Improved approximation bounds for the minimum constraint removal problem

As a fundamental problem of wireless sensor networks, the minimal exposure path problem corresponding to the sensor network's worst-case coverage plays an important role in the applications for detecting intrusions. However, most existing works about minimal exposure path are based on omnidirectional sensors. In contrast, this paper studies the minimal exposure path problem for two different types of directional sensing models: the binary sector model and directional sensitivity model. For the binary sector model, we construct a special Voronoi diagram, called sector centroids-based Voronoi diagram, to transform the minimal exposure path problem from a continuous geometric problem into a discrete geometric problem. By using the sector centroids-based Voronoi diagram, we develop an approximate algorithm to find the minimal exposure path in the sensors deployment field. For the directional sensitive model, we formulate the minimal exposure path problem by using two sensing intensity functions: all-sensor intensity function and maximum-sensor intensity function and then generate two weighted grids to convert the minimal exposure path problem into two discrete geometric problems. On the basis of the aforementioned weighted grids, we also develop two approximation algorithms to find the minimal exposure path for the directional sensitive model. We conduct extensive experiment simulations to validate and evaluate our proposed models and algorithms. Copyright © 2012 John Wiley & Sons, Ltd.

Linear Algebra in Dioids: A Survey of Recent Results

This chapter is concerned with the Boolean matrix approach to problems concerning the existence and determination of paths in directed graphs.

In the compiler literature, parsing algorithms for context-free grammars are presented using string rewriting systems or abstract machines such as pushdown automata. Unfortunately, the resulting descriptions can be baroque, and even a basic understanding of some parsing algorithms, such as Earley’s algorithm for general context-free grammars, can be elusive. In this paper, we present a graphical representation of context-free grammars called the Grammar Flow Graph (GFG) that permits parsing problems to be phrased as path problems in graphs; intuitively, the GFG plays the same role for context-free grammars that nondeterministic finite-state automata play for regular grammars. We show that the GFG permits an elementary treatment of Earley’s algorithm that is much easier to understand than previous descriptions of this algorithm. In addition, look-ahead computation can be expressed as a simple inter-procedural dataflow analysis problem, providing an unexpected link between front-end and back-end technologies in compilers. These results suggest that the GFG can be a new foundation for the study of context-free grammars.

Mining web log datasets has been extensively studied using Frequent Pattern Mining (FPM) and its various other forms. Identifyingfrequent patterns in different sequences can help in analyzing the most common sub-sequences (e.g., the pages visitedtogether). However, this approach would not be able to identify general structures spanning over multiple sequences. In responseto understanding general structures, we introduce a new form of sequential pattern mining called super-sequence frequent patternmining (SS-FPM). In contrast to sub-sequences determined by FPM, SS-FPM determines the super-sequences that can containthe common parts from different sequences. This can be useful in understanding the general behavior/flow of users in web usagemining, classifying web pages and users, making predictions etc. In essence, finding frequent super-sequence patterns turnsout to be related to the well-known heaviest (longest) path problem in graphs, which is known to be NP-hard. Accordingly,we transform a given sequential dataset into a sequence graph and formulate the problem as k-hop heaviest path problem. Wethen propose an efficient heuristic called sequence matrix method using dynamic programming techniques. We compared ourmethod to the existing Heavypath method. The results show that our method is more efficient especially on large datasets.

We introduce two generalizations of core percolation in graphs to hypergraphs, related to the minimum hyperedge cover problem and the minimum vertex cover problem on hypergraphs, respectively. We offer analytical solutions of these two core percolations for uncorrelated random hypergraphs whose vertex degree and hyperedge cardinality distributions are arbitrary but have nondiverging moments. We find that for several real-world hypergraphs their two cores tend to be much smaller than those of their null models, suggesting that covering problems in those real-world hypergraphs can actually be solved in polynomial time.

Given a set of geometric objects each associated with a time value, we wish to determine whether a given property is true for a subset of those objects whose time values fall within a query time window. We call such problems time-windowed decision problems, and they have been the subject of much recent attention, for instance studied by Bokal, Cabello, and Eppstein [SoCG 2015]. In this paper, we present new approaches to this class of problems that are conceptually simpler than Bokal et al.'s, and also lead to faster algorithms. For instance, we present algorithms for preprocessing for the time-windowed 2D diameter decision problem in O(n log n) time and the time-windowed 2D convex hull area decision problem in O(n alpha(n) log n) time (where alpha is the inverse Ackermann function), improving Bokal et al.'s O(n log^2 n) and O(n log n loglog n) solutions respectively. Our first approach is to reduce time-windowed decision problems to a generalized range successor problem, which we solve using a novel way to search range trees. Our other approach is to use dynamic data structures directly, taking advantage of a new observation that the total number of combinatorial changes to a planar convex hull is near linear for any FIFO update sequence, in which deletions occur in the same order as insertions. We also apply these approaches to obtain the first O(n polylog n) algorithms for the time-windowed 3D diameter decision and 2D orthogonal segment intersection detection problems.

