Bipartite Matching with Linear Edge Weights

Consider a complete weighted bipartite graph G in which each left vertex u has two real numbers intercept and slope, each right vertex v has a real number quality, and the weight of any edge (u, v) is defined as the intercept of u plus the slope of u times the quality of v. Let m (resp., n) denote the number of left (resp., right) vertices, and assume that m geq n. We develop a fast algorithm for computing a maximum weight matching (MWM) of such a graph. Our algorithm begins by computing an MWM of the subgraph induced by the n right vertices and an arbitrary subset of n left vertices; this step is straightforward to perform in O(n log n) time. The remaining m - n left vertices are then inserted into the graph one at a time, in arbitrary order. As each left vertex is inserted, the MWM is updated. It is relatively straightforward to process each such insertion in O(n) time; our main technical contribution is to improve this time bound to O(sqrt{n} log^2 n). This result has an application related to unit-demand auctions. It is well known that the VCG mechanism yields a suitable solution (allocation and prices) for any unit-demand auction. The graph G may be viewed as encoding a special kind of unit-demand auction in which each left vertex u represents a unit-demand bid, each right vertex v represents an item, and the weight of an edge (u, v) represents the offer of bid u on item v. In this context, our fast insertion algorithm immediately provides an O(sqrt{n} log^2 n)-time algorithm for updating a VCG allocation when a new bid is received. We show how to generalize the insertion algorithm to update (an efficient representation of) the VCG prices within the same time bound

What Butler Did

Sexual assault, along with the obscenity and criminal indecency provisions of the Criminal Code, is part of a family of offences directed at wrongful sexual objectification. Insofar as those offences all target pernicious forms of objectification, the ways in which each has been interpreted can reveal important things about the others. With that in mind, it is striking that the Supreme Court’s decisions in Butler and Labaye proceed on the basis that both section 163 and the offence of criminal indecency require proof that the conduct in question causes social harm. By emphasizing harm, the Court obscured the message that certain kinds of objectification are per se wrongful, whether or not we can point to any tangible harm. Perhaps more importantly, Butler and Labaye fail to provide the sort of sophisticated analysis of what makes conduct problematically objectifying in the first place. Even if these shortcomings produce no discernible effect on the way that courts decide particular cases before the m, it undermines the educative function of the criminal law. This is especially problematic in the context of sexual assault, where the law must not only reflect social values, but take a leadership role in transforming the m

What Butler Did

Sexual assault, along with the obscenity and criminal indecency provisions of the Criminal Code, is part of a family of offences directed at wrongful sexual objectification. Insofar as those offences all target pernicious forms of objectification, the ways in which each has been interpreted can reveal important things about the others. With that in mind, it is striking that the Supreme Court’s decisions in Butler and Labaye proceed on the basis that both section 163 and the offence of criminal indecency require proof that the conduct in question causes social harm. By emphasizing harm, the Court obscured the message that certain kinds of objectification are per se wrongful, whether or not we can point to any tangible harm. Perhaps more importantly, Butler and Labaye fail to provide the sort of sophisticated analysis of what makes conduct problematically objectifying in the first place. Even if these shortcomings produce no discernible effect on the way that courts decide particular cases before the m, it undermines the educative function of the criminal law. This is especially problematic in the context of sexual assault, where the law must not only reflect social values, but take a leadership role in transforming the m

Offence Definitions, Conclusive Presumptions, and Slot Machines

Canadian evidence scholars frequently claim that conclusive presumptions are nothing more than substantive offence definitions. This position reflects a persistent confusion, not about the function of legal presumptions in the law of evidence, but about the function of offence definitions beyond the law of evidence. Offence definitions, unlike conclusive presumptions, serve the normative function of defining wrongful conduct for citizens. This commentary argues that the language of conclusive presumptions allows us to distinguish the gravamen of a criminal offence from a means of facilitating proof of that wrong. It is, to that extent, worth preserving the distinction between conclusive presumptions and offence definitions

Offence Definitions, Conclusive Presumptions, and Slot Machines

Canadian evidence scholars frequently claim that conclusive presumptions are nothing more than substantive offence definitions. This position reflects a persistent confusion, not about the function of legal presumptions in the law of evidence, but about the function of offence definitions beyond the law of evidence. Offence definitions, unlike conclusive presumptions, serve the normative function of defining wrongful conduct for citizens. This commentary argues that the language of conclusive presumptions allows us to distinguish the gravamen of a criminal offence from a means of facilitating proof of that wrong. It is, to that extent, worth preserving the distinction between conclusive presumptions and offence definitions