynomial time algorithms with approximation guarantees of O(log ² k) for directed k-TSP, and O(log n) for directed orienteering (Chekuri and Pal in IEEE Symposium on Foundations in Computer Science, pp. 245–253, 2005). Using the algorithm for directed orienteering within the framework of Blum et al. (SIAM J. Comput. 37(2):653–670, 2007) and Bansal et al. (ACM Symposium on Theory of Computing, pp. 166–174, 2004), we also obtain poly-logarithmic approximation algorithms for the directed versions of discounted-reward TSP and vehicle routing problem with time-windows.     

Wavelength Assignment for Realizing Parallel FFT on Regular Optical Networks

Routing and wavelength assignment (RWA) is a central issue to increase efficiency and reduce cost in Wavelength Division Multiplexing (WDM) optical networks. In this paper, we address the problem of wavelength assignment for realizing parallel FFT on a class of regular optical WDM networks. We propose two methods for sequential mapping and shift-reversal mapping of FFT communication pattern to the optical WDM networks concerned. By sequential mapping, the numbers of wavelengths required to realize parallel FFT with 2ⁿ nodes on WDM linear arrays, rings, 2-D meshes and 2-D tori are 2^{n − 1}, 2^{n − 1}, 2^{max (k,n − k) − 1} and 2^{max (k,n − k) − 1} respectively. By shift-reversal mapping, the numbers of wavelengths required are max (3× 2^{n − 3},2), 2^{n − 2}, max (3× 2^{max (k,n − k) − 3},2) and 2^{max (k,n − k) − 2}. These results show that shift-reversal mapping outperforms sequential mapping. Our results have a clear significance for applications because FFT represents a common computation pattern shared by a large class of scientific and engineering problems and WDM optical networks as a promising technology in networking has an increasing popularity.     

Indexing Factors with Gaps

Indexing of factors or substrings is a widely used and useful technique in stringology and can be seen as a tool in solving diverse text algorithmic problems. A gapped-factor is a concatenation of a factor of length k, a gap of length d and another factor of length k′. Such a gapped factor is called a (k−d−k′)-gapped-factor. The problem of indexing the gapped-factors was considered recently by Peterlongo et al. (In: Stringology, pp. 182–196, 2006). In particular, Peterlongo et al. devised a data structure, namely a gapped factor tree (GFT) to index the gapped-factors. Given a text of length n over the alphabet Σ and the values of the parameters k, d and k′, the construction of GFT requires O(n|Σ|) time. Once GFT is constructed, a given (k−d−k′)-gapped-factor can be reported in O(k+k′+Occ) time, where Occ is the number of occurrences of that factor in  . In this paper, we present a new improved indexing scheme for the gapped-factors. The improvements we achieve come from two aspects. Firstly, we generalize the indexing data structure in the sense that, unlike GFT, it is independent of the parameters k and k′. Secondly, our data structure can be constructed in O(nlog ^1+ε n) time and space, where 0<ε<1. The only price we pay is a slight increase, i.e. an additional log log n term, in the query time. Preliminary version appeared in [29]. C.S. Iliopoulos is supported by EPSRC and Royal Society grants. M.S. Rahman is supported by the Commonwealth Scholarship Commission in the UK under the Commonwealth Scholarship and Fellowship Plan (CSFP). M.S. Rahman is on leave from Department of CSE, BUET, Dhaka 1000, Bangladesh.     

Construction of vertex-disjoint paths in alternating group networks

The existence of parallel node-disjoint paths between any pair of nodes is a desirable property of interconnection networks, because such paths allow tolerance to node and/or link failures along some of the paths, without causing disconnection. Additionally, node-disjoint paths support high-throughput communication via the concurrent transmission of parts of a message. We characterize maximum-sized families of parallel paths between any two nodes of alternating group networks. More specifically, we establish that in a given alternating group network AN _n , there exist n−1 parallel paths (the maximum possible, given the node degree of n−1) between any pair of nodes. Furthermore, we demonstrate that these parallel paths are optimal or near-optimal, in the sense of their lengths exceeding the internode distance by no more than four. We also show that the wide diameter of AN _n is at most one unit greater than the known lower bound D+1, where D is the network diameter.     

