Tropically convex constraint satisfaction

A semilinear relation S is max-closed if it is preserved by taking the componentwise maximum. The constraint satisfaction problem for max-closed semilinear constraints is at least as hard as determining the winner in Mean Payoff Games, a notorious problem of open computational complexity. Mean Payoff Games are known to be in the intersection of NP and co-NP, which is not known for max-closed semilinear constraints. Semilinear relations that are max-closed and additionally closed under translations have been called tropically convex in the literature. One of our main results is a new duality for open tropically convex relations, which puts the CSP for tropically convex semilinaer constraints in general into NP intersected co-NP. This extends the corresponding complexity result for scheduling under and-or precedence constraints, or equivalently the max-atoms problem. To this end, we present a characterization of max-closed semilinear relations in terms of syntactically restricted first-order logic, and another characterization in terms of a finite set of relations L that allow primitive positive definitions of all other relations in the class. We also present a subclass of max-closed constraints where the CSP is in P; this class generalizes the class of max-closed constraints over finite domains, and the feasibility problem for max-closed linear inequalities. Finally, we show that the class of max-closed semilinear constraints is maximal in the sense that as soon as a single relation that is not max-closed is added to L, the CSP becomes NP-hard.Comment: 29 pages, 2 figure

Values - reviewing the construct and drawing implications for values work in organisation and leadership. Kap. 2

I: H.Askeland, G. Espedal, B. Jelstad Løvaas & S. Sirris (Eds.), Understanding values work : Institutional perspectives in organizations and leadershipThis chapter outlines the trajectory of values, particularly within streams of organisational institutionalism, in order to analyse its application to values work in organisation and leadership. Conveying a frame for discussing values work, it aims at clarifying how to conceptualise the term values. Discussing classic and recent contributions, the chapter proposes seeing values as individual and collective conceptions of desirable trans-situational behaviours, objectives and ideals, serving to guide or valuate practice. Despite being an essential part of defining organisational institutionalism, and its sub-streams, values are seldom explicated. Utilising values in organisational and leadership research requires attention to their situatedness in contexts, and this chapter argues they are salient to organisations operating in pluralistic institutional environment. Studying values work, attention should be given to who and how such work is performed.publishedVersio