399: Travelling Salesman Problem

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Travelling Salesman Problem
What's the complexity class of the best linear programming cutting-plane techniques? I couldn't find it anywhere. Man, the Garfield guy doesn't have these problems...
Title text: What's the complexity class of the best linear programming cutting-plane techniques? I couldn't find it anywhere. Man, the Garfield guy doesn't have these problems...

[edit] Explanation

The travelling salesman problem is a classic problem in computer science. An intuitive way of stating this problem is that given a list of cities and their pairwise distances, the task is to find the shortest possible route that visits each city exactly once and then returns to the origin city. A naive solution solves the problem in O(n!) time (where n is the size of the list), simply by checking all possible routes, and selecting the shortest one. A more efficient dynamic programming approach yields a solution in O(n22n) time. These times are given using Big O notation, which is commonly used in computer science to show the efficiency or complexity of a solution or algorithm.

The joke is that the salesman selling online (say on eBay, Amazon Marketplace, or other virtual marketplace) does not have to worry about this problem since he does not need to travel, to which the travelling salesman angrily responds "shut the hell up".

The title text wonders about the time complexity of the cutting-plane method, which is sometimes used to solve optimization problems. The last sentence suggests the down side for Randall of writing comics about computer science; he sometimes encounters problems to which he cannot find the answer, whereas authors of simpler comics such as Garfield do not have this problem. This is also likely a reference to 78: Garfield, which parodies Garfield's simplicity.

The map almost certainly depicts the United States, with the locations highlighted suspected to be (from left to right): Seattle, San Francisco, Los Angeles, Phoenix, Denver, Minneapolis, Dallas, San Antonio, Houston, Chicago (cut off), Detroit, Atlanta, Miami, Washington D.C. Philadelphia, New York and Boston.

This is so far the only comic featuring the Brown Hat character.

Also see earlier strip 287: NP-Complete.

[edit] Transcript

[There is a linked black web, with a path in red; it appears to be a map of the United States.]
Brute-force solution:O(n!)
[The web continues in this one. A man with a hat and a case is drawing it.]
Dynamic programming algorithms: O(n22n)
[Another man, with a hat too, is at a computer, looking back over the chair.]
Selling on eBay: O(1)
eBay salesman: Still working on your route?
Drawing salesman: Shut the hell up.

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Does anyone remember if the Brown Hat appears in any other comics?

I'm not sure, so I created a category and page for him, let's see if that catches any others. --Jeff (talk) 22:04, 29 March 2013 (UTC)
According to the transcript we have two different Brown Hat Guys here. I'm working on this.--Dgbrt (talk) 21:49, 5 October 2013 (UTC)
I'm inclined to think that Brown Hat is specific to this comic, the brown hat being the 50's style homburg or fedora common to salesmen trying to look respectable... Randall likely added the hats to depict folks from a bygone era, (one of whom has caught up with the trend.) -- IronyChef (talk) 01:49, 10 January 2014 (UTC)
The second Brown Hat Guy seems similar to Black Hat both in personality and hat shape. Could he be the same character? Richmond tudor (talk) 03:22, 13 March 2015 (UTC)

It's probably not in the least important, but the network appears to be a collection of key cities in the US arranged by geographical location. 23:07, 9 March 2013 (UTC)

added a better explanation of the title text. -- Nick,5 Oct 2013 (talk) (please sign your comments with ~~~~)

Has anyone answered the question in the title text? --Ricketybridge (talk) 23:55, 9 January 2014 (UTC)

"it is bitter news that in the forty years since Held and Karp no better guarantee [than n^2.2^n] has been found for the problem" [1]. So whereas linear programming techinques tend to be quicker than other algorithms, they have not been shown to be better than O(n^2.2^n). 17:05, 17 August 2014 (UTC)

Doesn't someone at ebay still have to solve the TSP? I guess that's the entire point though. 08:48, 27 July 2014 (UTC)

No because you can send your sales information to all customers at once because they come to you, electronically. It takes no longer for you to be viewed by 100 people than by one person. Thus O(1). 17:05, 17 August 2014 (UTC)
I never used ebay so I don't know how it works and I'm probably missing something obvious. (Maybe it should be explained at the explanation?) If you wanted to personally sell about 17 items to 17 cities like the guy on the left, you have to visit each city by car or something. How does ebay visit the 17 cities to send the items? 06:34, 19 August 2014 (UTC)
You don't have to personally visit each buyer. You put a description of the item(s) you are selling online on eBay, and then people can decide to buy or not. If they do buy, they pay you online, any communication is done online, and you send them the item(s) in the mail. I don't think its necessary to have an explanation of how eBay works, as the majority of people would know. --Pudder (talk) 16:31, 8 December 2014 (UTC)

Oh god, wish I could vote this up... somehow... 15:43, 11 June 2015 (UTC)

It is curious that nobody here appears to have noticed that O(n!) is equivalent to O(n log(n)). Last time I checked, I seem to recall finding that O(n log(n)) was under O(n^2), and therefore under O(n^2.2^n). Or have I missed something? Certainly the worst-case can't be over O(n log(n)), as this is the class of the number of ways to sort a list of the cities. So, then, the O(n^2.2^n) must be better by some other multiplier. Come to that, O(n^2.2^n) seems to be within O(2^n). What am I missing here? 22:27, 9 August 2015 (UTC)

I find it more curious that you think O(n!) is equivalent to O(n log(n))… it most certainly isn’t. O(n log(n)) is (provably) the optimal complexity of a Comparison sort; O(n!) is the complexity of Bogosort, one of the stupidest sorting algorithms imaginable. — 21:23, 23 August 2015 (UTC)
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