Resource scheduling with variable requirements over time

The problem of scheduling resources for tasks with variable requirements over time can be stated as follows. We are given two sequences of vectors A=A ₁,…,A _n and R=R ₁,…,R _m . Sequence A represents resource availability during n time intervals, where each vector A _i has q elements. Sequence R represents resource requirements of a task during m intervals, where each vector R _i has q elements. We wish to find the earliest time interval i, termed latency, such that for 1≤k≤m, 1≤j≤q: A _i+k−1 ^j ≥R _k ^j , where A _i+k−1 ^j and R _k ^j are the jth elements of vectors A _i+k−1 and R _k , respectively. One application of this problem is I/O scheduling for multimedia presentations. The fastest known algorithm to compute the optimal solution of this problem has computation time (Amir and Farach, in Proceedings of the ACM-SIAM symposium on discrete algorithms (SODA), San Francisco, CA, pp. 212–223, 1991; Inf. Comput. 118(1):1–11, 1995). We propose a technique that approximates the optimal solution in linear time: . We evaluated the performance of our algorithm when used for multimedia I/O scheduling. Our results show that 95% of the time, our solution is within 5% of the optimal.     

Panconnectivity of Cartesian product graphs

A graph G of order n (≥2) is said to be panconnected if for each pair (x,y) of vertices of G there exists an xy-path of length ℓ for each ℓ such that d _G (x,y)≤ℓ≤n−1, where d _G (x,y) denotes the length of a shortest xy-path in G. In this paper, we consider the panconnectivity of Cartesian product graphs. As a consequence of our results, we prove that the n-dimensional generalized hypercube Q _n (k ₁,k ₂,…,k _n ) is panconnected if and only if k _i ≥3 (i=1,…,n), which generalizes a result of Hsieh et al. that the 3-ary n-cube Q³_nQ^{3}_{n} is panconnected.     

A Cubic Kernel for Feedback Vertex Set and Loop Cutset

The Feedback Vertex Set problem on unweighted, undirected graphs is considered. Improving upon a result by Burrage et al. (Proceedings 2nd International Workshop on Parameterized and Exact Computation, pp. 192–202, 2006), we show that this problem has a kernel with O(k ³) vertices, i.e., there is a polynomial time algorithm, that given a graph G and an integer k, finds a graph G′ with O(k ³) vertices and integer k′≤k, such that G has a feedback vertex set of size at most k, if and only if G′ has a feedback vertex set of size at most k′. Moreover, the algorithm can be made constructive: if the reduced instance G′ has a feedback vertex set of size k′, then we can easily transform a minimum size feedback vertex set of G′ into a minimum size feedback vertex set of G. This kernelization algorithm can be used as the first step of an FPT algorithm for Feedback Vertex Set, but also as a preprocessing heuristic for Feedback Vertex Set.     

On the number of transitive digraphs withn labeled vertices andk arcs

A computer determination of the numberT _n,k of transitive digraphs onn labeled vertices andk arcs is obtained for 2≤n≤6,0 ≤k≤n ²-n. Furthermore some formulae are given for determiningT _n,k, ∇n for extremal values ofk (namely, 0≤k≤6 andn ² -3n+4≤k≤n ²-n). Work carried out at Fondazione Ugo Bordoni under the agreements between Fondazione Ugo Bordoni and the Istituto Superiore P. T.     

Linear Time Algorithms for Exact Distance Transform

In 2003, Maurer et al. (IEEE Trans. Pattern Anal. Mach. Intell. 25:265–270, 2003) published a paper describing an algorithm that computes the exact distance transform in linear time (with respect to image size) for the rectangular binary images in the k-dimensional space ℝ ^k and distance measured with respect to L _p -metric for 1≤p≤∞, which includes Euclidean distance L ₂. In this paper we discuss this algorithm from theoretical and practical points of view. On the practical side, we concentrate on its Euclidean distance version, discuss the possible ways of implementing it as signed distance transform, and experimentally compare implemented algorithms. We also describe the parallelization of these algorithms and discuss the computational time savings associated with them. All these implementations will be made available as a part of the CAVASS software system developed and maintained in our group (Grevera et al. in J. Digit. Imaging 20:101–118, 2007). On the theoretical side, we prove that our version of the signed distance transform algorithm, GBDT, returns the exact value of the distance from the geometrically defined object boundary. We provide a complete proof (which was not given of Maurer et al. (IEEE Trans. Pattern Anal. Mach. Intell. 25:265–270, 2003) that all these algorithms work correctly for L _p -metric with 1<p<∞. We also point out that the precise form of the algorithm from Maurer et al. (IEEE Trans. Pattern Anal. Mach. Intell. 25:265–270, 2003) is not well defined for L ₁ and L _∞ metrics. In addition, we show that the algorithm can be used to find, in linear time, the exact value of the diameter of an object, that is, the largest possible distance between any two of its elements.     

