A. I. Barvinok. Two algorithmic results for the traveling salesman problem. Math. Oper. Res., 21:65--84, 1996.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....: i ng is connected and forms a cycle. In general, both the permanent and the hamiltonian are hard to compute and the best algorithms known so far are exponential in n, BCS97,Bur00] This applies also for the computational model due to Blum, Shub and Smale (BSS model) cf. BCSS98] Barvinok in [Bar96] has shown that if the (linear) rank of the matrix is bounded by r both the permanent and the hamiltonian can be computed in polynomial (not linear) time. Hence these problems are parametrically tractable in the sense of [DF99] Linear rank and treewidth are independent notions: The (n Theta n) ....

A.I. Barvinok. Two algorithmic results for the traveling salesman problem. Mathematics of Operations Research, 21:65--84, 1996.

....forms a cycle. In general, both the permanent and the hamiltonian are ]P hard to compute and the best algorithms known so far are exponential in n, BCS97,B ur99] This applies also for the computational model due to Blum, Shub and Smale (BSS model) cf. BCSS98] On the other hand, Barvinok in [Bar96] has shown that if the (linear) rank of the matrix is bounded by r, both the permanent and the hamiltonian can be computed in polynomial (not linear) time. Hence these problems are parametrically tractable in the sense of [DF99] Linear rank and tree width are independent notions: The (n n) matrix ....

A.I. Barvinok. Two algorithmic results for the traveling salesman problem. Mathematics of Operations Research, 21:65-84, 1996.

....cf. 66] but we shall not pursue this further. 1.2 Linear rank of a matrix Recall that the linear rank of M over a field or ring K is the maximum number of row (column) vectors of M which are linearly independent over K. It is interesting to compare our result with recent results by Barvinok, [6]. Theorem 2 (Barvinok 1996) There are real valued functions g per (r) and g ham (r) such that for (n Theta n) matrices M over Zof linear rank r (1) the permanent of M can be computed using g per (r) Delta O(n b ) arithmetic operations for some constant b = O(r) and (2) the hamiltonian of M ....

A.I. Barvinok. Two algorithmic results for the traveling salesman problem. Mathematics of Operations Research, 21:65--84, 1996.

....scheme for the traveling salesman problem on planar graphs with unit edge lengths; this result leads them to speculate that the Euclidean TSP has such an approximation scheme as well. Finally, improvements due to constraints on distances appear throughout this chapter. In a recent paper, Barvinok [Bar94] showed that the distance matrix implied by n points and 298 CHAPTER 8 GEOMETRIC PROBLEMS a polyhedral metric has low combinatorial rank, and exploited this fact to approximate the longest Euclidean traveling salesman tour arbitrarily well. The same constraints that aid in approximation, ....

....then compute the shortest path in the visibility graph of these points. Sophisticated techniques are required to keep the number of points polynomially bounded, and to analyze the necessary bit complexity. 8.7. 8 LONGEST SUBGRAPH PROBLEMS We have already mentioned the recent results by Barvinok [Bar94] on the Euclidean longest traveling salesman problem. Alon, Rajagopalan, and Suri [ARS93] consider several other longest subgraphproblems, with the additional constraint that the subgraph may not include any edge crossings. Shortest subgraphs matchings, trees, or tours satisfy this ....

A.I. Barvinok. Two algorithmic results for the traveling salesman problem. Manuscript, 1994.

....of this question arises from the fact that there are no polynomial bounds known on the accuracy that is necessary for comparing a sum of square roots to a given integer. See the book [12] for a discussion. The complexity of the Maximum TSP for geometric distances has been less clear. Barvinok [5] showed that there is a PTAS for the Maximum TSP under all metrics in IR d , for any fixed d. Very surprisingly, Barvinok, Johnson, Woeginger, and Woodroofe [6] showed that under polyhedral norms with a fixed number k of facets on the unit ball, the Maximum TSP is indeed polynomial. Making ....

A. I. Barvinok. Two algorithmic results for the traveling salesman problem. Mathematics of Operations Research, 21 (1996), 65--84.

....[95] but apparently the complexity of the degree four problem remains open. Open Problem 3. Is it NP hard to find the minimum weight degree four spanning tree of a planar point set It is also natural to consider maximization versions of these bounded degree spanning tree problems. Barvinok [14] shows that the adjacency matrix of a complete geometric graph (using a polyhedral approximation to the Euclidean distance function) has low combinatorial rank and uses this to approximate the maximum traveling salesman problem within any factor (1 #) in polynomial time. Little seems to be ....

