Modeling Stable Matching Problems with Answer Set Programming

The Stable Marriage Problem (SMP) is a well-known matching problem first introduced and solved by Gale and Shapley (1962). Several variants and extensions to this problem have since been investigated to cover a wider set of applications. Each time a new variant is considered, however, a new algorithm needs to be developed and implemented. As an alternative, in this paper we propose an encoding of the SMP using Answer Set Programming (ASP). Our encoding can easily be extended and adapted to the needs of specific applications. As an illustration we show how stable matchings can be found when individuals may designate unacceptable partners and ties between preferences are allowed. Subsequently, we show how our ASP based encoding naturally allows us to select specific stable matchings which are optimal according to a given criterion. Each time, we can rely on generic and efficient off-the-shelf answer set solvers to find (optimal) stable matchings.Comment: 26 page

Creation and the philosophy of science: Freedom, contingence and the modality of the natural sciences

Understanding creation through the theological loci of Christology and Trinitarian theology gives a view of the natural world as both contingent and free. This distinctively Christian view of the natural world carries implications for the natural sciences in terms of philosophical modality. This paper explores three such themes: (i) the nature of reason; (ii) the character of theories; and (iii) the relationship between discursivity and the logic of reality

Stable marriage and roommates problems with restricted edges: complexity and approximability

In the Stable Marriage and Roommates problems, a set of agents is given, each of them having a strictly ordered preference list over some or all of the other agents. A matching is a set of disjoint pairs of mutually acceptable agents. If any two agents mutually prefer each other to their partner, then they block the matching, otherwise, the matching is said to be stable. We investigate the complexity of finding a solution satisfying additional constraints on restricted pairs of agents. Restricted pairs can be either forced or forbidden. A stable solution must contain all of the forced pairs, while it must contain none of the forbidden pairs. Dias et al. (2003) gave a polynomial-time algorithm to decide whether such a solution exists in the presence of restricted edges. If the answer is no, one might look for a solution close to optimal. Since optimality in this context means that the matching is stable and satisfies all constraints on restricted pairs, there are two ways of relaxing the constraints by permitting a solution to: (1) be blocked by as few as possible pairs, or (2) violate as few as possible constraints n restricted pairs. Our main theorems prove that for the (bipartite) Stable Marriage problem, case (1) leads to View the MathML source-hardness and inapproximability results, whilst case (2) can be solved in polynomial time. For non-bipartite Stable Roommates instances, case (2) yields an View the MathML source-hard but (under some cardinality assumptions) 2-approximable problem. In the case of View the MathML source-hard problems, we also discuss polynomially solvable special cases, arising from restrictions on the lengths of the preference lists, or upper bounds on the numbers of restricted pairs

Some Effects of Prescribed Fire at Cedar Creek Natural History Area

On four oak savanna restoration compartments with a total area of 100 acres, annual burns (1965-1972) reduced the percent of milacre plots stocked with hazel to 39 compared with 65 on unburned areas. Four growing seasons after one and three fires the hazel distribution was not significantly different from the control. Annual burns increased the density of hazel stems in clones to 19.5 per .0001 acre compared to 11.0 on controls. Stem density four years after 1 and 3 burns averaged 10.0 and 8.0 per .0001 acre. The o.d. weight of live hazel stems per .0001 on annual burn areas was 16 percent of that on controls. Four years after 1 or 3 fires stem weight was not significantly different from the control. Stem height on annual burn areas averaged 17 inches compared with 33 inches on the con1rols. Maximum stem heights on annual burns averaged 24 inches compared wi1h 42 inches on controls. Four growing seasons after 1 or 3 fires average and maximum stem heights were not significantly different from controls

