Certain answers are a widely accepted semantics of query answering over incomplete databases. Since their computation is a coNP-hard problem, recent research has focused on developing evaluation algorithms with correc...
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Certain answers are a widely accepted semantics of query answering over incomplete databases. Since their computation is a coNP-hard problem, recent research has focused on developing evaluation algorithms with correctness guarantees, that is, techniques computing a sound but possibly incomplete set of certain answers. In this paper, we show how novel evaluation algorithms with correctness guarantees can be developed leveraging conditional tables and the conditional evaluation of queries, while retaining polynomial time data complexity.
The celebrated Cheeger's Inequality (Alon and Milman 1985;Alon 1986) establishes a bound on the edge expansion of a graph via its spectrum. This inequality is central to a rich spectral theory of graphs, based on ...
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The celebrated Cheeger's Inequality (Alon and Milman 1985;Alon 1986) establishes a bound on the edge expansion of a graph via its spectrum. This inequality is central to a rich spectral theory of graphs, based on studying the eigenvalues and eigenvectors of the adjacency matrix (and other related matrices) of graphs. It has remained open to define a suitable spectral model for hypergraphs whose spectra can be used to estimate various combinatorial properties of the hypergraph. In this article, we introduce a new hypergraph Laplacian operator generalizing the Laplacian matrix of graphs. In particular, the operator is induced by a diffusion process on the hypergraph, such that within each hyperedge, measure flows from vertices having maximum weighted measure to those having minimum. Since the operator is nonlinear, we have to exploit other properties of the diffusion process to recover the Cheeger's Inequality that relates hyperedge expansion with the "second eigenvalue" of the resulting Laplacian. However, we show that higher-order spectral properties cannot hold in general using the current framework. Since higher-order spectral properties do not hold for the Laplacian operator, we instead use the concept of procedural minimizers to consider higher-order Cheeger-like inequalities. For any k is an element of N, we give a polynomial-time algorithm to compute an O(log r)-approximation to the kth procedural minimizer, where r is the maximum cardinality of a hyperedge. We show that this approximation factor is optimal under the SSE hypothesis (introduced by Raghavendra and Steurer (2010)) for constant values of k. Moreover, using the factor-preserving reduction from vertex expansion in graphs to hypergraph expansion, we show that all our results for hypergraphs extend to vertex expansion in graphs.
We study the following geometric representation problem: Given a graph whose vertices correspond to axis-aligned rectangles with fixed dimensions, arrange the rectangles without overlaps in the plane such that two rec...
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We study the following geometric representation problem: Given a graph whose vertices correspond to axis-aligned rectangles with fixed dimensions, arrange the rectangles without overlaps in the plane such that two rectangles touch if the graph contains an edge between them. This problem is called Contact Representation of Word Networks (Crown) since it formalizes the geometric problem behind drawing word clouds in which semantically related words are close to each other. Crown is known to be NP-hard, and there are approximation algorithms for certain graph classes for the optimization version, Max-Crown, in which realizing each desired adjacency yields a certain profit. We present the first O(1)-approximation algorithm for the general case, when the input is a complete weighted graph, and for the bipartite case. Since the subgraph of realized adjacencies is necessarily planar, we also consider several planar graph classes (namely stars, trees, outerplanar, and planar graphs), improving upon the known results. For some graph classes, we also describe improvements in the unweighted case, where each adjacency yields the same profit. Finally, we show that the problem is APX-complete on bipartite graphs of bounded maximum degree.
We consider a well-studied multi-echelon (deterministic) inventory control problem, known in the literature as the one-warehouse multi-retailer (OWMR) problem. We propose a simple and fast 2-approximation algorithm fo...
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We consider a well-studied multi-echelon (deterministic) inventory control problem, known in the literature as the one-warehouse multi-retailer (OWMR) problem. We propose a simple and fast 2-approximation algorithm for this NP-hard problem, by recombining the solutions of single-echelon relaxations at the warehouse and at the retailers. We then show that our approach remains valid under quite general assumptions on the cost structures and under capacity constraints at some retailers. In particular, we present the first approximation algorithms for the OWMR problem with nonlinear holding costs, truckload discount on procurement costs, or with capacity constraints at some retailers. In all cases, the procedure is purely combinatorial and can be implemented to run in low polynomial time.
The projection games (aka Label Cover) problem is of great importance to the field of approximation algorithms, since most of the NP-hardness of approximation results we know today are reductions from Label Cover. In ...
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The projection games (aka Label Cover) problem is of great importance to the field of approximation algorithms, since most of the NP-hardness of approximation results we know today are reductions from Label Cover. In this paper we design several approximation algorithms for projection games: (1) A polynomial-time approximation algorithm that improves on the previous best approximation by Charikar et al. (Algorithmica 61(1):190-206, 2011). (2) A sub-exponential time algorithm with much tighter approximation for the case of smooth projection games. (3) A polynomial-time approximation scheme (PTAS) for projection games on planar graphs and a tight running time lower bound for such approximation schemes. The conference version of this paper had only the PTAS but not the running time lower bound.
