The Lovasz Local Lemma due to Erdos and Lovasz is a powerful tool in proving the existence of rare events. We present an extension of this lemma, which works well when the event to be shown to exist is a conjunction of individual events, each of which asserts that a random variable does not deviate much from its mean. As applications, we consider two classes of NP-hard integer programs: minimax and covering integer programs. A key technique, randomized rounding of linear relaxations, was developed by Raghavan and Thompson to derive good approximation algorithms for such problems. We use our extension of the Local Lemma to prove that randomized rounding produces, with non-zero probability, much better feasible solutions than known before, if the constraint matrices of these integer programs are column-sparse (e.g., routing using short paths, problems on hypergraphs with small dimension degree). This complements certain well-known results from discrepancy theory. We also generalize the method of pessimistic estimators due to Raghavan, to obtain constructive (algorithmic) versions of our results for covering integer programs.
For CIPs, the idea is to solve the LP relaxation, scale the components of @math suitably, and then perform randomized rounding; see for the details. Starting with this idea, the work of @cite_22 leads to certain approximation bounds; similar bounds are achieved through different means by Plotkin, Shmoys & Tardos @cite_14 . Work of this author @cite_10 improved upon these results by observing a correlation'' property of CIPs, getting an approximation ratio of @math . Thus, while the work of @cite_22 gives a general approximation bound for MIPs, the result of @cite_5 gives good results for sparse MIPs. For CIPs, the current-best results are those of @cite_10 ; however, no better results were known for sparse CIPs.
@cite_5 @cite_14 @cite_10 @cite_22
2064779029 2134422938 1993119087 2022191808
This article introduces constraint integer programming (CIP), which is a novel way to combine constraint programming (CP) and mixed integer programming (MIP) methodologies. CIP is a generalization of MIP that supports the notion of general constraints as in CP. This approach is supported by the CIP framework SCIP, which also integrates techniques from SAT solving. SCIP is available in source code and free for non-commercial use. We demonstrate the usefulness of CIP on two tasks. First, we apply the constraint integer programming approach to pure mixed integer programs. Computational experiments show that SCIP is almost competitive to current state-of-the-art commercial MIP solvers. Second, we employ the CIP framework to solve chip design verification problems, which involve some highly non-linear constraint types that are very hard to handle by pure MIP solvers. The CIP approach is very effective here: it can apply the full sophisticated MIP machinery to the linear part of the problem, while dealing with the non-linear constraints by employing constraint programming techniques. In a covering integer program (CIP), we seek an n-vector x of nonnegative integers, which minimizes cT ċ x, subject to Ax ≥ b, where all entries of A, b, c are nonnegative. In their most general form, CIPs include also multiplicity constraints of the type x ≤ d, i.e., arbitrarily large integers are not acceptable in the solution. The multiplicity constraints incur a dichotomy with respect to approximation between (0,1)-CIPs whose matrix A contains only zeros and ones and the general case. Let m denote the number of rows of A. The well known O(log m) cost approximation with respect to the optimum of the linear relaxation is valid for general CIPs, but multiplicity constraints can be dealt with effectively only in the (0,1) case. In the general case, existing algorithms that match the integrality gap for the cost objective violate the multiplicity constraints by a multiplicative O(log m) factor. We make progress by defining column-restricted CIPs, a strict superclass of (0,1)-CIPs, and showing how to find for them integral solutions of cost O(log m) times the LP optimum while violating the multiplicity constraints by a multiplicative O(1) factor. A citing sentence is one that appears in a scientific article and cites previous work. Citing sentences have been studied and used in many applications. For example, they have been used in scientific paper summarization, automatic survey generation, paraphrase identification, and citation function classification. Citing sentences that cite multiple papers are common in scientific writing. This observation should be taken into consideration when using citing sentences in applications. For instance, when a citing sentence is used in a summary of a scientific paper, only the fragments of the sentence that are relevant to the summarized paper should be included in the summary. In this paper, we present and compare three different approaches for identifying the fragments of a citing sentence that are related to a given target reference. Our methods are: word classification, sequence labeling, and segment classification. Our experiments show that segment classification achieves the best results. In a column-restricted covering integer program (CCIP), all the non-zero entries of any column of the constraint matrix are equal. Such programs capture capacitated versions of covering problems. In this paper, we study the approximability of CCIPs, in particular, their relation to the integrality gaps of the underlying 0,1-CIP. If the underlying 0,1-CIP has an integrality gap O(γ), and assuming that the integrality gap of the priority version of the 0,1-CIP is O(ω), we give a factor O(γ+ω) approximation algorithm for the CCIP. Priority versions of 0,1-CIPs (PCIPs) naturally capture quality of service type constraints in a covering problem. We investigate priority versions of the line (PLC) and the (rooted) tree cover (PTC) problems. Apart from being natural objects to study, these problems fall in a class of fundamental geometric covering problems. We bound the integrality of certain classes of this PCIP by a constant. Algorithmically, we give a polytime exact algorithm for PLC, show that the PTC problem is APX-hard, and give a factor 2-approximation algorithm for it.