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We study the problem of counting the number of popular matchings in a given instance. A popular matching instance consists of agents A and houses H, where each agent ranks a subset of houses according to their preferences. A matching is an assignment of agents to houses. A matching M is more popular than matching M ′ if the number of agents that prefer M to M ′ is more than the number of people that prefer M ′ to M . A matching M is called popular if there exists no matching more popular than M . McDermid and Irving gave a poly-time algorithm for counting the number of popular matchings when the preference lists are strictly ordered.We first consider the case of ties in preference lists. Nasre proved that the problem of counting the number of popular matching is #P-hard when there are ties. We give an FPRAS for this problem.We then consider the popular matching problem where preference lists are strictly ordered but each house has a capacity associated with it. We give a switching graph characterization of popular matchings in this case. Such characterizations were studied earlier for the case of strictly ordered preference lists (McDermid and Irving) and for preference lists with ties (Nasre). We use our characterization to prove that counting popular matchings in capacitated case is #P-hard.

Comfort and relax is the basic need needed eventually during the summer seasons and has amended an important need, of an hour with the use of such air conditioning and cooling system which is increasing rapidly. The power outages and high initial cost has made it unsuitable for the village areas. Photovoltaic systems are seen as a way for renewable energy in the future, as the solar air conditioning system provides many essential products to the village. The technology improves indoor air quality by monitoring product performance, improving ventilation and controlling humidity. Despite the growing number of design and management guidelines for energy use, the high electricity demand continues to rise and there is currently no solar air conditioning system for residential use at home, especially in cities, businesses and schools. The solar air conditioner with a freezer for food at home has been tested as part of the business, making it ideal for home use. The proposed work addresses this issue and is a design project aimed at maintaining a comfortable temperature inside a car parked in a parking lot or driveway. The proposed work addresses this issue and is a design project aimed at maintaining a comfortable temperature inside a car parked in a parking lot or driveway.

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