Approximability of (Simultaneous) Class Cover for Boxes

Bereg et al. (2012) introduced the Boxes Class Cover problem, which has its roots in classification and clustering applications: Given a set of n points in the plane, each colored red or blue, find the smallest cardinality set of axis-aligned boxes whose union covers the red points without covering any blue point. In this paper we give an alternative proof of APX-hardness for this problem, which also yields an explicit lower bound on its approximability. Our proof also directly applies when restricted to sets of points in general position and to the case where so-called half-strips are considered instead of boxes, which is a new result. We also introduce a symmetric variant of this problem, which we call Simultaneous Boxes Class Cover and can be stated as follows: Given a set S of n points in the plane, each colored red or blue, find the smallest cardinality set of axis-aligned boxes which together cover S such that all boxes cover only points of the same color and no box covering a red point intersects a box covering a blue point. We show that this problem is also APX-hard and give a polynomial-time constant-factor approximation algorithm

FPC: A New Approach to Firewall Policies Compression

Firewalls are crucial elements that enhance network security by examining the field values of every packet and deciding whether to accept or discard a packet according to the firewall policies. With the development of networks, the number of rules in firewalls has rapidly increased, consequently degrading network performance. In addition, because most real-life firewalls have been plagued with policy conflicts, malicious traffics can be allowed or legitimate traffics can be blocked. Moreover, because of the complexity of the firewall policies, it is very important to reduce the number of rules in a firewall while keeping the rule semantics unchanged and the target firewall rules conflict-free. In this study, we make three major contributions. First, we present a new approach in which a geometric model, multidimensional rectilinear polygon, is constructed for the firewall rules compression problem. Second, we propose a new scheme, Firewall Policies Compression (FPC), to compress the multidimensional firewall rules based on this geometric model. Third, we conducted extensive experiments to evaluate the performance of the proposed method. The experimental results demonstrate that the FPC method outperforms the existing approaches, in terms of compression ratio and efficiency while maintaining conflict-free firewall rules

Domain Ordering and Box Cover Problems for Beyond Worst-Case Join Processing

Join queries are a fundamental computational task in relational database management systems. For decades, complex joins were most often computed by decomposing the query into a query plan made of a sequence of binary joins. However, for cyclic queries, this type of query plan is sub-optimal. The worst-case run time of any such query plan exceeds the number of output tuples for any query instance. Recent theoretical developments in join query processing have led to join algorithms which are worst-case optimal, meaning that they run in time proportional to the worst-case output size for any query with the same shape and the same number of input tuples. Building on these results are a class of algorithms providing bounds which go beyond this worst-case output size by exploiting the structure of the input instance rather than just the query shape. One such algorithm, Tetris, is worst-case optimal and also provides an upper bound on its run time which depends on the minimum size of a geometric box certificate for the input query. A box certificate is a subset of a box cover whose union covers every tuple which is not present in the query output. A box cover is a set of n-dimensional boxes which cover all of the tuples not contained in the input relations. Many query instances admit different box certificates and box covers when the values in the attributes' domains are ordered differently. If we permute the input query according to a domain ordering which admits a smaller box certificate, use the permuted query as input to Tetris, then transform the result back with the inverse domain ordering, we can compute the query faster than was possible if the domain ordering was fixed. If we can efficiently compute an optimal domain ordering for a query, then we can state a beyond worst-case bound that is stronger than what is provided by Tetris. This paper defines several optimization problems over the space of domain orderings where the objective is to minimize the size of either the minimum box certificate or the minimum box cover for the given input query. We show that most of these problems are NP-hard. We also provide approximation algorithms for several of these problems. The most general version of the box cover minimization problem we will study, BoxMinPDomF, is shown to be NP-hard, but we can compute an approximation only a poly-logarithmic factor larger than K^(a*r), where K is the minimum box cover size under any domain ordering and r is the maximum number of attributes in a relation. This result allows us to compute join queries in time N+K^(a*r*(w+1))+Z, times a poly-logarithmic factor in N, where N is the number of input tuples, w is the treewidth of the query, and Z is the number of output tuples. This is a new beyond worst-case bound. There are queries for which this bound is exponentially smaller than any bound provided by Tetris. The most general version of the box certificate minimization problem we study, CertMinPDomF, is also shown to be NP-hard. It can be computed exactly if the minimum box certificate size is at most 3, but no approximation algorithm for an arbitrary minimum size is known. Finding such an approximation algorithm is an important direction for future research

Covering Problems via Structural Approaches

The minimum set cover problem is, without question, among the most ubiquitous and well-studied problems in computer science. Its theoretical hardness has been fully characterized--logarithmic approximability has been established, and no sublogarithmic approximation exists unless P=NP. However, the gap between real-world instances and the theoretical worst case is often immense--many covering problems of practical relevance admit much better approximations, or even solvability in polynomial time. Simple combinatorial or geometric structure can often be exploited to obtain improved algorithms on a problem-by-problem basis, but there is no general method of determining the extent to which this is possible. In this thesis, we aim to shed light on the relationship between the structure and the hardness of covering problems. We discuss several measures of structural complexity of set cover instances and prove new algorithmic and hardness results linking the approximability of a set cover problem to its underlying structure. In particular, we provide: - An APX-hardness proof for a wide family of problems that encode a simple covering problem known as Special-3SC. - A class of polynomial dynamic programming algorithms for a group of weighted geometric set cover problems having simple structure. - A simplified quasi-uniform sampling algorithm that yields improved approximations for weighted covering problems having low cell complexity or geometric union complexity. - Applications of the above to various capacitated covering problems via linear programming strengthening and rounding. In total, we obtain new results for dozens of covering problems exhibiting geometric or combinatorial structure. We tabulate these problems and classify them according to their approximability