Some problems around travelling salesmen, dart boards, and Euro-coins

Transcription

1 The Algorithmics Column by Gerhard JWoeginger Department of Mathematics and Computer Science Eindhoven University of Technology P.O. Box 53, 5600 MB Eindhoven, The Netherlands Some problems around travelling salesmen, dart boards, and Euro-coins Vladimir G. Deĭneko Gerhard J. Woeginger Abstract In 957 Fred Supnick investigated and solved a special case of the Travelling Salesman Problem. Since then, Supnick s results have been rediscovered many times by other researchers. This article discusses Supnick s results and some of the rediscoveries. The Travelling Salesman Problem The Travelling Salesman Problem (TSP, for short) is probably the most prominent and the most studied problem in combinatorial optimization. An instance of the TSP consists of n cities, 2,...,n whose distances d i, j are summarized in an n n orsvd@wbs.warwick.ac.uk. Warwick Business School, The University of Warwick, Coventry CV4 7AL, United Kingdom. gwoegi@win.tue.nl. Department of Mathematics and Computer Science, TU Eindhoven, P.O. Box 53, 5600 MB Eindhoven, The Netherlands. Bulletin of the EATCS no 90, pp , October 2006 c European Association for Theoretical Computer Science

2 BEATCS no 90 THE EATCS COLUMNS distance matrix D = (d i, j ). The goal is to find a shortest closed tour through these cities. In other words, a travelling salesman starts from his home city φ(), then visits each of the other n cities φ(2),φ(3),...,φ(n) exactly once, in the end returns to his home city φ(), and he does all this with the smallest possible amount of gas. Mathematically speaking, we wish to find a permutation φ of, 2,...,n that minimizes the value of n d φ(i),φ(i+) + d φ(n),φ() () i= The TSP models tons of situations that arise in robotics, production, scheduling, engineering, and many other areas. For more specific information on the TSP and its applications, we refer the reader to the book by Lawler, Lenstra, Rinnooy Kan & Shmoys (985). The TSP in its general formulation is an NP-hard problem (Garey & Johnson, 979), and hence computationally intractable. In this article, we will concentrate on a special case of the TSP where the underlying distance matrix is a so-called Supnick matrix. 2 Supnick matrices The following inequalities (2) go back to the eighteenth century, to the work of the French mathematician and Naval minister Gaspard Monge (78). An n n matrix D = (d i, j ) is called Monge matrix, if it satisfies the so-called Monge inequalities d i, j + d r,s d i,s + d r, j (2) for all i, j, r, s with i < r n and j < s n. In words: In every 2 2 sub-matrix the sum of the two entries on the main diagonal is less or equal to the sum of the two entries on the other diagonal. Burkard, Klinz & Rudolf (996) survey the role of Monge structures in combinatorial optimization. A Supnick matrix D = (d i, j ) is a symmetric Monge matrix. That is, a Supnick matrix satisfies (2), and it satisfies d i, j = d j,i for all i and j. Here is a small catalogue of Supnick matrices: Sum matrices: Let α,...,α n be real numbers. Then the sum matrix D with d i, j = α i + α j is a Supnick matrix. In fact, a sum matrix satisfies all inequalities (2) even with equality. Convex-function matrices: Let f :IR IR be a function that is symmetric (hence: f (x) = f ( x) for all 44

3 The Bulletin of the EATCS x) and convex (hence: f (x + δ) f (x) f (y + δ) f (y) for all x y and all δ 0). Let β β 2 β n be real numbers. Then matrix D with d i, j = f (β i β j ) is a Supnick matrix: Symmetry of f implies symmetry of D. Fori, j, r, s with i < r n and j < s n, we set x := β i β s, y := β r β s, and δ = β s β j. This yields x y and δ 0. Plugging these values into the convexity condition yields (2). LL-UR block matrices: Let x < y n be integers, and consider the Lower-Left Upper-Right block matrix D with d i, j = ifi x and j y or if i y and j x, and with d i, j = 0 in all other cases. It is easily verified that this matrix D is a Supnick matrix. It has a rectangular block of -entries in the lower left corner (below the main diagonal), a symmetric block of -entries in the upper right corner (above the main diagonal), and it has 0-entries everywhere else. See Figure for an illustration. 0 0 Figure : A Lower-Left Upper-Right block matrix. Note that the inequalities stated in (2) are linear inequalities, and that also the symmetry condition is a linear condition. Consequently, if we multiply a Supnick matrix by a non-negative real number, or if we add up two Supnick matrices, then the resulting matrix will again be a Supnick matrix: The Supnick matrices form a cone. Rudolf & Woeginger (995) took a closer look at the structure of this cone and its extremal rays, and they came up with the following simple characterization of Supnick matrices. Theorem. A matrix is a Supnick matrix, if and only if it can be written as the sum of a sum matrix S and a non-negative linear combination of LL-UR block matrices. 45

