The Hospital Residents problem with sizes (HRS) is a generalisation of the well-studied hospital residents (HR) problem. In the HRS problem, an agent a has a size s(a) and the agent occupies s(a) many positions of the hospital h when assigned to h. The notion of stability in this setting is suitably modified, and it is known that deciding whether an HRS instance admits a stable matching is NP-hard under severe restrictions. In this work, we explore a variation of stability, which we term occupancy-based stability. This notion was defined by McDermid and Manlove (J. of Comb. Opt. 2010) but remained unexplored to the best of our knowledge. In our work, we show that every HRS instance admits an occupancy-stable matching. We further show that computing a maximum-size occupancy-stable matching is NP-hard. We complement our hardness result by providing an approximation algorithm with a guarantee strictly better than 3 for the max-size occupancy-stable matching problem. Given that the classical notion of stability adapted for HRS is not guaranteed to exist in general, we show a practical restriction under which a stable matching is guaranteed to exist. We present an efficient algorithm to output a stable matching in the restricted HRS instances. We also provide an alternate NP-hardness proof for the decision version of the stable matching problem for HRS which imposes a severe restriction on the number of neighbours of non-unit sized agents.

We consider the problem of assigning agents to resources under the two-sided preference list setting where resources specify an upper-quota and a lower-quota, that is, respectively the maximum and minimum number of agents that can be assigned to it. Different notions of optimality including envy-freeness and relaxed stability are investigated for this setting. Krishnaa et al. (2020) show that in this setting, the problem of computing a maximum size envy-free matching ( MAXEFM ) or a maximum size relaxed stable matching ( MAXRSM ) that satisfies lower quotas is not approximable within a certain constant factor unless P = NP . In this work, we investigate the parameterized complexity of MAXEFM and MAXRSM . We show that MAXEFM is W [ 1 ] -hard and MAXRSM is para- NP -hard when parameterized on several natural parameters derived from the instance. We present kernelization results and FPT algorithms for both problems parameterized on other relevant parameters.

Tables are a universal idiom to present relational data. Billions of tables on Web pages express entity references, attributes and relationships. This representation of relational world knowledge is usually considerably better than completely unstructured, free-format text. At the same time, unlike manually-created knowledge bases, relational information mined from "organic" Web tables need not be constrained by availability of precious editorial time. Unfortunately, in the absence of any formal, uniform schema imposed on Web tables, Web search cannot take advantage of these high-quality sources of relational information. In this paper we propose new machine learning techniques to annotate table cells with entities that they likely mention, table columns with types from which entities are drawn for cells in the column, and relations that pairs of table columns seek to express. We propose a new graphical model for making all these labeling decisions for each table simultaneously, rather than make separate local decisions for entities, types and relations. Experiments using the YAGO catalog, DB-Pedia, tables from Wikipedia, and over 25 million HTML tables from a 500 million page Web crawl uniformly show the superiority of our approach. We also evaluate the impact of better annotations on a prototype relational Web search tool. We demonstrate clear benefits of our annotations beyond indexing tables in a purely textual manner.