Mathematical research on multicriteria optimization problems predominantly revolves around the set of Pareto optimal solutions. In practice, on the other hand, methods that output a single solution are more widespread...
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Mathematical research on multicriteria optimization problems predominantly revolves around the set of Pareto optimal solutions. In practice, on the other hand, methods that output a single solution are more widespread. Reference point methods are a successful example of this approach and are widely used in real-world multicriteria optimization. A reference point solution is the solution closest to a given reference point in the objective space. We study the connection between reference point methods and approximation algorithms for multicriteria optimization problems over discrete sets. In particular, we establish that, in terms of computational complexity, computing approximate reference point solutions is polynomially equivalent to approximating the Pareto set. Complementing these results, we show for a number of general algorithmic techniques in single criteria optimization how they can be lifted to reference point optimization. In particular, we lift the link between dynamic programming and FPTAS, as well as certain LP-rounding techniques. The latter applies, e.g., to SET COVER and several machine scheduling problems. (C) 2016 Elsevier B.V. All rights reserved.
Let c, k be two positive integers. Given a graph , the c-Load Coloring problem asks whether there is a c-coloring such that for every , there are at least k edges with both endvertices colored i. Gutin and Jones (Inf ...
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Let c, k be two positive integers. Given a graph , the c-Load Coloring problem asks whether there is a c-coloring such that for every , there are at least k edges with both endvertices colored i. Gutin and Jones (Inf Process Lett 114:446-449, 2014) studied this problem with . They showed 2-Load Coloring to be fixed-parameter tractable (FPT) with parameter k by obtaining a kernel with at most 7k vertices. In this paper, we extend the study to any fixed c by giving both a linear-vertex and a linear-edge kernel. In the particular case of , we obtain a kernel with less than 4k vertices and less than edges. These results imply that for any fixed , c-Load Coloring is FPT and the optimization version of c-Load Coloring (where k is to be maximized) has an approximation algorithm with a constant ratio.
In this paper we show a new way of constructing deterministic polynomial-time approximation algorithms for computing complex-valued evaluations of a large class of graph polynomials on bounded degree graphs. In partic...
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In this paper we show a new way of constructing deterministic polynomial-time approximation algorithms for computing complex-valued evaluations of a large class of graph polynomials on bounded degree graphs. In particular, our approach works for the Tutte polynomial and independence polynomial, as well as partition functions of complex-valued spin and edge-coloring models. More speci fi cally, we de fi ne a large class of graph polynomials C and show that if p is an element of C and there is a disk D centered at zero in the complex plane such that p (G) does not vanish on D for all bounded degree graphs G, then for each z in the interior of D there exists a deterministic polynomialtime approximation algorithm for evaluating p (G) at z. This gives an explicit connection between absence of zeros of graph polynomials and the existence of e ffi cient approximation algorithms, allowing us to show new relationships between well-known conjectures. Our work builds on a recent line of work initiated by Barvinok [Found. Comput. Math., 16 (2016), pp. 329-342;Theory Comput., 11 (2015), pp. 339-355;Computing the Partition Function of a Polynomial on the Boolean Cube, 2015;Discrete Anal., 2 (2017), 34pp], which provides a new algorithmic approach besides the existing Markov chain Monte Carlo method and the correlation decay method for these types of problems.
In an earlier paper (Bao and Liu [1]), we considered a version of the clustered traveling salesman problem (CTSP), in which both the starting and ending vertex of each cluster are free to be selected, and proposed a 2...
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In an earlier paper (Bao and Liu [1]), we considered a version of the clustered traveling salesman problem (CTSP), in which both the starting and ending vertex of each cluster are free to be selected, and proposed a 2.167-approximation algorithm. In this note, we first improve this approximation ratio to 1.9 by introducing a new method to define the inter node lengths for all the nodes in Step 2 of Algorithm A of Bao and Liu [1]. Based on the above method, we then provide a 2,5-approximation algorithm for another version of CTSP where the starting vertex of each cluster is given while the ending vertex is free to be selected, which improves the previous approximation ratio of 2.643 of Guttmann-Beck et al. [5]. (C) 2017 Elsevier B.V. All rights reserved.
In the past decades, there has been a burst of activity to simplify implementation of complex software systems. The solution framework in software engineering community for this problem is called component-based softw...
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In the past decades, there has been a burst of activity to simplify implementation of complex software systems. The solution framework in software engineering community for this problem is called component-based software design (CBSD), whereas in the modeling and simulation community it is called composability. Composability is a complex feature due to the challenges of creating components, selecting combinations of components, and integrating the selected components. In this paper, we address the challenge through the analysis of Component Selection (CS), the NP-complete process of selecting a minimal set of components to satisfy a set of objectives. Due to the computational complexity of CS, we consider approximation algorithms that make the component selection process practical. We define three variations of CS and present good approximation algorithms to find near optimal solutions. In spite of our creation of approximable variants of Component Selection, we prove that the general Component Selection problem is inapproximable.
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