4 BEATCS no 90 THE EATCS COLUMNS 3 Fred Supnick s theorem Now let us return to the travelling salesman problem. Fred Supnick (957) proved by a (somewhat involved) exchange argument that for the TSP with Supnick distance matrices, the optimal TSP tour is easy to find: The shortest tour is always given by the same permutation σ min. This permutation σ min first visits the odd cities in increasing order and then the even cities in decreasing order, and it constitutes a universally optimal solution for all instances of the Supnick TSP. Theorem 2. Let D = (d i, j ) be an n n Supnick matrix. The shortest TSP tour is given by the permutation σ min =, 3, 5, 7, 9,, 3,...4, 2, 0, 8, 6, 4, 2. (3) The longest TSP tour is given by the permutation σ max = n, 2, n 2, 4, n 4, 6,...,5, n 3, 3, n,. (4) (We write φ = φ(),φ(2),...,φ(n) to specify a permutation φ.) In the following paragraphs we will present a simple and quite straightforward argument for Supnick s result on the shortest tour. The argument is based on the additive characterization of Supnick matrices stated in Theorem. A similar argument can be used to prove Supnick s result on the longest tour. The TSP with a sum distance matrix D with d i, j = α i + α j is absolutely uninteresting: Since every city i contributes the value 2α i to the total tour length, every possible tour has the length 2 n i= α i. Every permutation φ minimizes (and simultaneously maximizes) the value of the expression in (). In particular, the Supnick permutation σ min yields a shortest tour for the TSP on sum matrices. The TSP on LL-UR block matrices is slightly more interesting. For technical reasons, we will now double every TSP tour and traverse it once in forward and once in backward direction. Since the distances are symmetric, this simply doubles the total tour length. Shortest solutions remain shortest, and non-shortest solutions remain non-shortest. Let us take a closer look at such a doubled tour corresponding to the permutation σ min : In the forward direction, the doubled tour runs from city to city 3, from city 3 to city 5, from 5 to 7 and so on. In the backward direction, it runs from city 2 to city 4, from 4 to 6, and so on. Hence, the doubled tour picks the entries d i,i+2 and d i+2,i for i =,...,n 2 together with the four entries d,2, d 2,, d n,n, d n,n out of the distance matrix, and it pays their total value. All the picked entries lie in the two diagonals above and in the two diagonals below the main diagonal of D. See Figure 2.0 for an illustration. Now let us argue that the doubled tour for σ min is the shortest doubled tour for any LL-UR block matrix D. We distinguish three cases that depend on the size and position of the two rectangular blocks of -entries. We recall that the lower left corner of the upper right block is the matrix element d x,y with x < y n. 46

5 The Bulletin of the EATCS 47 (0) () (2) (3) Figure 2: Illustrations for the proof of Supnick s result. Case : If y x 3, then the rectangular blocks in matrix D do not touch the doubled tour σ min ; see Figure 2.. Then the corresponding cost is 0, which clearly is minimum. Case 2: If y x =, then the rectangular blocks in D cover four of the entries picked by the doubled tour σ min, and the corresponding cost equals 4. See Figure 2.2 for an illustration. In this case the cities in G = {,...,x} are pairwise at distance 0, and the cities in G 2 = {x +,...,n} are pairwise at distance 0. The distance between any city in G and any city in G 2 equals. Any TSP tour must go at least once from G into G 2, and at least once back from G 2 into G. Hence, in this case any tour has cost at least 2 and any

6 BEATCS no 90 THE EATCS COLUMNS doubled tour has cost at least 4. Again, σ min yields a shortest solution. Case 3: The last case y x = 2 is illustrated in Figure 2.3. Now the cost of the doubled tour σ min equals 2. Every TSP tour must contain some move from a city with number x to a city with number y, or a move from a city y to a city x. Therefore any tour has cost at least, any doubled tour has cost at least 2, and also in this case σ min yields a shortest solution. Summarizing, we have shown that permutation σ min yields the shortest TSP tour for every distance matrix that is an LL-UR block matrix and for every distance matrix that is a sum matrix. Then σ min also yields the shortest TSP tour for any non-negative linear combination of such matrices, and by Theorem these combinations are exactly the Supnick matrices. The argument is complete. 4 Rediscoveries of Supnick s theorem During the Cold War, scientific results were often discovered on one side of the Iron Curtain and rediscovered independently on the other side. This also happened to Theorem 2. Supnick derived his result in 957, when he worked at the City University of New York. Supnick s result was rediscovered once by Rubinshtein (97) in Russia and once by Michalski (987) in Poland. Both rediscoveries state Supnick s result in its full generality and in the language of the TSP. Other researchers only rediscovered special cases of Theorem 2. For instance, Chao & Liang (992) derived the special case of Theorem 2 where the underlying matrix is a convex-function matrix. Another rediscovery is problem B-3 of the 57th William Lowell Putnam Mathematical Competition, which reads as follows: Given that {x, x 2,...,x n } = {, 2,...,n}, find the largest possible value of x x 2 + x 2 x x n x n + x n x as a function of n 2. This Putnam problem asks for the longest TSP tour in the distance matrix D = (d i, j ) with d i, j = ij. Matrix D itself is not a Supnick matrix, but matrix D is a Supnick matrix. Since the longest tour for D corresponds to the shortest tour for D, the problem is solved by permutation σ min. Theorem 2 then yields the answer 6 (2n3 + 3n 2 n + 8). Dart boards. Now let us turn to a number of rediscoveries of Supnick s result that are centered around the game of darts. The arrangement of the numbers, 2,...,20 on a modern dart board was devised in 896 by Brian Gamlin, a carpenter from Bury in the County of Lancashire. See the left half of Figure 3 for an illustration. Gamlin s arrangement reduces the element of chance and encourages accurate play, since large numbers (good scores) are always placed between 48

