Bipartite Matching Is in NC

by amichail • 1 month ago

126 18 1 month ago

amirhirsch3• 26 days ago

This is an awesome result.

For those unfamiliar: NC is the class of problems which can be solved in polylogarthmic depth with polynomial number of logic gates. It is unproven if NC != P similar to P != NP.

gignico• 26 days ago

Yes, but logic gates with constant fan-in, crucially, otherwise that's called AC.

amirhirsch3• 26 days ago

Yes.

There is a beautiful proof of the disjunction between AC0 and NC showing parity cannot be done in AC0 using harmonic analysis of Boolean functions

amluto• 26 days ago

I never studied these specific classes, but my immediate intuition is that an n-input fan-in AND or OR gate can be reduced to a tree of 2-input gates with depth O(log(n)), which preserves polylog complexity, so surely AC = NC.

Wikipedia agrees :)

If you specify the exponent of the log, you get a different answer.

osti3• 26 days ago

So is it a class of problems that can be parallelized well?

adgjlsfhk1• 26 days ago

no (in both directions). lots of np/exp problems paralize well and you can be in NC and parallelize really inefficiently (e.g. you can get a 10x speedup, but you need 1000000x the hardware). the better framing is that NC is the class of efficient algorithms that can be sped up near arbitrarily by parallelization

vintermann• 26 days ago

Many years ago, on Boardgamegeek, there was a game trading system called "Math Trades", where participants would list a number of trades they were willing to make, and they ran some complicated calculations to figure out how to let as many as possible trade.

CS professor Chris Okasaki, known for his book on purely functional data structures, also played board games and he came across this phenomenon. He realized that it could be modelled as a bipartite matching problem, and wrote a MUCH faster program to manage math trades.

https://okasaki.blogspot.com/2008/03/what-heck-is-math-trade...

I guess it can be made even faster now in theory.

emil-lp• 26 days ago

The kidney exchange problem isn't bipartite matching but a cycle packing problem (or disjoint cycle cover).

mirashii• 26 days ago

The math trades still happen regularly at cons, e.g. Origins had one just last week.

sigbottle• 26 days ago

Chris okasaki! Was into functional data structures in college, great book and great dude

throwaway81523• 26 days ago

I don't think this new result is supposed to be a speedup. It might even be slower than the existing method. Rather, it's a way to get rid of the random number generator that the old algorithm relied on, so it's deterministic unlike the old way. I'm not even sure that it's guaranteed to find the answer, as opposed to finding it with high probability.

It's mostly of theoretical interest except for some possible niche applications, I'd say. For a math trade type of problem, you'd just go ahead and use the old algorithm with an RNG.

Another famous result of this type was AKS primality testing. Randomized algorithms like Miller-Rabin were known for a long time, very reliable, and quite simple to implement, but AKS was an important theoretical advance because it didn't use random inputs. I think everyone still uses Miller-Rabin in practice.

klipt• 25 days ago

Of course the easiest solution to the problem is money. You sell the games you don't want for money. You buy the games you want with money.

Much simpler than algorithms for bartering.

Bipartite Matching Is in NC

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