Reactive Local Search for the Maximum Clique Problem<Superscript><Emphasis Type="Italic">1</Emphasis></Superscript>

A new Reactive Local Search (\RLS ) algorithm is proposed for the solution of the Maximum-Clique problem. \RLS is based on local search complemented by a feedback (history-sensitive) scheme to determine the amount of diversification. The reaction acts on the single parameter that decides the temporary prohibition of selected moves in the neighborhood, in a manner inspired by Tabu Search. The performance obtained in computational tests appears to be significantly better with respect to all algorithms tested at the the second DIMACS implementation challenge. The worst-case complexity per iteration of the algorithm is O(max {n,m}) where n and m are the number of nodes and edges of the graph. In practice, when a vertex is moved, the number of operations tends to be proportional to its number of missing edges and therefore the iterations are particularly fast in dense graphs. Received September 11, 1997; revised February 5, 1998.     

An Augmented Lagrangian Method for <Emphasis Type="Italic">TV</Emphasis><Subscript><Emphasis Type="Italic">g</Emphasis></Subscript>+<Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>-norm Minimization

In this paper, the minimization of a weighted total variation regularization term (denoted TV _g ) with L ¹ norm as the data fidelity term is addressed using the Uzawa block relaxation method. The unconstrained minimization problem is transformed into a saddle-point problem by introducing a suitable auxiliary unknown. Applying a Uzawa block relaxation method to the corresponding augmented Lagrangian functional, we obtain a new numerical algorithm in which the main unknown is computed using Chambolle projection algorithm. The auxiliary unknown is computed explicitly. Numerical experiments show the availability of our algorithm for salt and pepper noise removal or shape retrieval and also its robustness against the choice of the penalty parameter. This last property is useful to attain the convergence in a reduced number of iterations leading to efficient numerical schemes. The specific role of the function g in TV _g is also investigated and we highlight the fact that an appropriate choice leads to a significant improvement of the denoising results. Using this property, we propose a whole algorithm for salt and pepper noise removal (denoted UBR-EDGE) that is able to handle high noise levels at a low computational cost. Shape retrieval and geometric filtering are also investigated by taking into account the geometric properties of the model.     

An O(2<Superscript>O(k)</Superscript>n<Superscript>3</Superscript>) FPT Algorithm for the Undirected Feedback Vertex Set Problem

We describe an algorithm for the Feedback Vertex Set problem on undirected graphs, parameterized by the size k of the feedback vertex set, that runs in time O(c^kn³) where c = 10.567 and n is the number of vertices in the graph. The best previous algorithms were based on the method of bounded search trees, branching on short cycles. The best previous running time of an FPT algorithm for this problem, due to Raman, Saurabh and Subramanian, has a parameter function of the form 2^{O(k log k /log log k)}. Whether an exponentially linear in k FPT algorithm for this problem is possible has been previously noted as a significant challenge. Our algorithm is based on the new FPT technique of iterative compression. Our result holds for a more general form of the problem, where a subset of the vertices may be marked as forbidden to belong to the feedback set. We also establish "exponential optimality" for our algorithm by proving that no FPT algorithm with a parameter function of the form O(2^o(k)) is possible, unless there is an unlikely collapse of parameterized complexity classes, namely FPT = M[1].     

A Flexible Algorithm for Generating All the Spanning Trees in Undirected Graphs

In this paper we propose an algorithm for generating all the spanning trees in undirected graphs. The algorithm requires O (n+m+ τ n) time where the given graph has n vertices, m edges, and τ spanning trees. For outputting all the spanning trees explicitly, this time complexity is optimal. Our algorithm follows a special rooted tree structure on the skeleton graph of the spanning tree polytope. The rule by which the rooted tree structure is traversed is irrelevant to the time complexity. In this sense, our algorithm is flexible. If we employ the depth-first search rule, we can save the memory requirement to O (n+m). A breadth-first implementation requires as much as O (m+ τ n) space, but when a parallel computer is available, this might have an advantage. When a given graph is weighted, the best-first search rule provides a ranking algorithm for the minimum spanning tree problem. The ranking algorithm requires O (n+ m + τ n) time and O (m+ τ n) space when we have a minimum spanning tree. Received January 21, 1995; revised February 19, 1996.     

