Most recently, Anatoly Panyukov claimed to construct an algorithm that solves the Hamiltonian Circuit Problem and has a polynomial complexity. So a manuscript was sent out to colleagues early. We could even do sum two by two. Vinay Deolalikar is standing by his claim and proof. So, a perhaps more objective assessment of this piece of text over pages in 12pt, yet more like 70 pages in 10pt might be:.
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Today we will discuss cake cutting and more. Aziz and Mackenzie have solved an open problem concerning how to cut cakes. The real answer probably is it is a beautiful math problem.
It is easy to state without lots of background. Cake cutting is hard. We like problems that strike back: problems that are not easy to solve. This is a strange view. In real life we might prefer problems that we can easily solve. But not in math. We like problems that are not trivial. The cake cutting problem is hard, so we like it. Cutting Cakes We are theorists so our cakes are one-dimensional line segments.
The problem involves a finite set of agents, say Alice, Bob, and so on. They want to divide the cake, the line segment, into a finite number of pieces. The pieces are then allocated to the agents. The goal is to get a fair division of the cake. Often agents will not have the same tastes: Some like icing more than others, some like the end pieces, while others do not. The fact that the agents assign different values to a piece of the cake is what makes the problem challenging.
If there are two agents the problem has long been solved. Let Bob divide the cake into two pieces, so that he is happy to get either of these pieces. Then have Alice chose which piece she wants. It is easy to see that both Bob and Alice are happy. Both are envy-free: neither would exchange their piece for the others piece. There is a large literature on the cake-cutting problem. We have taken the quote from an article on Medium that neatly conveys details on various protocols. Some main results are: The Selfridge-Conway discrete procedure produces an envy-free division for people using at most The Brams-Taylor-Zwicker moving knives procedure produces an envy-free division for people using at most cuts.
Three different procedures produce an envy-free division for people. Both algorithms require a finite but unbounded number of cuts.
That is to say, the number of cuts may depend on details of their preference functions. The procedure by Aziz and Mackenzie finds an envy-free division for people in a bounded number of cuts. The last is the result in the CACM paper.
Note, the number of cuts can be large: Even for this is immense, galactic. This should be compared to the best lower bound that is order. This gap is even larger than the usual gaps we find in complexity theory. This has started me thinking: what exactly is the relationship between this and proof complexity?
The latter has well-established relationships to complexity-class questions. See for instance these slides by Sam Buss and notes that were scribed by Ken and others. What I am puzzled by is that in most cases the blowup is only one or two exponentials.
The setting with cake-cutting is different, but how different? Easy Cases The Aziz and Mackenzie algorithm takes a long time. It is a nontrivial result, but not one that applies in any practical case. It always takes way too long. The cake will be stale by the time the agents have agreed on their pieces. This raises a question, that also applies to many computational problems.
Is there a way cut a cake faster on some interesting examples? We can explain this by the analogy to sorting. The fastest sorting algorithms run in time where there are objects. But what happens if the objects are already in sorted order? Or at least close to sorted order? The answer is it depends: Some sorting algorithms always take the same time, independent of the input structure. There are other sorting algorithms that can take advantage of the nature of the input.
That is some sorting algorithms can run say in linear time if the input is almost sorted. For the cake cutting problem we ask: Is there a way to cut cakes that is envy-free when the agents have some property? We do not know the answer, but we think it is an interesting question. Here is an example. Suppose that the agents have the same measures. That is, they evaluate every piece of cake in the same way.
If we know this—and if we continue our supposition above that Bob can cut with exact precision—then there is an easy answer: Have Bob do the cuts. Then all agents will be equally happy since they have the same measures. The question is, what if we do not know? I believe there should be some theorem like this: Theorem 1 Conjecture There is an envy-free algorithm that operates in time the algorithm so that either: It yields an envy-free solution, or It determines that some agents have different measures.
In the second case the cake will be cut as before. Open Problems I originally planned on discussing size and complexity of proofs. This is driven by the complexity of the cake cutting algorithms. They tend to have lots of cases and are difficult to understand. They are also difficult to find—this is why cake cutting questions have been resistance to progress.
More on this in the future.
DEOLALIKAR PROOF PDF
Still no final chapter, lessons learned Vinay Deolalikar just over a year ago claimed that he had a proof that. Today I want to make a short comment on the status of his claim. I recently received an email asking: Has the anniversary slipped your notice? But there is not a lot to say about the situation—for better or for worse. Here is all that we know publicly: His web page still claims the result, and explains that it is out to a journal for the standard refereeing process. He still believes that he has a proof, and reiterated that it is being checked by referees. He has expanded and updated the paper, and claims to have answered all the issues that were raised on the web here and elsewhere.
P versus NP problem
Today we will discuss cake cutting and more. Aziz and Mackenzie have solved an open problem concerning how to cut cakes. The real answer probably is it is a beautiful math problem. It is easy to state without lots of background. Cake cutting is hard.
Deolalikar’s Claim: One Year Later
Malanris This can be contrasted with k-SAT, that looks weather there is any satisfying assignment. Thomson Course Technology, But in either case, there is a limit as to how much good expository guidelines can help in preventing these sorts of events from occurring. Not that it is just deolalokar to find a single solution. Most to the point, these competitions have lists of hard instances on which to test the algorithm. At deolalikwr point you can see that there might be several NDTM solving SAT0, and we are in trouble as we do not know whose acceptance path distribution we should consider.