Computational Complexity And Algorithms For Dirty Data Evaluation And Repairing

In this dissertation, we study the dirty data evaluation and repairing problem in relational database. Dirty data is usually inconsistent, inaccurate, incomplete and stale. Existing methods and theories of consistency describe using integrity constraints, such as data dependencies. However, integrity constraints are good at detection but not at evaluating the degree of data inconsistency and cannot guide the data repairing. This dissertation first studies the computational complexity of and algorithms for the database inconsistency evaluation. We define and use the minimum tuple deletion to evaluate the database inconsistency. For such minimum tuple deletion problem, we study the relationship between the size of rule set and its computational complexity. We show that the minimum tuple deletion problem is NP-hard to approximate the minimum tuple deletion within 17/16 if given three functional dependencies and four attributes involved. A near optimal approximated algorithm for computing the minimum tuple deletion is proposed with a ratio of 2 − 1/2r , where r is the number of given functional dependencies. To guide the data repairing, this dissertation also investigates the data repairing method by using query feedbacks, formally studies two decision problems, functional dependency restricted deletion and insertion propagation problem, corresponding to the feedbacks of deletion and insertion. A comprehensive analysis on both combined and data complexity of the cases is provided by considering different relational operators and feedback types. We have identified the intractable and tractable cases to picture the complexity hierarchy of these problems, and provided the efficient algorithm on these tractable cases. Two improvements are proposed, one focuses on figuring out the minimum vertex cover in conflict graph to improve the upper bound of tuple deletion problem, and the other one is a better dichotomy for deletion and insertion propagation problems at the absence of functional dependencies from the point of respectively considering data, combined and parameterized complexities

The reducibility of optimal 1-planar graphs with respect to the lexicographic product

A graph is called 1-planar if it can be drawn on the plane (or on the sphere) such that each edge is crossed at most once. A 1-planar graph $G$ is called optimal if it satisfies $|E(G)| = 4|V(G)|-8$. If $G$ and $H$ are graphs, then the lexicographic product $G\circ H$ has vertex set the Cartesian product $V(G)\times V(H)$ and edge set $\{(g_1,h_1) (g_2,h_2): g_1 g_2 \in E(G),\,\, \text{or}\,\, g_1=g_2 \,\, \text{and}\,\, h_1 h_2 \in E(H)\}$. A graph is called reducible if it can be expressed as the lexicographic product of two smaller non-trivial graphs. In this paper, we prove that an optimal 1-planar graph $G$ is reducible if and only if $G$ is isomorphic to the complete multipartite graph $K_{2,2,2,2}$. As a corollary, we prove that every reducible 1-planar graph with $n$ vertices has at most $4n-9$ edges for $n=6$ or $n\ge 9$. We also prove that this bound is tight for infinitely many values of $n$. Additionally, we give two necessary conditions for a graph $G\circ 2K_1$ to be 1-planar.Comment: 23 pages, 14 fugure

Drawings of Complete Multipartite Graphs up to Triangle Flips

For a drawing of a labeled graph, the rotation of a vertex or crossing is the cyclic order of its incident edges, represented by the labels of their other endpoints. The extended rotation system (ERS) of the drawing is the collection of the rotations of all vertices and crossings. A drawing is simple if each pair of edges has at most one common point. Gioan's Theorem states that for any two simple drawings of the complete graph Kn with the same crossing edge pairs, one drawing can be transformed into the other by a sequence of triangle flips (a.k.a. Reidemeister moves of Type 3). This operation refers to the act of moving one edge of a triangular cell formed by three pairwise crossing edges over the opposite crossing of the cell, via a local transformation. We investigate to what extent Gioan-type theorems can be obtained for wider classes of graphs. A necessary (but in general not sufficient) condition for two drawings of a graph to be transformable into each other by a sequence of triangle flips is that they have the same ERS. As our main result, we show that for the large class of complete multipartite graphs, this necessary condition is in fact also sufficient. We present two different proofs of this result, one of which is shorter, while the other one yields a polynomial time algorithm for which the number of needed triangle flips for graphs on n vertices is bounded by O(n16). The latter proof uses a Carathéodory-type theorem for simple drawings of complete multipartite graphs, which we believe to be of independent interest. Moreover, we show that our Gioan-type theorem for complete multipartite graphs is essentially tight in the following sense: For the complete bipartite graph Km, n minus two edges and Km, n plus one edge for any m, n ≥ 4, as well as Kn minus a 4-cycle for any n ≥ 5, there exist two simple drawings with the same ERS that cannot be transformed into each other using triangle flips. So having the same ERS does not remain sufficient when removing or adding very few edges