Modulo Constraints and the Complexity of Typechecking XML Views

The typechecking problem for transformations of relational data into tree data is the following: given a relational-to-XML transformation P, and an XML type d, decide whether for every database instance the result of the transformation P on satisfies d. TreeQL programs with projection-free conjunctive queries (see Alon et al. in ACM Trans. Comput. Log. 4(3):315–354, 2003) are considered as transformations and DTDs with arbitrary regular expressions as XML types. A non-elementary upper bound for the typechecking problem was already given by Alon et al. (ACM Trans. Comput. Log. 4(3):315–354, 2003) (although in a more general setting, where equality and negation in projection-free conjunctive queries and additional universal integrity constraints are allowed). In this paper we show that the typechecking problem is coNEXPTIME-complete. As an intermediate step we consider the following problem, which can be formulated independently of XML notions. Given a set of triples of the form (φ,k,j), where φ is a projection-free conjunctive query and k,j are natural numbers, decide whether there exists a database such that, for each triple (φ,k,j) in the set, there exists a natural number α, such that there are exactly k+j*α tuples satisfying the query φ in . Our main technical contribution consists of a NEXPTIME algorithm for the last problem. Partially supported by Polish Ministry of Science and Higher Education research project N206 022 31/3660, 2006/2009. This paper is an extended version of 20, where the coNEXPTIME upper bound was shown.     

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.     

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.     

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.     

Quadratic Kernelization for Convex Recoloring of Trees

The Convex Recoloring (CR) problem measures how far a tree of characters differs from exhibiting a so-called “perfect phylogeny”. For an input consisting of a vertex-colored tree T, the problem is to determine whether recoloring at most k vertices can achieve a convex coloring, meaning by this a coloring where each color class induces a subtree. The problem was introduced by Moran and Snir (J. Comput. Syst. Sci. 73:1078–1089, 2007; J. Comput. Syst. Sci. 74:850–869, 2008) who showed that CR is NP-hard, and described a search-tree based FPT algorithm with a running time of O(k(k/log k) ^k n ⁴). The Moran and Snir result did not provide any nontrivial kernelization. In this paper, we show that CR has a kernel of size O(k ²).     

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.     

Improved on-line broadcast scheduling with deadlines

We study an on-line broadcast scheduling problem in which requests have deadlines, and the objective is to maximize the weighted throughput, i.e., the weighted total length of the satisfied requests. For the case where all requested pages have the same length, we present an online deterministic algorithm named BAR and prove that it is 4.56-competitive. This improves the previous algorithm of (Kim, J.-H., Chwa, K.-Y. in Theor. Comput. Sci. 325(3):479–488, 2004) which is shown to be 5-competitive by (Chan, W.-T., et al. in Lecture Notes in Computer Science, vol. 3106, pp. 210–218, 2004). In the case that pages may have different lengths, we give a ( )-competitive algorithm where Δ is the ratio of maximum to minimum page lengths. This improves the (4Δ+3)-competitive algorithm of (Chan, W.-T., et al. in Lecture Notes in Computer Science, vol. 3106, pp. 210–218, 2004). We also prove an almost matching lower bound of Ω(Δ/log Δ). Furthermore, for small values of Δ we give better lower bounds. The work described in this paper was fully supported by grants from the Research Grants Council of the Hong Kong SAR, China [CityU 1198/03E, HKU 7142/03E, HKU 5172/03E], an NSF Grant of China [No. 10371094], and a Nuffield Foundation Grant of UK [NAL/01004/G].     

