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We show that the determination of a minimum cut-set of odd cardinality in a graph with even and odd vertices can be dealt with by a minor modification of the polynomially bounded algorithm of Gomory and Hu for multi-terminal networks. We connect this problem to the problem of identifying a matching (or blossom) constraint that chops off a point which is not contained in the convex hull of matchings or proving that no such inequality exists. Both the b-matching problems without and with upper bounds are considered. We discuss how the results of this paper can be used in conjunction with commercial LP packages lo solve b-matching problems.

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ObjectivesTo compare the effects of the Rajiv Aarogyasri Health Insurance Scheme of Andhra Pradesh (AP) with health financing innovations including the Rashtriya Swasthya Bima Yojana (RSBY) in Maharashtra (MH) over time on access to and out-of-pocket expenditure (OOPE) on hospital inpatient care.Study designA difference-in-differences (DID) study using repeated cross-sectional surveys with parallel control.SettingNational Sample Survey Organisation of India (NSSO) urban and rural ‘first stratum units’, 863 in AP and 1008 in MH.MethodsWe used two cross-sectional surveys: as a baseline, the data from the NSSO 2004 survey collected before the Aarogyasri and RSBY schemes were launched; and as postintervention, a survey using the same methodology conducted in 2012.Participants8623 households in AP and 10 073 in MH.Main outcome measuresAverage OOPE, large OOPE and large borrowing per household per year for inpatient care, hospitalisation rate per 1000 population per year.ResultsAverage expenditure, large expenditures and large borrowings on inpatient care had increased in MH and AP, but the increase was smaller in AP across these three measures. DIDs for average expenditure and large borrowings were significant and in favour of AP for the rural and the poorest households. Hospitalisation rates also increased in both states but more so in AP, although the DID was not significant and the subgroup analysis presented a mixed picture.ConclusionsHealth innovations in AP had a greater beneficial effect on inpatient care-related expenditures than innovations in MH. The Aarogyasri scheme is likely to have contributed to these impacts in AP, at least in part. However, OOPE increased in both states over time. Schemes such as the Aarogyasri and RSBY may result in some positive outcomes, but additional interventions may be required to improve access to care for the most vulnerable sections of the population.