A CONJECTURE ON A CONTINUOUS OPTIMIZATION MODEL FOR THE GOLOMB RULER PROBLEM

. A Golomb Ruler (GR) is a set of integer marks along an imaginary ruler such that all the distances of the marks are diﬀerent. Computing a GR of minimum length is associated to many applications (from astronomy to information theory). Although not yet demonstrated to be NP-hard, the problem is computationally very challenging. This brief note proposes a new continuous optimization model for the problem and, based on a given theoretical result and some computational experiments, we conjecture that an optimal solution of this model is also a solution to an associated GR of minimum length.


Introduction
Given a positive integer n, the Golomb Ruler Problem (GRP) asks for locating n integer marks along an imaginary ruler such that all distances between pairs of marks are distinct and the length of the ruler is minimum [3,6].The GRP has many applications, including astronomy [4], communications [1], and information theory [18].
Historically, the GRP first appeared related to Sidon sets [21], i.e. sets of integers so that all pairwise sums of elements of the set are distinct.Lower bounds for the GRP and connections to number theory are discussed, respectively, in [7,14,22,23].
We say that we have a Golomb Ruler (GR), given n marks, when the associated distances are all different, and an Optimal Golomb Ruler (OGR), when a GR has the smallest length.
Although not yet demonstrated to be NP-hard, solving the GRP exactly proved to be very difficult [5,19].Solutions for all n ≤ 27 are available on Wikipedia ("Golomb Ruler", acessed on February 15th, 2021).
In [13], a linear programming model is proposed for the GRP and, more recently, authors in [8] compare different optimization approaches to certify optimality to the GRP.
We present a new model to the GRP based on a continuous optimization formulation (Sect.2.1).In Section 2.2, we give a theoretical result and conjecture that a solution of this model is also a solution to the associated GRP.Computational experiments are given in Section 3. Section 4 concludes the paper with new research directions.
2. Models, a theoretical result and a conjecture for the GRP Formally, a GR consists of a set of integers R = {r 1 , . . ., r n }, with r 1 < r 2 < . . .< r n , such that for each positive integer d, there is at most one pair {r i , r j } ⊂ R, i < j, that satisfies The set R represents positions of n marks on the ruler with length L = r n − r 1 .Without loss of generality, we fix r 1 = 0.

A new model for the GRP
In order to present the first model to the GRP (first described in [8]), we need an upper bound L u for the length of the ruler, given also as input of the problem, in addition to n marks on the ruler: x i = n − 1, Since we are minimizing t, inequalities for i = 1, . . ., L u , "force" the length of the ruler to be minimum.At the same time, equation Lu i=1 x i = n − 1 requires that n − 1 of the variables x i are equal to 1, and inequalities for j = 1, . . ., L u − 1, guarantee that all distances between pairs of marks are distinct.
A solution (x 1 , . . ., x Lu ) of model ( 2.1) provides a GRP solution given by (ix i ), for i = 1, . . ., L u , whose minimum length of the associated ruler is the optimal value found for t.
In the literature, there are many heuristics for calculating upper bounds for the GRP (e.g.see [8]).We propose a simple greedy heuristic whose pseudocode is given below

A theorem and a conjecture for the GRP
The next result establishes that, if the optimal solution (x 1 , . . ., x Lu ) of model (2.2) is such that x i ∈ {0, 1}, for i = 1, . . ., L u , in fact, (x 1 , . . ., x Lu ) is also a solution for the associated GRP.
Computational experiments provided in the next section indicate that the opposite direction of the above result is also true, which we state as a conjecture, given as follows.
Conjecture 2.2.Given an upper bound L u for model (2.1), if L is the optimal value associated to an optimal solution x = (x 1 , . . ., x Lu ), then x = (x 1 , . . ., x Lu ) is an optimal solution for model (2.2) with optimal value equal to L.

Computational results
For n = 5, . . ., 28, the values for L u were obtained using the greedy heuristic as described in the previous section.Table 1 gives such values, comparing to two conjectured upper bounds for the GRP [8].
Model (2.2) was solved using the software "Knitro 12.3.0",via Neos Server Version 6.02 , and "Cplex 12.8.0"was used to solve model (2.1), since it is better than Knitro to combinatorial problems.All the instances run on a Lenovo laptop with 6 GB RAM, intel celeron 1.6 GHz and 64 bit windows 10.
The limit time was set to 120 000 s when the solver Cplex was used.For the solver Knitro, we used 5000 random initial solutions.
From Table 2, we see that the computational time required for solving model (2.1) increases exponentially and no solution is found for n > 7.

Conclusions
Using a very modest computer, small-size instances of the GRP were solved, with related computational time increasing sub-exponentially (differently from the exponential cost of the combinatorial model), which may suggest promising research directions based on this new approach.Indeed, it is possible to develop an ad hoc nonlinear optimization algorithm for the mathematical model proposed,in order to improve the computational results.This possibility can be considered the main advantage of our proposal, compared to the other approaches given in literature.
Compared to two conjectured upper bounds for the GRP, the proposed greedy heuristic also presented good results as observed in Table 1.
Our approach is based on a continuous optimization model and we conjecture that an optimal solution of this model is also a solution to the related GRP.
We emphasize that GRP solutions for n = 24, 25, 26, 27 have been obtained by a massively parallel computational system (www.distributed.net)and the problem for n = 28 is currently under way, whose search started in 2014.
Motivated by the application of the GRP on wireless localization [17] and Partial Digest Problem [15], the connection with Distance Geometry [2,[9][10][11][12]16] should be exploited, also considering the variants of the GRP in 2D or higher dimensions [20].Please help to maintain this journal in open access!Encourage your library to subscribe or verify its subscription by contacting subscribers@edpsciences.org We are thankful to our subscribers and sponsors for making it possible to publish the journal in open access, free of charge for authors.More information and list of sponsors: https://www.edpsciences.org/en/maths-s2o-programme

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Algorithm Greedy Heuristic.Input: number of marks n Set d = [0, 0, . ..], mark = [0, 0, . ..],I = {0, 1, 2, 3, . ..},J = ∅, L = 0; Put a mark in position 0, set nmark = 1 and mark[0] = 1; while nmark < n do Define k = min i∈I−J {i} such that d[i] = 0 and set This journal is currently published in open access with no charge for authors under a Subscribe-to-Open model (S2O).Open access is the free, immediate, online availability of research articles combined with the rights to use these articles fully in the digital environment.S2O is one of the transformative models that aim to move subscription journals to open access.Every year, as long as the minimum amount of subscriptions necessary to sustain the publication of the journal is attained, the content for the year is published in open access.Ask your library to support open access by subscribing to this S2O journal.
and nmark = nmark − 1; end nmark = nmark + 1 end Output: upper bound L 0 and vector mark To solve larger instances of the GRP, compared to model (2.1), we propose below a continuous version of this model, with a simple but fundamental modification on the first constraint:

Table 2 .
Computational time in seconds and optimal values.