Il metodo di Treffiz-Fichera per i semigruppi. potenziale di tipo potenza

A variation of the Trefftz-Fichera method is presented to compute lower bounds for the eigenvalues of a positive self-adjoint operator with discrete spectrum with grow at least in a logarithmic way as the index diverges. As suggested by Barnes et al. [2] to compute ground state, the semigroupe ^−βH, β>0, is used rather than the iterated resolvent(H+β) ⁻ⁿ,n=1,2,... As an example, the method is applied to the operatorH=−Δ+|x|^γ. inL ²(R), 1≤γ≤4.      

Sublinear Fully Distributed Partition with Applications

We present new efficient deterministic and randomized distributed algorithms for decomposing a graph with n nodes into a disjoint set of connected clusters with radius at most k−1 and having O(n ^1+1/k ) intercluster edges. We show how to implement our algorithms in the distributed CONGEST\mathcal{CONGEST} model of computation, i.e., limited message size, which improves the time complexity of previous algorithms (Moran and Snir in Theor. Comput. Sci. 243(1–2):217–241, 2000; Awerbuch in J. ACM 32:804–823, 1985; Peleg in Distributed Computing: A Locality-Sensitive Approach, 2000) from O(n) to O(n ^1−1/k ). We apply our algorithms for constructing low stretch graph spanners and network synchronizers in sublinear deterministic time in the CONGEST\mathcal{CONGEST} model.     

A Unified Primal-Dual Algorithm Framework Based on Bregman Iteration

In this paper, we propose a unified primal-dual algorithm framework for two classes of problems that arise from various signal and image processing applications. We also show the connections to existing methods, in particular Bregman iteration (Osher et al., Multiscale Model. Simul. 4(2):460–489, 2005) based methods, such as linearized Bregman (Osher et al., Commun. Math. Sci. 8(1):93–111, 2010; Cai et al., SIAM J. Imag. Sci. 2(1):226–252, 2009, CAM Report 09-28, UCLA, March 2009; Yin, CAAM Report, Rice University, 2009) and split Bregman (Goldstein and Osher, SIAM J. Imag. Sci., 2, 2009). The convergence of the general algorithm framework is proved under mild assumptions. The applications to ℓ ¹ basis pursuit, TV−L ² minimization and matrix completion are demonstrated. Finally, the numerical examples show the algorithms proposed are easy to implement, efficient, stable and flexible enough to cover a wide variety of applications.     

Fast Evaluation of Interlace Polynomials on Graphs of Bounded Treewidth

We consider the multivariate interlace polynomial introduced by Courcelle (Electron. J. Comb. 15(1), 2008), which generalizes several interlace polynomials defined by Arratia, Bollobás, and Sorkin (J. Comb. Theory Ser. B 92(2):199–233, 2004) and by Aigner and van der Holst (Linear Algebra Appl., 2004). We present an algorithm to evaluate the multivariate interlace polynomial of a graph with n vertices given a tree decomposition of the graph of width k. The best previously known result (Courcelle, Electron. J. Comb. 15(1), 2008) employs a general logical framework and leads to an algorithm with running time f(k)⋅n, where f(k) is doubly exponential in k. Analyzing the GF(2)-rank of adjacency matrices in the context of tree decompositions, we give a faster and more direct algorithm. Our algorithm uses 2^3k2+O(k)·n2^{3k^{2}+O(k)}\cdot n arithmetic operations and can be efficiently implemented in parallel.     

A randomized clustering of anonymous wireless ad hoc networks with an application to the initialization problem

The Distributed Mobility-Adaptive Clustering (DMAC) due to Basagni partitions the nodes of a mobile ad hoc network into clusters, thus giving the network a hierarchical organization. This algorithm supports the mobility of the nodes, even during the cluster formation. The main feature of DMAC is that in a weighted network (in which two or more nodes cannot have the same weight), nodes have to choose the clusterheads taking into account only the node weight, i.e. the mobility when a node weight is the inverse of its speed. In our approach many nodes may have the same speed and hence the same weight. We assume that nodes have no identities and the number of nodes, say n, is the only known parameter of the network. After the randomized clustering, we show that the initialization problem can be solved in a multi-hop ad hoc wireless network of n stations in O(k ^1/2log ^1/2 k)+D _b −1+O(log (max (P _i )+log ²max (P _i )) broadcast rounds with high probability, where k is the number of clusters, D _b is the blocking diameter and max (P _i ), 1≤i≤k, is the maximum number of nodes in a cluster. Thus the initialization protocol presented here uses less broadcast rounds than the one in Ravelemanana (IEEE Trans. Parallel Distributed Syst. 18(1):17–28 2007).     