Solving Connected Dominating Set Faster than 2<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

In the connected dominating set problem we are given an n-node undirected graph, and we are asked to find a minimum cardinality connected subset S of nodes such that each node not in S is adjacent to some node in S. This problem is also equivalent to finding a spanning tree with maximum number of leaves. Despite its relevance in applications, the best known exact algorithm for the problem is the trivial Ω(2 ⁿ ) algorithm that enumerates all the subsets of nodes. This is not the case for the general (unconnected) version of the problem, for which much faster algorithms are available. Such a difference is not surprising, since connectivity is a global property, and non-local problems are typically much harder to solve exactly. In this paper we break the 2 ⁿ barrier, by presenting a simple O(1.9407 ⁿ ) algorithm for the connected dominating set problem. The algorithm makes use of new domination rules, and its analysis is based on the Measure and Conquer technique. An extended abstract of this paper appeared in the proceedings of FSTTCS’06. Fedor V. Fomin was additionally supported by the Research Council of Norway.     

Algorithms for P<Subscript>4</Subscript>-Comparability Graph Recognition and Acyclic P<Subscript>4</Subscript>-Transitive Orientation

We consider two problems pertaining to P₄-comparability graphs, namely, the problem of recognizing whether a simple undirected graph is a P₄-comparability graph and the problem of producing an acyclic P₄-transitive orientation of a P₄-comparability graph. These problems have been considered by Hoàng and Reed who described O(n⁴)- and O(n⁵)-time algorithms for their solution, respectively, where n is the number of vertices of the input graph. Faster algorithms have recently been presented by Raschle and Simon, and by Nikolopoulos and Palios; the time complexity of these algorithms for either problem is O(n + m²), where m is the number of edges of the graph. In this paper we describe O(n m)-time and O(n + m)-space algorithms for the recognition and the acyclic P₄-transitive orientation problems on P₄-comparability graphs. The algorithms rely on properties of the P₄-components of a graph, which we establish, and on the efficient construction of the P₄-components by means of the BFS-trees of the complement of the graph rooted at each of its vertices, without however explicitly computing the complement. Both algorithms are simple and use simple data structures.     

Robust polynomial classifier using <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>-<Emphasis Type="Italic">norm</Emphasis> minimization

In this paper we present a robust polynomial classifier based on L ¹-norm minimization. We do so by reformulating the classifier training process as a linear programming problem. Due to the inherent insensitivity of the L ¹-norm to influential observations, class models obtained via L ¹-norm minimization are much more robust than their counterparts obtained by the classical least squares minimization (L ²-norm). For validation purposes, we apply this method to two recognition problems: character recognition and sign language recognition. Both are examined under different signal to noise ratio (SNR) values of the test data. Results show that L ¹-norm minimization provides superior recognition rates over L ²-norm minimization when the training data contains influential observations especially if the test dataset is noisy.     

Equivalence of <Emphasis Type="Italic">ε</Emphasis> -Approximate Separation and Optimization in Fixed Dimensions<Superscript><Emphasis Type="Italic">1</Emphasis></Superscript>

Given a closed, convex set X\subseteq \bbR ⁿ , containing the origin, we consider the problem (P) : max {c^\T x\colon x ∈ X} . We show that, for a fixed dimension, n , and fixed \eps , 0 <\eps<1 , the existence of a combinatorial, strongly polynomial \eps -approximation separation algorithm for the set X is equivalent to the existence of a combinatorial, strongly polynomial \eps -approximation optimization algorithm for the problem (P) . Received June 5, 1996; revised September 25, 1997.     

Fault Identification in System-Level Diagnosis: a Logic-Based Framework and an <Emphasis Type="Italic">O</Emphasis>(<Emphasis Type="Italic">n</Emphasis><Superscript>2</Superscript>\sqrt{τ}/{\log <Emphasis Type="Italic">n</Emphasis>}) Algorithm

Much research has been devoted to system-level diagnosis—SLD. Two issues have been addressed. The first of these is diagnosability, i.e., provide necessary and sufficient conditions for a system of n units to be diagnosable provided that the number of faulty units does not exceed τ . The second is the design of fault identification algorithms, assuming that the system being considered is diagnosable. This paper focuses on the second of these concerns, discussing several algorithms of which the most efficient runs in O(n ^2.5 ) . By considering a logical framework, this paper investigates the process of fault identification and proposes a fault identification algorithm which runs in O( n ² \sqrt τ / \sqrt log n ) , τ < n/2 . Received January 10, 2000; revised August 3, 2000.     