Manipulation Strategies for the Rank Maximal Matching Problem

We consider manipulation strategies for the rank-maximal matching problem. In the rank-maximal matching problem we are given a bipartite graph $G = (A \cup P, E)$ such that $A$ denotes a set of applicants and $P$ a set of posts. Each applicant $a \in A$ has a preference list over the set of his neighbours in $G$, possibly involving ties. Preference lists are represented by ranks on the edges - an edge $(a,p)$ has rank $i$, denoted as $rank(a,p)=i$, if post $p$ belongs to one of $a$'s $i$-th choices. A rank-maximal matching is one in which the maximum number of applicants is matched to their rank one posts and subject to this condition, the maximum number of applicants is matched to their rank two posts, and so on. A rank-maximal matching can be computed in $O(\min(c \sqrt{n},n) m)$ time, where $n$ denotes the number of applicants, $m$ the number of edges and $c$ the maximum rank of an edge in an optimal solution. A central authority matches applicants to posts. It does so using one of the rank-maximal matchings. Since there may be more than one rank- maximal matching of $G$, we assume that the central authority chooses any one of them randomly. Let $a_1$ be a manipulative applicant, who knows the preference lists of all the other applicants and wants to falsify his preference list so that he has a chance of getting better posts than if he were truthful. In the first problem addressed in this paper the manipulative applicant $a_1$ wants to ensure that he is never matched to any post worse than the most preferred among those of rank greater than one and obtainable when he is truthful. In the second problem the manipulator wants to construct such a preference list that the worst post he can become matched to by the central authority is best possible or in other words, $a_1$ wants to minimize the maximal rank of a post he can become matched to

Socially stable matchings in the hospitals / residents problem

In the Hospitals/Residents (HR) problem, agents are partitioned into hospitals and residents. Each agent wishes to be matched to an agent in the other set and has a strict preference over these potential matches. A matching is stable if there are no blocking pairs, i.e., no pair of agents that prefer each other to their assigned matches. Such a situation is undesirable as it could lead to a deviation in which the blocking pair form a private arrangement outside the matching. This however assumes that the blocking pair have social ties or communication channels to facilitate the deviation. Relaxing the stability definition to take account of the potential lack of social ties between agents can yield larger stable matchings. In this paper, we define the Hospitals/Residents problem under Social Stability (HRSS) which takes into account social ties between agents by introducing a social network graph to the HR problem. Edges in the social network graph correspond to resident-hospital pairs in the HR instance that know one another. Pairs that do not have corresponding edges in the social network graph can belong to a matching M but they can never block M. Relative to a relaxed stability definition for HRSS, called social stability, we show that socially stable matchings can have different sizes and the problem of finding a maximum socially stable matching is NP-hard, though approximable within 3/2. Furthermore we give polynomial time algorithms for three special cases of the problem

Key Issues Generated from the XI International Rangeland Congress 2021: Summary and Way Forward

The important issues, knowledge gaps, and evolving research approaches for the global rangelands are summarised in this review of submissions to the Joint XXIV International Grasslands and XI International Rangelands Congress (IGC/IRC). In the big picture, it is concluded that stand-alone studies of livestock production are becoming rare compared to that of the past International Rangelands Congresses (IRC). Rather, added effort is now being directed at understanding the fuller context of social-ecological systems (SESs) on rangelands in a quest to improve the prospects for sustainable resource management as well as the enhancement of human welfare. Although climate change is upon us, there was still a dearth of papers that dealt with broad- scaled climate-adaptation per se; opportunities to improve local drought response were the default topics here with a focus on implementing better drought early warning systems and integrating perspectives among producers and scientists. Invasive species challenges remain as prominent global concerns, and woody encroachment is viewed as a major contributor to rangeland degradation. Treatments to combat rangeland degradation can involve innovative layering methods incorporating grazing management and use of prescribed fire. While there is an important backdrop concerning ecosystem services from rangelands, research in this area is still in its infancy. Analysing trade-offs between production and conservation for services such as carbon sequestration loom large going forward. There were relatively few papers concerning wildlife, tourism, and associated issues; successes and challenges for natural resource conservancies were noted, in particular. These are topics that merit more creative research and development attention in the future. Some contributions highlighted the important issue of landscape conversion from rangelands to cultivation; in conjunction with human population growth, loss of such key resources can be very negative for wildlife and associated values. In terms of pastoralism and related sub-themes, while it was noted that the majority of studies now embrace SESs and integrated, participatory, action- oriented approaches, there is little effort to standardize methodologies. A focus on repeatable methods can help grow sustainability science on rangelands, and this is a challenge for research and outreach education. The volume of studies submitted overall indicated a decided numerical advantage for the Global South over the Global North. Why this is the case remains unclear, however. Disciplinary research traditions in wealthier nations may not yet provide the incentives needed to spur innovative SES work. Finally, policy makers are seen by many investigators as being ignorant of rangeland development issues. It is argued, however, that this view has not changed for 40 years. How to better engage policy makers in comprehensive SES projects is an important future goal. Policy makers themselves can then also become human research subjects in the overall process. Based on our review the future for IRC stakeholders is clear: Continue the expansion of interdisciplinary SES and action-based approaches and increase attention to climate-change adaptation/mitigation, ecosystem services, community-based development, human empowerment, market development, poverty mitigation, and creation of effective policy frameworks