Graph Spanners in the Streaming Model: An Experimental Study

This article reports the results of an extensive experimental analysis of efficient algorithms for computing graph spanners in the data streaming model, where an (α,β)-spanner of a graph G is a subgraph S⊆G such that for each pair of vertices the distance in S is at most α times the distance in G plus β. To the best of our knowledge, this is the first computational study of graph spanner algorithms in a streaming setting. We compare experimentally the randomized algorithms proposed by Baswana () and by Elkin (In: Proceedings of the 34th International Colloquium on Automata, Languages and Programming (ICALP 2007), Wroclaw, Poland, pp. 716–727, 9–13 July 2007) for general stretch factors with the deterministic algorithm presented by Ausiello et al. (In: Proceedings of the 15th Annual European Symposium on Algorithms (ESA 2007), Engineering and Applications Track, Eilat, Israel, 8–10 October 2007. LNCS, vol. 4698, pp. 605–617, 2007), designed for building small stretch spanners. All the algorithms we implemented work in a data streaming model where the input graph is given as a stream of edges in arbitrary order, and all of them need a single pass over the data. Differently from the algorithm in Ausiello et al., the algorithms in Baswana () and Elkin (In: Proceedings of the 34th International Colloquium on Automata, Languages and Programming (ICALP 2007), Wroclaw, Poland, pp. 716–727, 9–13 July 2007) need to know in advance the number of vertices in the graph. The results of our experimental investigation on several input families confirm that all these algorithms are very efficient in practice, finding spanners with stretch and size much smaller than the theoretical bounds and comparable to those obtainable by off-line algorithms. Moreover, our experimental findings confirm that small values of the stretch factor are the case of interest in practice, and that the algorithm by Ausiello et al. tends to produce spanners of better quality than the algorithms by Baswana and Elkin, while still using a comparable amount of time and space resources. Work partially supported by the Italian Ministry of University and Research under Project MAINSTREAM “Algorithms for Massive Information Structures and Data Streams”. A preliminary version of this paper was presented at the 15th Annual European Symposium on Algorithms (ESA 2007) 5.     

On the Minimum Feedback Vertex Set Problem: Exact and Enumeration Algorithms

We present a time algorithm finding a minimum feedback vertex set in an undirected graph on n vertices. We also prove that a graph on n vertices can contain at most 1.8638 ⁿ minimal feedback vertex sets and that there exist graphs having 105 ^n/10≈1.5926 ⁿ minimal feedback vertex sets. Preliminary extended abstracts of this paper appeared in the proceedings of SWAT’06 [29] and IWPEC’06 [18]. Additional support of F.V. Fomin, S. Gaspers and A.V. Pyatkin by the Research Council of Norway. The work of A.V. Pyatkin was partially supported by grants of the Russian Foundation for Basic Research (project code 05-01-00395), INTAS (project code 04–77–7173). I. Razgon is supported by Science Foundation Ireland (Grant Number 05/IN/I886).     

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.     

Applying Approximate Counting for Computing the Frequency Moments of Long Data Streams

This paper takes up a remark in the well-known paper of Alon, Matias, and Szegedy (J. Comput. Syst. Sci. 58(1):137–147, 1999) about the computation of the frequency moments of data streams and shows in detail how any F _k with k≥1 can be approximately computed using space O(km ^1−1/k (k+log m+log log  n)) based on approximate counting. An important building block for this, which may be interesting in its own right, is a new approximate variant of reservoir sampling using space O(log log  n) for constant error parameters.     

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).     

Path Hitting in Acyclic Graphs

An instance of the path hitting problem consists of two families of paths, and ℋ, in a common undirected graph, where each path in ℋ is associated with a non-negative cost. We refer to and ℋ as the sets of demand and hitting paths, respectively. When p∈ℋ and share at least one mutual edge, we say that p hits q. The objective is to find a minimum cost subset of ℋ whose members collectively hit those of . In this paper we provide constant factor approximation algorithms for path hitting, confined to instances in which the underlying graph is a tree, a spider, or a star. Although such restricted settings may appear to be very simple, we demonstrate that they still capture some of the most basic covering problems in graphs. Our approach combines several novel ideas: We extend the algorithm of Garg, Vazirani and Yannakakis (Algorithmica, 18:3–20, 1997) for approximate multicuts and multicommodity flows in trees to prove new integrality properties; we present a reduction that involves multiple calls to this extended algorithm; and we introduce a polynomial-time solvable variant of the edge cover problem, which may be of independent interest. An extended abstract of this paper appeared in Proceedings of the 14th Annual European Symposium on Algorithms, 2006. This work is part of D. Segev’s Ph.D. thesis prepared at Tel-Aviv University under the supervision of Prof. Refael Hassin.     

Lower Bounds for Kernelizations and Other Preprocessing Procedures

We first present a method to rule out the existence of parameter non-increasing polynomial kernelizations of parameterized problems under the hypothesis P≠NP. This method is applicable, for example, to the problem Sat parameterized by the number of variables of the input formula. Then we obtain further improvements of corresponding results in (Bodlaender et al. in Lecture Notes in Computer Science, vol. 5125, pp. 563–574, Springer, Berlin, 2008; Fortnow and Santhanam in Proceedings of the 40th ACM Symposium on the Theory of Computing (STOC’08), ACM, New York, pp. 133–142, 2008) by refining the central lemma of their proof method, a lemma due to Fortnow and Santhanam. In particular, assuming that the polynomial hierarchy does not collapse to its third level, we show that every parameterized problem with a “linear OR” and with NP-hard underlying classical problem does not have polynomial self-reductions that assign to every instance x with parameter k an instance y with |y|=k ^O(1)⋅|x|^1−ε (here ε is any given real number greater than zero). We give various applications of these results. On the structural side we prove several results clarifying the relationship between the different notions of preprocessing procedures, namely the various notions of kernelizations, self-reductions and compressions.     

