n-Queens Completion is NP-Complete

Peter Nightingale and Ian Gent at Falkland Palace, Wednesday, 17 August 2017.
©Stuart Nicol Photography, 2017

Ian Gent, Christopher Jefferson and Peter Nightingale have shown that a classic chess puzzle is NP-Complete. Their paper “Complexity of n-Queens Completion” was published in the Journal of Artificial Intelligence Research on August 30.

The n-Queens puzzle is a classic chess problem: given a chessboard of size n by n, can you place n queens so that no two queens attack each other?  That is, can you place the queens with no two queens are on the same row, column, or diagonal? The n-Queens puzzle has long been known to be simple to solve:  you can solve the problem for all n except 2 and 3, and solutions for all other n can be described in a few lines.  This very simplicity has led to repeated controversy in Artificial Intelligence (AI). The n-Queens puzzle is often used as a benchmark problem, but good results on the problem can always be challenged because the problem is so simple to solve without using AI methods.

The new work follows a challenge on Facebook on New Year’s Day, 2015, when a friend of Ian’s asked him how hard n-Queens is if some queens were already placed on the board.  It turns out, this version (dating from 1850) of the puzzle is only two years younger than the more famous n-Queens problem. The picture shows Peter (left) and Ian (right) with queens on the board at positions suggested by Nauck in 1850, the squares b4 and d5.  Can you put another 6 queens on the board so that the entire board is a solution of 8-Queens?  The general version with some number of queens preplaced on an n by n board is the n-Queens Completion puzzle.

 

With queens at b4 and d5, can you place 6 more queens to get a solution to the 8-queens puzzle? ©Stuart Nicol Photography, 2017

Ian, Christopher and Peter have shown that the n-Queens puzzle is in fact hard, not simple.  It belongs to the complexity classes NP-Complete and #P-Complete. Other NP-Complete problems include the “travelling salesperson problem”, finding cliques in graphs, and many other important problems, from scheduling to circuit layout. This puts n-Queens Completion at the centre of the most important theoretical problem in computer science — it has long been known that either all NP-complete problems are easy, or none of them are. Most computer scientists believe that this means there is no efficient algorithm possible for solving this problem, compared to the very simple techniques long known for n-Queens.
The importance of this work is that it provides researchers with a benchmark that can be used for evaluating AI techniques. Moreover, it helps to explain why so many AI techniques have been used on the n-Queens puzzle. Most techniques do most of their work with some queens partially placed, using some form of (hopefully intelligent) trial and error. In fact it turns out that many researchers – in order to solve a simple problem – have solved it by turning the simple problem of n-Queens into the hard problem of n-Queens Completion.
It does seem that AI researchers should not use n-Queens as a benchmark, but the very closely related n-Queens Completion puzzle is a valid benchmark. As well as the theoretical results, the paper shows how example instances can be generated which appear to be hard in practice. Some caution is still needed, though. It does seem to be quite hard to generate hard instances of n-Queens Completion.
The University has also issued an article on the same paper, under the title “Simple” chess puzzle holds key to $1m prize

Seminar: Propagation and Reification: SAT and SMT in Prolog (continued)

Event details

  • When: 23rd June 2017 13:00 - 14:00
  • Where: Cole 1.33a
  • Series: AI Seminar Series
  • Format: Seminar

Jacob Howe, City University, London

Abstract: This talk will recap how a watched literal DPLL based SAT solver can be succinctly coded in 20 lines of Prolog. The focus of the talk will be the extension of this solver to an SMT solver which will be discussed with a particular focus on the case where the theory is that of rational-tree constraints, and its application in a reverse engineering problem.
[Note change of time from that previously advertised]

Success in the Laidlaw Undergraduate Internship Programme in Research and Leadership

Congratulations to Patrick Schrempf and Billy Brown who have been successful in their applications for a Laidlaw Undergraduate Internship in Research and Leadership for 2017. You can read further details about Billy and Patrick below.

Billy Brown:

I’m a fourth year Computer Science student from Belgium with too much interest for the subject. I play and referee korfball for the university, and I am fascinated by Old English and Norse history and mythology. I plan on using the Laidlaw Internship programme to get into the field of Computer Science research.

Project summary:

The Essence Domain Inference project aims to improve automated decision making by optimising the understanding of the statements used to define a problem specification. As part of the compilation of the high level Essence specification language, this project would tighten the domains to which a specified problem applies, with a domain inference algorithm.

The work is very much in the context of the recently-announced EPSRC grant working on automated constraint modelling in an attempt to advance the state of the art in solving complex combinatorial search problems. The modelling pipeline is akin to a compiler in that we refine a specification in the Essence language Billy mentions down to a number of powerful solving formalisms. The work Billy plan is to improve the refinement process and therefore the performance of the solvers, leading to higher quality solutions more quickly.

Patrick Schrempf:
I am currently a third year Computer Science student from Vienna. After enjoying doing research with the St Andrews Computer Human Interaction (SACHI) group last year, I am looking forward to the Laidlaw Internship Programme. Apart from research and studying, I enjoy training and competing with the Triathlon Club and the Pool Society.
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