Space Efficient Dynamic Orthogonal Range Reporting

In this paper we present new space efficient dynamic data structures for orthogonal range reporting. The described data structures support planar range reporting queries in time O(log n+klog log (4n/(k+1))) and space O(nlog log n), or in time O(log n+k) and space O(nlog  ^ε n) for any ε>0. Both data structures can be constructed in O(nlog n) time and support insert and delete operations in amortized time O(log ² n) and O(log nlog log n) respectively. These results match the corresponding upper space bounds of Chazelle (SIAM J. Comput. 17, 427–462, 1988) for the static case. We also present a dynamic data structure for d-dimensional range reporting with search time O(log  ^d−1 n+k), update time O(log  ^d n), and space O(nlog  ^d−2+ε n) for any ε>0. The model of computation used in our paper is a unit cost RAM with word size log n. A preliminary version of this paper appeared in the Proceedings of the 21st Annual ACM Symposium on Computational Geometry 2005. Work partially supported by IST grant 14036 (RAND-APX).     

Approximating Buy-at-Bulk and Shallow-Light <Emphasis Type="Italic">k</Emphasis>-Steiner Trees

We study two related network design problems with two cost functions. In the buy-at-bulk k-Steiner tree problem we are given a graph G(V,E) with a set of terminals T⊆V including a particular vertex s called the root, and an integer k≤|T|. There are two cost functions on the edges of G, a buy cost b:E→ℝ⁺ and a distance cost r:E→ℝ⁺. The goal is to find a subtree H of G rooted at s with at least k terminals so that the cost ∑ _e∈H b(e)+∑ _t∈T−s dist(t,s) is minimized, where dist(t,s) is the distance from t to s in H with respect to the r cost. We present an O(log ⁴ n)-approximation algorithm for the buy-at-bulk k-Steiner tree problem. The second and closely related one is bicriteria approximation algorithm for Shallow-light k-Steiner trees. In the shallow-light k-Steiner tree problem we are given a graph G with edge costs b(e) and distance costs r(e), and an integer k. Our goal is to find a minimum cost (under b-cost) k-Steiner tree such that the diameter under r-cost is at most some given bound D. We develop an (O(log n),O(log ³ n))-approximation algorithm for a relaxed version of Shallow-light k-Steiner tree where the solution has at least terminals. Using this we obtain an (O(log ² n),O(log ⁴ n))-approximation algorithm for the shallow-light k-Steiner tree and an O(log ⁴ n)-approximation algorithm for the buy-at-bulk k-Steiner tree problem. Our results are recently used to give the first polylogarithmic approximation algorithm for the non-uniform multicommodity buy-at-bulk problem (Chekuri, C., et al. in Proceedings of 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS’06), pp. 677–686, 2006). A preliminary version of this paper appeared in the Proceedings of 9th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX) 2006, LNCS 4110, pp. 153–163, 2006. M.T. Hajiaghayi supported in part by IPM under grant number CS1383-2-02. M.R. Salavatipour supported by NSERC grant No. G121210990, and a faculty start-up grant from University of Alberta.     