7 The Bulletin of the EATCS Figure 3: A classical dart board to the left. The most difficult dart board to the right. small numbers (bad scores). For instance, the large number 20 at the top of the dart board is placed between the small numbers and 5. If you are aiming for the segment 20, then a poor shot is penalized by a low score of or 5. Why did Gamlin select this particular arrangement? Is there some simple quality criterion that uniquely singles out the Gamlin arrangement from all possible arrangements? Unfortunately, no such simple quality criterion is known. Some Mathematicians are worried about this, and over the years they have proposed and analyzed a considerable number of reasonable quality criteria; see for instance Singmaster (980) and Lipscombe & Sangalli (2000). The most popular quality criterion is the so-called L p -criterion: Here the penalty incurred for missing a segment x and hitting the adjacent segment y instead equals x y p. The best (that is, most difficult) dart board with respect to the L p -criterion is a permutation of the numbers, 2,...,20 (or generally of the numbers, 2,...,n) that maximizes the total penalty of all segments. The reader will have little difficulty to recognize that the most difficult dart board for the L p -criterion corresponds to the longest TSP tour in the convex-function matrix with f (x) = x p and β i = i for i =,...,n. Theorem 2 yields that the best number arrangement is σ max. For n = 20 this arrangement is 20, 2, 8, 4, 6, 6, 4, 8, 2, 0,, 9, 3, 7, 5, 5, 7, 3, 9,, (5) and the corresponding most difficult dart board is depicted in the right half of Figure 3. Selkirk (976) was probably the first to discuss the L and the L 2 -criterion for dart boards. He correctly identifies the permutation σ max, and he states that for L 2 49

8 BEATCS no 90 THE EATCS COLUMNS the highest possible total penalty equals 3 (n3 4n+3) if n is odd and 3 (n3 4n+6) if n is even. Selkirk states a number of assertions all of which are correct, but none of which are proved. Eiselt & Laporte (99) formulate L and L 2 as a maximum cost TSP, and then compute an optimal solution by using a branch-and-bound code. They only consider the case n = 20, and their computer program indeed comes up with the arrangement in (5). Everson & Bassom show that permutation σ max maximizes the L -criterion for arbitrary values of n. Cohen & Tonkes (200) prove by an exchange argument that permutation σ max maximizes the total penalty under the L p -criterion for every integer p. Curtis (2004) reproves the results of Cohen & Tonkes (200), but by a different approach that is based on a certain greedy algorithm. Curtis also discusses so-called hoopla boards, and thereby rediscovers Supnick s result for convex-function matrices with f (x) = x p and arbitrary values β β 2 β n. Problem 0725 in the American Mathematical Monthly (Mihai & Woltermann, 200) reads as follows: Fix a positive integer n. Given a permutation φ of {, 2,...,n}, let F(φ) = n i=(φ(i) φ(i+)) 2, where φ(n+) = φ(). Find the extreme values of F(φ)asφ ranges over all permutations. Obviously, this problem asks for the easiest and for the most difficult dart board under the L 2 -criterion. The minimum of F(φ) is4n 6, and the maximum is 3 (n3 4n + 3) if n is odd and 3 (n3 4n + 6) if n is even (as also observed by Selkirk, 976). Euro-coins. The diameter of a -Euro coin is mm, and the diameter of a 2- Euro coin is mm. There are only two essentially different ways of arranging two 2-Euro coins and three -Euro coins in a ring, so that each coin is tangent to two others while all five coins are externally tangent to a disk inside the ring. See Figure 4 for an illustration. For which of the two arrangements is the diameter of the inner disk larger? This puzzle goes back to Joe Konhauser, and it is discussed in problem 43 of the book Which Way Did the Bicycle Go? by Konhauser, Velleman & Wagon (996). It turns out that the arrangement with adjacent 2- Euro coins has a slightly larger central disk. Duncan, Velleman & Wagon (996) discuss a generalization of this puzzle. Suppose we are given n 3 disks of radii r r 2 r n. We wish to place them in some order around a central disk so that each given disk touches the central disk and its two immediate neighbors. If the given disks are of widely different sizes (such as 00, 00, 00, 00, ), we allow a disk to overlap other given disks that are not immediate 50

