4,720 research outputs found

    Collapsibility to a subcomplex of a given dimension is NP-complete

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    In this paper we extend the works of Tancer and of Malgouyres and Franc\'es, showing that (d,k)(d,k)-collapsibility is NP-complete for d≥k+2d\geq k+2 except (2,0)(2,0). By (d,k)(d,k)-collapsibility we mean the following problem: determine whether a given dd-dimensional simplicial complex can be collapsed to some kk-dimensional subcomplex. The question of establishing the complexity status of (d,k)(d,k)-collapsibility was asked by Tancer, who proved NP-completeness of (d,0)(d,0) and (d,1)(d,1)-collapsibility (for d≥3d\geq 3). Our extended result, together with the known polynomial-time algorithms for (2,0)(2,0) and d=k+1d=k+1, answers the question completely

    The Complexity of Quantified Constraint Satisfaction: Collapsibility, Sink Algebras, and the Three-Element Case

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    The constraint satisfaction probem (CSP) is a well-acknowledged framework in which many combinatorial search problems can be naturally formulated. The CSP may be viewed as the problem of deciding the truth of a logical sentence consisting of a conjunction of constraints, in front of which all variables are existentially quantified. The quantified constraint satisfaction problem (QCSP) is the generalization of the CSP where universal quantification is permitted in addition to existential quantification. The general intractability of these problems has motivated research studying the complexity of these problems under a restricted constraint language, which is a set of relations that can be used to express constraints. This paper introduces collapsibility, a technique for deriving positive complexity results on the QCSP. In particular, this technique allows one to show that, for a particular constraint language, the QCSP reduces to the CSP. We show that collapsibility applies to three known tractable cases of the QCSP that were originally studied using disparate proof techniques in different decades: Quantified 2-SAT (Aspvall, Plass, and Tarjan 1979), Quantified Horn-SAT (Karpinski, Kleine B\"{u}ning, and Schmitt 1987), and Quantified Affine-SAT (Creignou, Khanna, and Sudan 2001). This reconciles and reveals common structure among these cases, which are describable by constraint languages over a two-element domain. In addition to unifying these known tractable cases, we study constraint languages over domains of larger size

    Simpson's Paradox and Collapsibility

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    Simpson's paradox and collapsibility are two closely related concepts in the context of data analysis. While the knowledge about the occurrence of Simpson's paradox helps a statistician to draw correct and meaningful conclusions, the concept of collapsibility deals with dimension-reduction aspects, when Simpson's paradox does not occur. We discuss in this paper in some detail the nature and the genesis of Simpson's paradox with respect to well-known examples and also various concepts of collapsiblity. The main aim is to bring out the close connections between these two phenomena, especially with regard to the analysis of contingency tables, regression models and a certain measure of association or a dependence function. There is a vast literature on these topics and so we focus only on certain aspects, recent developments and some important results in the above-mentioned areas.Comment: 19 page

    Effects of variation in water content, clay fraction and sodium carbonate additions on the synthetic moulding properties of Igbokoda clay and silica sand

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    Loto, C.A. and Adebayo, H., 1990. Effects of variation in water content, clay fraction and sodium carbonate additions on the synthetic moulding properties of Igbokoda clay and silica sand. Appl. Clay Sci., 5: 165-181. In this investigation, the Igbokoda pure silica sand is characterised and used as a base sand for the Igbokoda clay- sand mixture in an attempt to develop an efficient synthetic moulding sand. The mechanical properties of the clay-sand-water mixture were determined. The effect of additions ofNa2C03 to the moulding sand was examined with particular attention to its influence on mechanical properties of the synthetic moulding sand. The fineness number, screen bulk fraction, total coarse fraction and the total fine fraction of Igbokoda silica sand were found to be within the range that could give good mouldable properties. The values obtained for the green compressive strength, the dry compressive strength, the green and dry shear strengths, collapsibility and toughness indicate that Igbokoda clay has good values as a binder for synthetic moulding sand. In general, the addition of Na2C03 gives improved properties to the moulding sand though with a tendency towards impaired collapsibility values
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