Model Checking Weighted Integer Reset Timed Automata

Weighted timed automata (WTA), introduced in Alur et al. (Proceedings of HSCC’01, LNCS, vol. 2034, pp. 49–62, Springer, Berlin, 2001), Behrmann et al. (Proceedings of HSCC’01, LNCS, vol. 2034, pp. 147–161, Springer, Berlin, 2001) are an extension of Alur and Dill (Theor. Comput. Sci. 126(2):183–235, 1994) timed automata, a widely accepted formalism for the modelling and verification of real time systems. Weighted timed automata extend timed automata by allowing costs on the locations and edges. There has been a lot of interest Bouyer et al. (Inf. Process. Lett. 98(5):188–194, 2006), Bouyer et al. (Log. Methods Comput. Sci. 4(2):9, 2008), Brihaye et al. (Proceedings of FORMATS/FTRTFT’04, LNCS, vol. 3253, pp. 277–292, Springer, Berlin, 2004), Brihaye et al. (Inf. Comput. 204(3):408–433, 2006) in studying the model checking problem of weighted timed automata. The properties of interest are written using logic weighted CTL (WCTL), an extension of CTL with costs. It has been shown Bouyer et al. (Log. Methods Comput. Sci. 4(2):9, 2008) that the problem of model checking WTAs with a single clock using WCTL with no external cost variables is decidable, while 3 clocks render the problem undecidable Bouyer et al. (Inf. Process. Lett. 98(5):188–194, 2006). The question of 2 clocks is open. In this paper, we introduce a subclass of weighted timed automata called weighted integer reset timed automata (WIRTA) and study the model checking problem. We give a clock reduction technique for WIRTA. Given a WIRTA A\mathcal{A} with n≥1 clocks, we show that a single clock WIRTA A￠\mathcal{A}' preserving the paths and costs of A\mathcal{A} can be obtained. This gives us the decidability of model checking WIRTA with n≥1 clocks and m≥1 costs using WCTL with no external cost variables. We then show that for a restricted version of WCTL with external cost variables, the model checking problem is undecidable for WIRTA with 3 stopwatch costs and 1 clock. Finally, we show that model checking WTA with 2 clocks and 1 stopwatch cost against WCTL with no external cost variables is undecidable, thereby answering a question that has remained long open.     

On the Planar Piecewise Quadratic 1-Center Problem

In this paper we introduce a minimax model unifying several classes of single facility planar center location problems. We assume that the transportation costs of the demand points to the serving facility are convex functions {Q _i }, i=1,…,n, of the planar distance used. Moreover, these functions, when properly transformed, give rise to piecewise quadratic functions of the coordinates of the facility location. In the continuous case, using results on LP-type models by Clarkson (J. ACM 42:488–499, 1995), Matoušek et al. (Algorithmica 16:498–516, 1996), and the derandomization technique in Chazelle and Matoušek (J. Algorithms 21:579–597, 1996), we claim that the model is solvable deterministically in linear time. We also show that in the separable case, one can get a direct O(nlog n) deterministic algorithm, based on Dyer (Proceedings of the 8th ACM Symposium on Computational Geometry, 1992), to find an optimal solution. In the discrete case, where the location of the center (server) is restricted to some prespecified finite set, we introduce deterministic subquadratic algorithms based on the general parametric approach of Megiddo (J. ACM 30:852–865, 1983), and on properties of upper envelopes of collections of quadratic arcs. We apply our methods to solve and improve the complexity of a number of other location problems in the literature, and solve some new models in linear or subquadratic time complexity.     

Approximability of Sparse Integer Programs

The main focus of this paper is a pair of new approximation algorithms for certain integer programs. First, for covering integer programs {min cx:Ax≥b,0≤x≤d} where A has at most k nonzeroes per row, we give a k-approximation algorithm. (We assume A,b,c,d are nonnegative.) For any k≥2 and ε>0, if P≠NP this ratio cannot be improved to k−1−ε, and under the unique games conjecture this ratio cannot be improved to k−ε. One key idea is to replace individual constraints by others that have better rounding properties but the same nonnegative integral solutions; another critical ingredient is knapsack-cover inequalities. Second, for packing integer programs {max cx:Ax≤b,0≤x≤d} where A has at most k nonzeroes per column, we give a (2k ²+2)-approximation algorithm. Our approach builds on the iterated LP relaxation framework. In addition, we obtain improved approximations for the second problem when k=2, and for both problems when every A _ij is small compared to b _i . Finally, we demonstrate a 17/16-inapproximability for covering integer programs with at most two nonzeroes per column.     

Solving MAX-<Emphasis Type="Italic">r</Emphasis>-SAT Above a Tight Lower Bound

We present an exact algorithm that decides, for every fixed r≥2 in time O(m)+2^O(k2)O(m)+2^{O(k^{2})} whether a given multiset of m clauses of size r admits a truth assignment that satisfies at least ((2 ^r −1)m+k)/2 ^r clauses. Thus Max-r-Sat is fixed-parameter tractable when parameterized by the number of satisfied clauses above the tight lower bound (1−2^−r )m. This solves an open problem of Mahajan et al. (J. Comput. Syst. Sci. 75(2):137–153, 2009).