A. I. Barvinok. Two algorithmic results for the traveling salesman problem. Manuscript, 1994.

....optimal tour by opt. The problem is Max SNP hard [3] and therefore cannot have a polynomial time approximation scheme unless P=NP. Several polynomial algorithms with a constant performance guarantee are known for it [7, 8, 9, 10] a polynomial approximation scheme is known for a geometric version [2], while polynomially solvable cases are described in [3, 5, 6] Fisher, Nemhauser and Wolsey [7] showed that the greedy (see also [9] the best neighbor, and the 2 interchange algorithms produce tours whose weights are at least 0.5opt. The 2 matching algorithm of Fisher, Nemhauser and Wolsey [7] ....

A. I. Barvinok, "Two algorithmic results for the traveling salesman problem," Mathematics of Operations Research 21 (1996),65-84.

....enumerate the solutions but we count them 1.2 Linear rank of a matrix Recall that the linear rank of M over a field or ring K is the maximal number of row (column) vectors of M which are linearly independent over K. It is interesting to compare our result with recent results due to Barvinok, [Bar96]. Theorem 2 ( Barvinok 1996) There are functions g per (r) and g ham (r) such that for (n Theta n) matrices M over Zof linear rank r (i) the permanent of M can be computed using g per (r) Delta O(n b ) arithmetic operations for some constant b = O(r) and (ii) the Hamiltonian of M can be ....

A.I. Barvinok. Two algorithmic results for the traveling salesman problem. Mathematics of Operations Research, 21:65--84, 1996.

....R. Bhatia has recently shown that their algorithm gives a bound of 19=27 instead of the claimed 5=7. However, neither of these algorithms works for our maximum scatter problem. Also, there exists a polynomial time approximation scheme for geometric (e.g. Euclidean) versions of the MAX TSP [4]. The non crossing version of geometric MAX TSP is considered in [2] Very recently, since the original appearance of our own paper [3] Barvinok et al. 5] show that the MAX TSP can be solved exactly in polynomial time, for points in fixed dimension, under a (constant complexity) polyhedral ....

A. I. Barvinok. Two algorithmic results for the traveling salesman problem. Math. Oper. Res., 21:65--84, 1996.

....that achieves a 2=3 approximation factor, and a recent algorithm of [12] that achieves a 5=7 approximation factor. However, neither of these algorithms works for our maximum scatter problem. Also, there exists a polynomial time approximation scheme for the Euclidean version of the MAX TSP [3]. Nevertheless, as with our maximum scatter TSP for points in the plane (and Euclidean distances) the complexity of the Euclidean version of the MAX TSP is open [2] The most directly related work on the maximum scatter TSP has been done by Penavic [15] who studied the optimal firing sequences ....

A. I. Barvinok. Two algorithmic results for the traveling salesman problem. Math. Oper. Res., 21:65--84, 1996.

....of this question arises from the fact that there are no polynomial bounds known on the accuracy that is necessary for comparing a sum of square roots to a given integer. See the book [7] for a discussion. The complexity of the Maximum TSP for geometric distances has been less clear. Barvinok [4] showed that there is a PTAS for the Maximum TSP under most geometric norms in IR d , for any fixed d. Very surprisingly, Barvinok, Johnson, Woeginger, and Woodroofe [5] showed that under polyhedral norms with a fixed number k of facets on the unit ball, the Maximum TSP is indeed polynomial. ....

A. I.Barvinok. Two algorithmic results for the traveling salesman problem. Mathematics of Operations Research, 21 (1996), 65--84.

....to the best of our knowledge, no non trivial class of matrices for which permanents are computable in polynomial time is known from the literature. Most recently, several new papers appeared which discuss the complexity of computing the permanent for various subclasses of matrices. Barvinok [Bar96] shows that for matrices of fixed rank the problem of computing the permanent is polynomial, but exponential in the rank. A.P. Il ichev, G.P. Kogan and V.N. Shevchenko [IKS97] showed polynomial algorithms for computing the permanent of certain matrices with symmetry conditions for rows. W. McCuaig ....

A.I. Barvinok. Two algorithmic results for the traveling salesman problem. Mathematics of Operations Research, 21:65--84, 1996.

....enumerate the solutions but we count them 1.2 Linear rank of a matrix Recall that the linear rank of M over a field or ring K is the maximal number of row (column) vectors of M which are linearly independent over K. It is interesting to compare our result with recent results due to Barvinok, [Bar96]. Theorem2 (Barvinok 1996) There are functions g per (r) and g ham (r) such that for (n Theta n) matrices M over Zof linear rank r (i) the permanent of M can be computed using g per (r) Delta O(n b ) arithmetic operations for some constant b = O(r) and (ii) the Hamiltonian of M can be ....

