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The MPs' recommendations come in a Home Affairs select committee report that claims alcohol-related public disorder is placing a "heavy burden" on police resources.
The committee suggests a ban on supermarket 'loss leader' deals, in which beer and other drinks are sold at a huge discount to attract shoppers, and is calling for restrictions on happy hours and other drinks promotions.
It also revealed policymakers were considering raising the age for buying alcohol in supermarkets and shops to 21.
The report, Policing in the 21st Century, claims the biggest problem facing police forces is violence and disorder caused by excessive drinking of cheap alcohol.
It also criticises Whitehall-imposed targets for distorting police priorities leading them to focus on "trivial misdemeanours" so that forces were "hitting the targets but missing the point".
MPs have said one possible solution in England and Wales would be setting a minimum price on alcohol, an option already being considered in Scotland.
Keith Vaz, the committee's Labour chairman, said: "We cannot have, on one hand, a world of alcohol promotions for profit that fuels surges of crime and disorder and, on the other, the police diverting all their resources to cope with it."
Last week drinks firms and retailers launched a social marketing campaign called Project 10 aimed at tackling alcohol misuse.
The campaign is intended to influence attitudes toward the social acceptability of drunkenness and so challenge the minority to reconsider their behaviour.