The class Steiner minimal tree problem is an extension of the standard Steiner minimal tree problem in graphs, motivated by the problem of wire routing in the area of physical design of very large scale integration (VLSI). This problem is NP-hard, even if there are no Steiner nodes and the graph is a tree; moreover, there exists no polynomial time approximation algorithm with a constant bound on the relative error under the hypothesis that P \(\neq\) NP [16]. Hence, fast and good heuristic algorithms are needed in practice. In this paper, we present an integer programming formulation of the problem. Using Lagrangean relaxation and subgradient optimization, we derive a lower bound. In order to test the lower bound, we present a procedure for generating test problems for the class Steiner minimal tree problem that have known optimal solutions. The computational experiments for the test problems demonstrate that the lower bound is very tight and differs from the optimal solutions by only a few percent on average for sparse graphs.

We study truthful mechanisms for allocation problems in graphs, both for the minimization (i.e., scheduling) and maximization (i.e., auctions) setting. The minimization problem is a special case of the well-studied unrelated machines scheduling problem, in which every given task can be executed only by two pre-specified machines in the case of graphs or a given subset of machines in the case of hypergraphs. This corresponds to a multigraph whose nodes are the machines and its hyperedges are the tasks. This class of problems belongs to multidimensional mechanism design, for which there are no known general mechanisms other than the VCG and its generalization to affine minimizers. We propose a new class of truthful mechanisms that have significantly better performance than affine minimizers in many settings. Specifically, we provide upper and lower bounds for truthful mechanisms for general multigraphs, as well as special classes of graphs such as stars, trees, planar graphs, k -degenerate graphs, and graphs of a given treewidth. We also consider the objective of minimizing or maximizing the L p -norm of the values of the players, a generalization of the makespan minimization that corresponds to p = ∞, and extend the results to any p > 0.

This paper addresses the minmax regret 1-sink location problem on dynamic flow path networks with parametric weights. We are given a dynamic flow network consisting of an undirected path with positive edge lengths, positive edge capacities, and nonnegative vertex weights. A path can be considered as a road, an edge length as the distance along the road and a vertex weight as the number of people at the site. An edge capacity limits the number of people that can enter the edge per unit time. We consider the problem of locating a sink in the network, to which all the people evacuate from the vertices as quickly as possible. In our model, each weight is represented by a linear function in a common parameter t, and the decision maker who determines the location of a sink does not know the value of t. We formulate the sink location problem under such uncertainty as the minmax regret problem. Given t and a sink location x, the cost of x under t is the sum of arrival times at x for all the people determined by t. The regret for x under t is the gap between the cost of x under t and the optimal cost under t. The task of the problem is formulated as the one to find a sink location that minimizes the maximum regret over all t. For the problem, we propose an \(O(n^4 2^{\alpha (n)} \alpha (n) \log n)\) time algorithm where n is the number of vertices in the network and \(\alpha (\cdot )\) is the inverse Ackermann function. Also for the special case in which every edge has the same capacity, we show that the complexity can be reduced to \(O(n^3 2^{\alpha (n)} \alpha (n) \log n)\).

An augmented metric space (X, d_X, f_X) is a metric space (X, d_X) equipped with a function f_X: X → ℝ. It arises commonly in practice, e.g, a point cloud X in ℝ^d where each point x∈ X has a density function value f_X(x) associated to it. Such an augmented metric space naturally gives rise to a 2-parameter filtration. However, the resulting 2-parameter persistence module could still be of wild representation type, and may not have simple indecomposables. In this paper, motivated by the elder-rule for the zeroth homology of a 1-parameter filtration, we propose a barcode-like summary, called the elder-rule-staircode, as a way to encode the zeroth homology of the 2-parameter filtration induced by a finite augmented metric space. Specifically, given a finite (X, d_X, f_X), its elder-rule-staircode consists of n = |X| number of staircase-like blocks in the plane. We show that the fibered barcode, the fibered merge tree, and the graded Betti numbers associated to the zeroth homology of the 2-parameter filtration induced by (X, d_X, f_X) can all be efficiently computed once the elder-rule-staircode is given. Furthermore, for certain special cases, this staircode corresponds exactly to the set of indecomposables of the zeroth homology of the 2-parameter filtration. Finally, we develop and implement an efficient algorithm to compute the elder-rule-staircode in O(n²log n) time, which can be improved to O(n²α(n)) if X is from a fixed dimensional Euclidean space ℝ^d, where α(n) is the inverse Ackermann function.

