Newsgroups: comp.constraints
Path: cantaloupe.srv.cs.cmu.edu!europa.chnt.gtegsc.com!howland.reston.ans.net!gatech!EU.net!news2.EUnet.fr!news.fnet.fr!ilog!puget
From: puget@ilog.ilog.fr (Jean-Francois Puget)
Subject: Benchmark
Message-ID: <1995May12.141354.22674@ilog.fr>
Sender: news@ilog.fr
Nntp-Posting-Host: laumiere
Organization: ILOG S.A., France
Date: Fri, 12 May 95 14:13:54 GMT
Lines: 72

We recently worked on a surprisingly difficult combinatorial
problem, called the Golomb problem. We propose it as a benchmark,
since it only involves simple constraints (substraction and 
difference).

This
problem arose in a radio astronomical application, where the relative
positions of several radio telescopes had to be computed. The
problem is to find a set of values $n$ representing the graduations 
of a rule such that the differences between two graduations are all
distinct, and such that the length of the rule is minimal. The ith
graduation is represented by a finite domain variable $x_i$. The
constraints are:

x_1 = 0
x_i < x_{i+1}

The differences (x_i - x_j) are distinct, ie, for all pairs
(i, j), (k,l) where i <> j, k <> l :

x_i - x_j <> x_k - x_l  

The latter constraint can expressed as an alldiff constraint on the set
of expressions x_i-x_j.

The goal is to minimize the length of the rule, i.e. to minimize the
value of $x_n$.

Note that symmetries can be removed by orienting the rule, adding the 
constraint that the first variable is smaller than the difference
between the last one and the next to last:

x_2-x_1 < x_n-x_{n-1}

For instance, an optimal solution for n=5 is:

0  2  7  10  11


We directly encode the constraints and use a static order (choosing
the first graduation) for the choice of the variable to enumerate and
a standard minimization procedure. 

The time to find and prove the optimality of the solution increases very
quickly with the size of the problem.

An optimal solution for Golomb 12 is:

0  2  6  24  29  40  43  55  68  75  76  85

It is found after more than 2042000 backtracks.
The total number of backtracks (or fails) including the proof of
optimality is 3143316.

We recently solved Golomb 13. An optimal solution is:
0  2  5  25  37  43  59  70  85  89  98  99  106

it was found after 156478 seconds on a sparc 20 and more than 
26969000 fails. The total number of fails including the proof for
optimality is 53108352. It took 330643 seconds on a sparc 20, i.e.
almost 4 days.

The results reported here were obtained with Ilog Solver.


Jean-Francois Puget and Michel Leconte

-- 
  Jean-Francois Puget		 net : puget@ilog.fr
  ILOG S.A.                      url : http://www.ilog.fr
  2 Avenue Gallieni - BP 85	 tel : +33 1 46 63 66 66
  F-94253 Gentilly Cedex FRANCE	 fax : +33 1 46 63 15 82
