This is a problem with all programs involving the impoverished.  It's NOT a money thing, it's a PEOPLE thing.  This isn't simply about the bottom line and perhaps savings, it's about saving lives, preventing or at least delaying deaths and disabilities, preventing or delaying suffering - and not just of the one(s) directly involved but in most cases also many of their friends and family who must witness their suffering.

Even if you want to look JUST at the money, you can't look ONLY at the money in the program.  You'd need to look further, at the money saved by the people who'd be able to work longer, return to work sooner or need less intensive care of one sort or another.  You'd also need to look at things like how less PRODUCTION would be lost to their extended inability to work (and the extension of their friends' and families' inability to work because of how the problem of one often impacts others) and even to how much less INCOME would be lost through the lost work of all these people.

So ultimately what we have through such "savings" is a mix of short-sighted "penny wise, pound foolishness" and a total lack of compassion for the greater society.  For society extends far beyond corporations and the top 5 or so percentage in terms of income and assets.

The problem of homelessness is not one dimensional. It has multiple dimensions that require more than one single solution.
Money is only one dimension of the problem. Even People is over simplifying the different diverse types of people who are homeless.
Reading the original Indypendent article, I realize the objection here is not so much the reduction of funding, but the "rules" to keep funding that has been previously granted without, or with less conditions. If I understand correctly, the new rules require a funded "faith-based" agency to accept homeless not just one day a week, but five days a week. And instead of having a bus to pick up a group of them at the same time, the homeless will be given bus fare.
I don't know much about the system or the people who make us of it, but I don't think the new rules are workable, and the threat of cutting funding based on non-compliance with unworkable rules sound unreasonable.

It would be better to allow different types of shelters to operate as best they can, and be funded using a simple formula based on per person-night occupancy. There can be further enhancements for the types of clients, i.e. disabled, addiction, etc.
Again, I think the key is to increase capabilities to serve, not reduction. To focus only on monetary efficiency is very narrow-minded thinking, and as you said, Danetta, penny wise but pound foolish.

Taking the clients into account is a very important point.  Many shelters - at least here on the West Coast (I'll admit I've never been to the East Coast) - make decisions about whom they will or won't allow into the shelter based on characteristics like disabilities DESPITE all the applicable laws to the contrary.  There are quite often programs that are set up SPECIFICALLY for those with addictions though some may require sobriety or at least not having anything in the person's possession while in the shelter, but it can be anywhere from difficult to impossible to find a spot in a shelter no matter what the circumstances if you've got a disability and particularly so if you use a service animal that's not a guide dog for the blind trained and certified by some well known program.  This is a problem even women run into when trying to escape domestic violence, shelters get picky to the point of cherry picking who becomes a resident.

Again, I'll admit I don't know about the East Coast, but out here in Seattle, moving all the homeless around the city by bus wouldn't fly.  Too many of the city's residents would get their knickers in a twist about the people dragging their belongings with them, about the ones who are clearly intoxicated, about the ones who have obvious mental health issues (often stated as if mental illness is contagious or as if the mentally ill are all violent time bombs waiting to go off when what was observed was often something like the homeless person talking to someone who wasn't there) or even complaining about their hygiene.  I've even seen it happen a time or two where someone took advantage of the homeless person's captivity to try to lecture them about the importance of getting a job - as if it was easy to do and would miraculously solve all the homeless person's problems.

And the per person per night occupancy rate has to be figured reasonably, not delusionally.  Sticking with Seattle, the rate for a person per day in a boarding house for the mentally ill (basically like a shelter but with supervision to make sure they take all their medications) is $32/day (yet they - the government, families, etc. - complain because the houses close and the mentally ill end up on the streets as a result).  I can't board my DOG for that.  How is any agency supposed to adequately house and feed PEOPLE for that unless they're kept in tents and fed the modern equivalent of gruel?  To underfund the agencies then complain because they can't do their jobs is absurd.

The answers aren't nearly as difficult as one might think, even though the complexity of homelessness is vast.  It's all about priorities, folks and until we as a nation begin to put the wellbeing, education, employment opportunities and living conditions of our citizens first and foremost above all other issues facing us, we will continue to quibble over chump change while those in charge of the pursestrings continue to throw breathtaking amounts of money at what they consider their own priorities.  A tough job to accomplish, no doubt about it, but only because we've stood apathetically by for so long we've lost control of our government.  Obama is the first hope and sign I've seen in many years that real change is coming but it's up to us to keep the priorities focused upon US, folks.