We introduce new proof systems for propositional logic, simple deduction Frege systems, general deduction Frege systems, and nested deduction Frege systems, which augment Frege systems with variants of the deduction rule. We give upper bounds on the lengths of proofs in Frege proof systems compared to lengths in these new systems. As applications we give near-linear simulations of the propositional Gentzen sequent calculus and the natural deduction calculus by Frege proofs. The length of a proof is the number of lines (or formulas) in the proof.A general deduction Frege proof system provides at most quadratic speedup over Frege proof systems. A nested deduction Frege proof system provides at most a nearly linear speedup over Frege system where by “nearly linear” is meant the ratio of proof lengths is O(α(n)) where α is the inverse Ackermann function. A nested deduction Frege system can linearly simulate the propositional sequent calculus, the tree-like general deduction Frege calculus, and the natural deduction calculus. Hence a Frege proof system can simulate all those proof systems with proof lengths bounded by O(n . α(n)). Also we show that a Frege proof of n lines can be transformed into a tree-like Frege proof of O(n log n) lines and of height O(log n). As a corollary of this fact we can prove that natural deduction and sequent calculus tree-like systems simulate Frege systems with proof lengths bounded by O(n log n).

We consider the implementation of meldable priority queues with integer keys in the RAM model. We present two new general techniques for transforming non-meldable priority queues into meldable ones. These transformations can be described symbolically as:non-meldable priority queue +union-find ← meldable priority queuenon-meldable priority queue +slow meldable priority queue ←faster meldable priority queueUsing the first transformation to combine a recent non-meldable RAM priority queue of Thorup with the classical union-find data structure we obtain a meldable RAM priority queue with an amortized cost of O(log log n·α(n)) per operation, where α(n) = α(n, n) is the inverse Ackermann function. Using instead a randomized priority queue of Han and Thorup we obtain an expected amortized cost of O(√(log log n) · α(n)) per operation. The second transformation yields slower meldable priority queues, but the obtained queues can support the insert, find-min and decrease-key operations in constant time. In particular, by combining a randomized atomic-heap of Thorup with, e.g., the classical Fibonacci heaps of Fredman and Tarjan, we obtain, for every fixed e > 0, a meldable priority queue with an expected amortized cost of O(1) for each insert, find-min and decrease-key operation, and an expected amortized cost of O((log n)1/2+e) for each delete or meld operation.Using the meldable priority queues of the first type, we obtain improved algorithms for finding minimum directed spanning trees in graphs with integer edge weights: a deterministic O(m · log log n · α(n)) time algorithm and a randomized O(m · √(log log n) · α(n)) expected time algorithm. These bounds improve, for very sparse graphs, on the O(m + n log n) running time of an algorithm by Gabow, Galil, Spencer and Tarjan that works for arbitrary edge weights.

Let an arrangement of blue lines in 3-space be fixed, and imagine a movable red line entangled in the arrangement. We show an O(n4a(n)) algorithm for building a data structure that permits enumeration of mutually inaccessible classes of such red lines, where a(n) is the inverse Ackermann function. The core of the algorithm is a construction of O(n2) 2-D arrangement of hyperbolas, each in O(n2a(n)) time.The algorithm is applied to stabbing 3-polytopes, enumerating pairwise-visible face pairs, enumerating 2-D projections of convex 4-polytopes, and other problems, resulting in O(n4a(n)) algorithms in each case.

Given a partition of an n element set into equivalence classes, we study the problem of assigning unique labels to these elements in order to support the query that asks whether the elements corresponding to two given labels belong to the same equivalence class. This has various applications including for testing whether two vertices are in the same connected component in an undirected graph or in the same strongly connected component in a directed graph. We consider the problem in several models. Concerning labeling schemes where we assign labels to elements and the query is to be answered just by examining the labels of the queried elements (without any extra space): if each vertex is required to have a unique label, then we show that a label space of \(\sum _{i=1}^n \lfloor {n \over i} \rfloor \) is necessary and sufficient. In other words, \(\lg n + \lg \lg n + O(1)\) bits of space are necessary and sufficient for representing each of the labels. This slightly strengthens the known lower bound and is in contrast to the known necessary and sufficient bound of \(\lceil \lg n \rceil \) for the label length, if each vertex need not get a unique label. Concerning succinct data structures for the problem when the n elements are to be uniquely assigned labels from label set \(\{1,\ldots , n\}\), we first show that \(\varTheta (\sqrt{n})\) bits are necessary and sufficient to represent the equivalence class information. This space includes the space for implicitly encoding the vertex labels. We can support the query in such a structure in O(1) time in the standard word RAM model. We also develop a dynamic structure that uses \(O(\sqrt{n} \lg n)\) bits to support equivalence queries and unions in \(O(\lg n/\lg \lg n)\) worst case time or \(O(\alpha (n))\) expected amortized time where \(\alpha (n)\) is the inverse Ackermann function. Concerning succinct data structures for the problem when the n elements are to be uniquely assigned labels from label set \(\{1,\ldots , cn\}\) for any constant \(c > 1\), we show that \(\varTheta (\lg n)\) bits are necessary and sufficient to represent the equivalence class information. Moreover, we can support the query in such a structure in O(1) time in the standard word RAM model. We believe that our work can trigger further work on tradeoffs between label space and auxiliary data structure space for other labeling problems.