Your wording choice leaves a lot of freedom in the answers. Hopefully some of these spark an extended discussion, and possibly more questions:
No Relationships, as in no humans are aware of each other
The strictest of definitions I can imagine is one in which humans are simply not allowed to be aware of each-other's minds. As this occurs automatically in society, the government would have to stop it.
First off, an assumption I will make the entire time: the government never forces anyone so much as coerces them to play along. If the government is forcing the destruction of interpersonal relationships, humanity itself will fight back. The government will have to smoothly adjust the human mind.
The Problem of Other Minds is a class issue in psychology and philosophy. We become aware of "other minds" at roughly the age of 2 or 3, and begin to treat them as Others. This would be a relationship, so the government needs to make sure nobody ever gets an opportunity to be aware of other minds.
One solution is homogeneity. In a normal toddler development, there are very clear minds: parents, the cat, and maybe even the pot that is so remarkably kind as to protect their head during their quest to slay the dragons and/or get the cookies off the counter without mommy noticing. The toddler eventually notices there is something different about some of these minds, and categorizes them as Others. If the government could set up each person's environment to be more homogenous, it could remove the need to identify Others until that part of the mind doesn't function. As an example, consider Stack Exchange without names, icons, or scores. Just a flat Q/A database.
One other solution is disruption. Other minds may be hostile, and one of the easiest ways to generate artificial hostility is a language barrier. If each human is trained different, linguistically, the government could ensure that they have no desire to talk to each other, even if they recognize the others as minds.
No Relationships, as in no monogamous relationships
Less exacting would be the desire to overcome monogamy. Marriage is a very powerful institution today, and I could see a distopian government wanting to squelch it and take the power for itsef.
This would take more balance. Groupings form naturally, whether pairs, trios, or any other grouping. Effort would have to be taken to make this undesirable. One solution involves some number games. In Chinese culture, some numbers are considered unlucky. If the government could make the number two tremendously unlucky, anything in 2's would be avoided subconsciously.
This could have very interesting effects on AI. While binary is the easiest way to build a computer using electrical circuits, there's no reason you could not develop one using trinary logic. Trinary logic is more conducive to AI because it allows for either "maybe" or "unknown," both of which are useful for AIs.
Defunct relationships, with little caring
The definition of a relationship is tricky. How does the government know it got them all? Another approach might be to simply allow relationships, but ensure that when they do occur, they don't matter much.
This is the most distopian case in our current environment. As suggested by others, computers are leeching our attention away from each other continuously. All we have to do is make sure the humans lose the ability to be soft enough to truly support one another better than the machines do, and the relationships become unimportant.
Bright flashy lights, issues that need immediate attention, intelligence puzzles rather than intuitive puzzles, these would all condition us to a style of interaction which ensures there is a barrier between everyone.