Maximizing bichromatic reverse nearest neighbor for <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-norm in two- and three-dimensional spaces

Bichromatic reverse nearest neighbor (BRNN) has been extensively studied in spatial database literature. In this paper, we study a related problem called MaxBRNN: find an optimal region that maximizes the size of BRNNs for L _p -norm in two- and three- dimensional spaces. Such a problem has many real-life applications, including the problem of finding a new server point that attracts as many customers as possible by proximity. A straightforward approach is to determine the BRNNs for all possible points that are not feasible since there are a large (or infinite) number of possible points. To the best of our knowledge, there are no existing algorithms which solve MaxBRNN for any L _p -norm space of two- and three-dimensionality. Based on some interesting properties of the problem, we come up with an efficient algorithm called MaxOverlap for to solve this problem. Extensive experiments are conducted to show that our algorithm is efficient.     

A Repeated-Update Problem in the DeltaBlue Algorithm

We observe a repeated-update problem in the process of updating walkabout strengths of the DeltaBlue algorithm, which is known as a fast constraint solving method based on local propagation. According to the basic references on the DeltaBlue algorithm, the time complexity of the planning phase is described as O(MN) for a constraint problem such that the number of constraints is N and the maximum number of methods a constraint has is M . We, however, point out that the time complexity is not O(MN) using a very simple example. In this example, the time complexity is exponential order for N . Then we present a corrected DeltaBlue algorithm whose time complexity is O(EN ²) for a constraint problem such that the number of constraints is N and the maximum number of variables a constraint has is E . Finally we measure the performance of the corrected DeltaBlue algorithm using two benchmarks.     

Efficient Solution of Multi-Term Fractional Differential Equations Using P(EC)<Superscript><Emphasis Type="Italic">m</Emphasis></Superscript>E Methods

We investigate strategies for the numerical solution of the initial value problem with initial conditions where 0<₁<₂<<. Here y ⁽ _j ) denotes the derivative of order _j >0 (not necessarily _j ) in the sense of Caputo. The methods are based on numerical integration techniques applied to an equivalent nonlinear and weakly singular Volterra integral equation. The classical approach leads to an algorithm with very high arithmetic complexity. Therefore we derive an alternative that leads to lower complexity without sacrificing too much precision. Mathematics Subject Classification: Primary 65L05; Secondary 65L06, 65L20     

A Hierarchical Extension of the <Emphasis Type="Italic">D</Emphasis><Superscript>*</Superscript> Algorithm

In this paper a contribution to the practice of path planning using a new hierarchical extension of the D^* algorithm is introduced. A hierarchical graph is stratified into several abstraction levels and used to model environments for path planning. The hierarchical D^* algorithm uses a down-top strategy and a set of pre-calculated trajectories in order to improve performance. This allows optimality and specially lower computational time. It is experimentally proved how hierarchical search algorithms and on-line path planning algorithms based on topological abstractions can be combined successfully.     

Some Aperture-Angle Optimization Problems

Abstract. Let P and Q be two disjoint convex polygons in the plane with m and n vertices, respectively. Given a point x in P , the aperture angle of x with respect to Q is defined as the angle of the cone that: (1) contains Q , (2) has apex at x and (3) has its two rays emanating from x tangent to Q . We present algorithms with complexities O(n log m) , O(n + n log (m/n)) and O(n + m) for computing the maximum aperture angle with respect to Q when x is allowed to vary in P . To compute the minimum aperture angle we modify the latter algorithm obtaining an O(n + m) algorithm. Finally, we establish an Ω(n + n log (m/n)) time lower bound for the maximization problem and an Ω(m + n) bound for the minimization problem thereby proving the optimality of our algorithms.     

Consistency techniques for continuous constraints

We consider constraint satisfaction problems with variables in continuous, numerical domains. Contrary to most existing techniques, which focus on computing one single optimal solution, we address the problem of computing a compact representation of the space of all solutions admitted by the constraints. In particular, we show how globally consistent (also called decomposable) labelings of a constraint satisfaction problem can be computed.Our approach is based on approximating regions of feasible solutions by 2 ^k -trees, a representation commonly used in computer vision and image processing. We give simple and stable algorithms for computing labelings with arbitrary degrees of consistency. The algorithms can process constraints and solution spaces of arbitrary complexity, but with a fixed maximal resolution.Previous work has shown that when constraints are convex and binary, path-consistency is sufficient to ensure global consistency. We show that for continuous domains, this result can be generalized to ternary and in fact arbitrary n-ary constraints using the concept of (3,2)-relational consistency. This leads to polynomial-time algorithms for computing globally consistent labelings for a large class of constraint satisfaction problems with continuous variables.     