9 The Bulletin of the EATCS neighbors. In what order should the given disks be arranged so as to maximize the radius of the central disk? Figure 4: There are two ways of arranging three -Euro coins and two 2-Euro coins around a central disk: 2-Euro coins together (left) or 2-Euro coins apart (right). In the right picture the central disk is slightly larger than in the left picture. Let us first consider a central disk with a fixed radius R. Look at a single tangent configuration made up of the central disk, and two disks of radius x and y. The three centers form a triangle with sides R + x, R + y, and x + y. Applying the law of cosines and simplifying gives that in this triangle, the angle at the center of the disk with radius R is θ R (x, y) = arccos ( 5 ) 2xy. (6) R 2 + Rx + Ry + xy It can be checked that the distances d i, j = θ R (r i, r j ) are symmetric and satisfy the inequalities (2). Therefore, the corresponding distance matrix D is a Supnick matrix, and Theorem 2 can be applied. Because of the minus-sign in the definition of the d i, j, the permutation σ min in this case yields the arrangement with largest overall angle θ(r). If the overall angle θ(r) is strictly less than 2π, then the radius R was chosen too large and must be decreased. If the overall angle θ(r) is strictly greater than 2π, then the radius R was chosen too small and must be increased. To summarize, the arrangement σ min maximizes and the arrangement σ max minimizes the radius of the central disk. This has been (re)discovered by Duncan, Velleman & Wagon (996). If we go back to the puzzle of the five Euro-coins with radii r = 23.25, r 2 = 23.25, r 3 = 23.25, r 4 = 25.75, r 5 = 25.75, we see that the maximizing permutation σ min =, 3, 5, 4, 2 indeed puts the two larger coins next to each other.

10 BEATCS no 90 THE EATCS COLUMNS References [] R.E. Burkard, B. Klinz, and R. Rudolf (996). Perspectives of Monge properties in optimization. Discrete Applied Mathematics 70, [2] C.C. Chao and W.Q. Liang (992). Arranging n distinct numbers on a line or a circle to reach extreme total variations European Journal of Combinatorics 3, [3] G.L. Cohen and E. Tonkes (200). Dartboard arrangements. Electronic Journal of Combinatorics 8(2), R4. [4] S.A. Curtis (2004). Darts and hoopla board design. Information Processing Letters 92, [5] J. Duncan, D. Velleman, and St. Wagon (996). Solution to Problem Crux Mathematicorum 22, [6] H.A. Eiselt and G. Laporte (99). A combinatorial optimization problem arising in dartboard design. The Journal of the Operational Research Society 42, 3 8. [7] P.J. Everson and A.P. Bassom (994/5). Optimal arrangements for a dartboard. Mathematical Spectrum 27, [8] M.R. Garey and D.S. Johnson (979). Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, San Francisco. [9] J.D.E. Konhauser, D. Velleman, and St. Wagon (996). Problem 43 in Which Way Did the Bicycle Go? Dolciani Mathematical Expositions. [0] E.L. Lawler, J.K. Lenstra, A.H.G. Rinnooy Kan, and D.B. Shmoys (eds.) (985). The travelling salesman problem. John Wiley, Chichester. [] T. Lipscombe and A. Sangalli (2000). The devil s dartboard. Crux Mathematicorum 26, [2] M. Michalski (987). On a class of polynomially solvable travelling salesman problems. Zastosowania Matematyki 9, [3] V. Mihai and M. Woltermann (200). Problem 0725: The smoothest and roughest permutations. American Mathematical Monthly 08, [4] G. Monge (78). Mémoire sur la théorie des déblais et des remblais. Histoire de l Académie Royale des Science (Année 78, avec les Mémoires de Mathématique et de Physique, pour la même Année), [5] M.I. Rubinshtein (97). On the symmetric travelling salesman problem. Automation and Remote Control 32, [6] R. Rudolf and G.J. Woeginger (995). The cone of Monge matrices: Extremal rays and applications. Mathematical Methods of Operations Research 42, [7] K. Selkirk (976). Re-designing the dartboard. Mathematical Gazette 60, [8] D. Singmaster (980). Arranging a dartboard. Bulletin of the Institute of Mathematics and its Applications 6, [9] F. Supnick (957). Extreme Hamiltonian lines. Annals of Mathematics 66,