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A standard approach to accelerating shortest path algorithms on networks is the bidirectional search, which explores the graph from the start and the destination, simultaneously. In practice this strategy performs particularly well on scale-free real-world networks. Such networks typically have a heterogeneous degree distribution (e.g., a power-law distribution) and high clustering (i.e., vertices with a common neighbor are likely to be connected themselves). These two properties can be obtained by assuming an underlying hyperbolic geometry. <br /> To explain the observed behavior of the bidirectional search, we analyze its running time on hyperbolic random graphs and prove that it is (O) over tilde (n(2-1/alpha) + n(1/(2 alpha)) + delta(max)) with high probability, where alpha is an element of (1/2, 1) controls the power-law exponent of the degree distribution, and dmax is the maximum degree. This bound is sublinear, improving the obvious worst-case linear bound. Although our analysis depends on the underlying geometry, the algorithm itself is oblivious to it.
Multi-column dependencies in relational databases come associated with two different computational tasks. The detection problem is to decide whether a dependency of a certain type and size holds in a given database, the discovery problem asks to enumerate all valid dependencies of that type. We settle the complexity of both of these problems for unique column combinations (UCCs), functional dependencies (FDs), and inclusion dependencies (INDs). We show that the detection of UCCs and FDs is W[2]-complete when parameterized by the solution size. The discovery of inclusion-wise minimal UCCs is proven to be equivalent under parsimonious reductions to the transversal hypergraph problem of enumerating the minimal hitting sets of a hypergraph. The discovery of FDs is equivalent to the simultaneous enumeration of the hitting sets of multiple input hypergraphs. We further identify the detection of INDs as one of the first natural W[3]-complete problems. The discovery of maximal INDs is shown to be equivalent to enumerating the maximal satisfying assignments of antimonotone, 3-normalized Boolean formulas.
We consider the subset selection problem for function f with constraint bound B that changes over time. Within the area of submodular optimization, various greedy approaches are commonly used. For dynamic environments we observe that the adaptive variants of these greedy approaches are not able to maintain their approximation quality. Investigating the recently introduced POMC Pareto optimization approach, we show that this algorithm efficiently computes a phi=(alpha(f)/2)(1 - 1/e(alpha)f)-approximation, where alpha(f) is the submodularity ratio of f, for each possible constraint bound b <= B. Furthermore, we show that POMC is able to adapt its set of solutions quickly in the case that B increases. Our experimental investigations for the influence maximization in social networks show the advantage of POMC over generalized greedy algorithms. We also consider EAMC, a new evolutionary algorithm with polynomial expected time guarantee to maintain phi approximation ratio, and NSGA-II with two different population sizes as advanced multi-objective optimization algorithm, to demonstrate their challenges in optimizing the maximum coverage problem. Our empirical analysis shows that, within the same number of evaluations, POMC is able to perform as good as NSGA-II under linear constraint, while EAMC performs significantly worse than all considered algorithms in most cases.
A core operator of evolutionary algorithms (EAs) is the mutation. Recently, much attention has been devoted to the study of mutation operators with dynamic and non-uniform mutation rates. Following up on this area of work, we propose a new mutation operator and analyze its performance on the (1 + 1) Evolutionary Algorithm (EA). Our analyses show that this mutation operator competes with pre-existing ones, when used by the (1 + 1) EA on classes of problems for which results on the other mutation operators are available. We show that the (1 + 1) EA using our mutation operator finds a (1/3)-approximation ratio on any non-negative submodular function in polynomial time. We also consider the problem of maximizing a symmetric submodular function under a single matroid constraint and show that the (1 + 1) EA using our operator finds a (1/3)-approximation within polynomial time. This performance matches that of combinatorial local search algorithms specifically designed to solve these problems and outperforms them with constant probability. Finally, we evaluate the performance of the (1 + 1) EA using our operator experimentally by considering two applications: (a) the maximum directed cut problem on real-world graphs of different origins, with up to 6.6 million vertices and 56 million edges and (b) the symmetric mutual information problem using a four month period air pollution data set. In comparison with uniform mutation and a recently proposed dynamic scheme, our operator comes out on top on these instances.
Unique column combinations (UCCs) are a fundamental concept in relational databases. They identify entities in the data and support various data management activities. Still, UCCs are usually not explicitly defined and need to be discovered. State-of-the-art data profiling algorithms are able to efficiently discover UCCs in moderately sized datasets, but they tend to fail on large and, in particular, on wide datasets due to run time and memory limitations. <br /> In this paper, we introduce HPIValid, a novel UCC discovery algorithm that implements a faster and more resource-saving search strategy. HPIValid models the metadata discovery as a hitting set enumeration problem in hypergraphs. In this way, it combines efficient discovery techniques from data profiling research with the most recent theoretical insights into enumeration algorithms. Our evaluation shows that HPIValid is not only orders of magnitude faster than related work, it also has a much smaller memory footprint.
Large real-world networks typically follow a power-law degree distribution. To study such networks, numerous random graph models have been proposed. However, real-world networks are not drawn at random. Therefore, Brach et al. (27th symposium on discrete algorithms (SODA), pp 1306-1325, 2016) introduced two natural deterministic conditions: (1) a power-law upper bound on the degree distribution (PLB-U) and (2) power-law neighborhoods, that is, the degree distribution of neighbors of each vertex is also upper bounded by a power law (PLB-N). They showed that many real-world networks satisfy both properties and exploit them to design faster algorithms for a number of classical graph problems. We complement their work by showing that some well-studied random graph models exhibit both of the mentioned PLB properties. PLB-U and PLB-N hold with high probability for Chung-Lu Random Graphs and Geometric Inhomogeneous Random Graphs and almost surely for Hyperbolic Random Graphs. As a consequence, all results of Brach et al. also hold with high probability or almost surely for those random graph classes. In the second part we study three classical NP-hard optimization problems on PLB networks. It is known that on general graphs with maximum degree Delta, a greedy algorithm, which chooses nodes in the order of their degree, only achieves a Omega (ln Delta)-approximation forMinimum Vertex Cover and Minimum Dominating Set, and a Omega(Delta)-approximation forMaximum Independent Set. We prove that the PLB-U property with beta>2 suffices for the greedy approach to achieve a constant-factor approximation for all three problems. We also show that these problems are APX-hard even if PLB-U, PLB-N, and an additional power-law lower bound on the degree distribution hold. Hence, a PTAS cannot be expected unless P = NP. Furthermore, we prove that all three problems are in MAX SNP if the PLB-U property holds.
Random walks are frequently used in randomized algorithms. We study a derandomized variant of a random walk on graphs called the rotor-router model. In this model, instead of distributing tokens randomly, each vertex serves its neighbors in a fixed deterministic order. For most setups, both processes behave in a remarkably similar way: Starting with the same initial configuration, the number of tokens in the rotor-router model deviates only slightly from the expected number of tokens on the corresponding vertex in the random walk model. The maximal difference over all vertices and all times is called single vertex discrepancy. Cooper and Spencer [Combin. Probab. Comput., 15 (2006), pp. 815-822] showed that on Z(d), the single vertex discrepancy is only a constant c(d). Other authors also determined the precise value of c(d) for d = 1, 2. All of these results, however, assume that initially all tokens are only placed on one partition of the bipartite graph Z(d). We show that this assumption is crucial by proving that, otherwise, the single vertex discrepancy can become arbitrarily large. For all dimensions d >= 1 and arbitrary discrepancies l >= 0, we construct configurations that reach a discrepancy of at least l.