A permuted factors approach for the linearization of polynomial matrices

In Antoniou and Vologiannidis (Electron J Linear Algebra 11:78–87, 2004; 15:107–114, 2006), a new family of companion forms associated with a regular polynomial matrix T (s) has been presented, using products of permutations of n elementary matrices, generalizing similar results presented in Fiedler (Linear Algebra Its Appl 371:325–331, 2003) where the scalar case was considered. In this paper, extending this “permuted factors” approach, we present a broader family of companion-like linearizations, using products of up to n(n−1)/2 elementary matrices, where n is the degree of the polynomial matrix. Under given conditions, the proposed linearizations can be shown to consist of block entries where the coefficients of the polynomial matrix appear intact. Additionally, we provide a criterion for those linearizations to be block symmetric. We also illustrate several new block symmetric linearizations of the original polynomial matrix T (s), where in some of them the constraint of nonsingularity of the constant term and the coefficient of maximum degree are not a prerequisite.     

A note on the alternating group network

The class of alternating group networks was introduced in the late 1990’s as an alternative to the alternating group graphs as interconnection networks. Recently, additional properties for the alternating group networks have been published. In particular, Zhou et al., J. Supercomput (2009), doi:, was published very recently in this journal. We show that this so-called new interconnection topology is in fact isomorphic to the (n,n−2)-star, a member of the well-known (n,k)-stars, 1≤k≤n−1, a class of popular networks proposed earlier for which a large amount of work have already been done. Specifically, the problem in Zhou et al., J. Supercomput (2009), doi:, was addressed in Lin and Duh, Inf. Sci. 178(3), 788–801, 2008, when k = n−2.     

Almost k-wise independence versus k-wise independence

We say that a distribution over {0,1}ⁿ is (ε,k)-wise independent if its restriction to every k coordinates results in a distribution that is ε-close to the uniform distribution. A natural question regarding (ε,k)-wise independent distributions is how close they are to some k-wise independent distribution. We show that there exist (ε,k)-wise independent distributions whose statistical distance is at least n^O(k)·ε from any k-wise independent distribution. In addition, we show that for any (ε,k)-wise independent distribution there exists some k-wise independent distribution, whose statistical distance is n^O(k)·ε.     

Optimal Error Estimates of the Local Discontinuous Galerkin Method for Surface Diffusion of Graphs on Cartesian Meshes

In (Xu and Shu in J. Sci. Comput. 40:375–390, 2009), a local discontinuous Galerkin (LDG) method for the surface diffusion of graphs was developed and a rigorous proof for its energy stability was given. Numerical simulation results showed the optimal order of accuracy. In this subsequent paper, we concentrate on analyzing a priori error estimates of the LDG method for the surface diffusion of graphs. The main achievement is the derivation of the optimal convergence rate k+1 in the L ² norm in one-dimension as well as in multi-dimensions for Cartesian meshes using a completely discontinuous piecewise polynomial space with degree k≥1.     

On Covering Problems of Rado

T. Rado conjectured in 1928 that if ℱ is a finite set of axis-parallel squares in the plane, then there exists an independent subset ℐ⊆ℱ of pairwise disjoint squares, such that ℐ covers at least 1/4 of the area covered by ℱ. He also showed that the greedy algorithm (repeatedly choose the largest square disjoint from those previously selected) finds an independent set of area at least 1/9 of the area covered by ℱ. The analogous question for other shapes and many similar problems have been considered by R. Rado in his three papers (in Proc. Lond. Math. Soc. 51:232–264, 1949; 53:243–267, 1951; and J. Lond. Math. Soc. 42:127–130, 1968) on this subject. After 45 years, Ajtai (in Bull. Acad. Polon. Sci. Sér. Sci. Math. Astron. Phys. 21:61–63, 1973) came up with a surprising example disproving T. Rado’s conjecture. We revisit Rado’s problem and present improved upper and lower bounds for squares, disks, convex bodies, centrally symmetric convex bodies, and others, as well as algorithmic solutions to these variants of the problem.     

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.