Approximate Matching of Run-Length Compressed Strings

Abstract. We focus on the problem of approximate matching of strings that have been compressed using run-length encoding. Previous studies have concentrated on the problem of computing the longest common subsequence (LCS) between two strings of length m and n , compressed to m' and n' runs. We extend an existing algorithm for the LCS to the Levenshtein distance achieving O(m'n+n'm) complexity. Furthermore, we extend this algorithm to a weighted edit distance model, where the weights of the three basic edit operations can be chosen arbitrarily. This approach also gives an algorithm for approximate searching of a pattern of m letters (m' runs) in a text of n letters (n' runs) in O(mm'n') time. Then we propose improvements for a greedy algorithm for the LCS, and conjecture that the improved algorithm has O(m'n') expected case complexity. Experimental results are provided to support the conjecture.     

Augmenting the Rigidity of a Graph in <Emphasis Type="Italic">R</Emphasis><Superscript>2</Superscript>

Given a Laman graph G, i.e. a minimally rigid graph in R ², we provide a Θ(n ²) algorithm to augment G to a redundantly rigid graph, by adding a minimum number of edges. Moreover, we prove that this problem of augmenting is NP-hard for an arbitrary rigid graph G in R ².     

New algorithms for max restricted path consistency

Max Restricted Path Consistency (maxRPC) is a local consistency for binary constraints that enforces a higher order of consistency than arc consistency. Despite the strong pruning that can be achieved, maxRPC is rarely used because existing maxRPC algorithms suffer from overheads and redundancies as they can repeatedly perform many constraint checks without triggering any value deletions. In this paper we propose and evaluate techniques that can boost the performance of maxRPC algorithms by eliminating many of these overheads and redundancies. These include the combined use of two data structures to avoid many redundant constraint checks, and the exploitation of residues to quickly verify the existence of supports. Based on these, we propose a number of closely related maxRPC algorithms. The first one, maxRPC3, has optimal O(end ³) time complexity, displays good performance when used stand-alone, but is expensive to apply during search. The second one, maxRPC3 ^rm , has O(en ² d ⁴) time complexity, but a restricted version with O(end ⁴) complexity can be very efficient when used during search. The other algorithms are simple modifications of maxRPC3 ^rm . All algorithms have O(ed) space complexity when used stand-alone. However, maxRPC3 has O(end) space complexity when used during search, while the others retain the O(ed) complexity. Experimental results demonstrate that the resulting methods constantly outperform previous algorithms for maxRPC, often by large margins, and constitute a viable alternative to arc consistency on some problem classes.     

Extension of <Emphasis Type="Italic">O</Emphasis>(<Emphasis Type="Italic">n</Emphasis> log <Emphasis Type="Italic">n</Emphasis>) Filtering Algorithms for the Unary Resource Constraint to Optional Activities

Scheduling is one of the most successful application areas of constraint programming mainly thanks to special global constraints designed to model resource restrictions. Among these global constraints, edge-finding and not-first/not-last are the most popular filtering algorithms for unary resources. In this paper we introduce new O(n log n) versions of these two filtering algorithms and one more O(n log n) filtering algorithm called detectable precedences. These algorithms use a special data structures Θ-tree and Θ-Λ-tree. These data structures are especially designed for “what-if” reasoning about a set of activities so we also propose to use them for handling so called optional activities, i.e. activities which may or may not appear on the resource. In particular, we propose new O(n log n) variants of filtering algorithms which are able to handle optional activities: overload checking, detectable precedences and not-first/not-last.     

An <Emphasis Type="Italic">R</Emphasis> || <Emphasis Type="Italic">C</Emphasis><Subscript>max</Subscript> Quantum Scheduling Algorithm

Grover’s search algorithm can be applied to a wide range of problems; even problems not generally regarded as searching problems, can be reformulated to take advantage of quantum parallelism and entanglement, and lead to algorithms which show a square root speedup over their classical counterparts. In this paper, we discuss a systematic way to formulate such problems and give as an example a quantum scheduling algorithm for an R||C_max problem. R||C_max is representative for a class of scheduling problems whose goal is to find a schedule with the shortest completion time in an unrelated parallel machine environment. Given a deadline, or a range of deadlines, the algorithm presented in this paper allows us to determine if a solution to an R||C_max problem with N jobs and M machines exists, and if so, it provides the schedule. The time complexity of the quantum scheduling algorithm is while the complexity of its classical counterpart is .