A.I. Barvinok. Two algorithmic results for the traveling salesman problem. Mathematics of Operations Research, 21:65--84, 1996.

....algorithm that achieves a 2=3 approximation factor, and a recent algorithm of [6] that achieves a 5=7 approximation factor. However, neither of these algorithms works for our maximum scatter problem. Also, there exists a polynomial time approximation scheme for the Euclidean version of the MAX TSP [2]. Nevertheless, as with our maximum scatter TSP for points in the plane (and Euclidean distances) the complexity of the Euclidean version of the MAX TSP is open [1] The most directly related work on the maximum scatter TSP has been done by Penavic, who studied the optimal firing sequences for a ....

A. I. Barvinok. Two algorithmic results for the traveling salesman problem. Math. Oper. Res., 21:65--84, 1996.

....et al. use a maximum weight 2 matching in place of the previous algorithm s cycle cover. Since a 2 matching is just a cycle cover containing only cycles of size three or more, the same analysis yields a 2 3 approximation for the undirected case. We are also aware of a recent result by Barvinok [2] for graphs whose edge weights are distances between points in a Euclidean metric space. He describes an algorithm that, for any fixed #, produces an estimate # est for the length of a longest tour; this estimate is guaranteed to satisfy (1 #)# opt # # est # (1 #)# opt , where # opt is the ....

A. I. Barvinok. Two algorithmic results for the traveling salesman problem, April 1994. Unpublished manuscript.

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A. Barvinok. Two algorithmic results for the traveling salesman problem. Mathematics of Operations Research, 21:65--84, 1996.

....of polynomial time algorithms A k , 1 k 1, where A k is guaranteed to nd a tour whose length is within a ratio of 1 (1=k) of optimal. The situation for geometric versions of the Maximum TSP has been less clear than for its minimum counterpart. Serdyukov [26] and, independently, Barvinok [6], have shown that once again polynomial time approximation schemes exist for all xed dimensions d and all L p or polyhedral norms (and in a sense for any xed norm; see [6] Until now, however, the complexity of the optimization problems themselves when d is xed has remained open: For no xed ....

....of the Maximum TSP has been less clear than for its minimum counterpart. Serdyukov [26] and, independently, Barvinok [6] have shown that once again polynomial time approximation schemes exist for all xed dimensions d and all L p or polyhedral norms (and in a sense for any xed norm; see [6]) Until now, however, the complexity of the optimization problems themselves when d is xed has remained open: For no xed dimension d and L p or polyhedral norm was the problem of determining the maximum tour length known either to be NP hard or to be polynomial time solvable. In this paper, we ....

Barvinok, A.I., \Two algorithmic results for the traveling salesman problem," Math. Op. Res., 21 (1996), 65-84.

....of a 0 1 matrix is small (bounded by a polynomial in the size n of the matrix) it can be computed in polynomial time, see Section 7. 3 of [Minc 78] and [Grigoriev and Karpinski 87] Finally, the permanent of matrices (real or complex) of a small (fixed) rank is computable in polynomial time [Barvinok 96] Since the exact computation is difficult, the next goal is to find a very good approximation algorithm. A fully polynomial time (randomized) approximation scheme is a (probabilistic) algorithm that for any given ffl 0 approximates the desired quantity within relative error ffl in time ....

A. Barvinok, Two algorithmic results for the traveling salesman problem, Math. Oper. Res. 21 (1996), 65--84.

....approximation scheme (PTAS) exists, i.e. a sequence of polynomial time algorithms A k , 1 k 1, where A k is guaranteed to find a tour whose length is within a ratio of 1 (1=k) of optimal. The situation for geometric versions of the Maximum TSP is less completely resolved. Barvinok [3] has shown that once again polynomial time approximation schemes exist for all fixed dimensions d and all L p or polyhedral norms (and in a sense for any fixed norm; see [3] Until now, however, the complexity of the optimization problems themselves when d is fixed has remained open. For no ....

....of 1 (1=k) of optimal. The situation for geometric versions of the Maximum TSP is less completely resolved. Barvinok [3] has shown that once again polynomial time approximation schemes exist for all fixed dimensions d and all L p or polyhedral norms (and in a sense for any fixed norm; see [3]) Until now, however, the complexity of the optimization problems themselves when d is fixed has remained open. For no fixed dimension d and L p or polyhedral norm was the problem of determining the maximum tour length known either to be NP hard or to be polynomial time solvable. In this paper, ....

[Article contains additional citation context not shown here]

A.I. Barvinok, "Two algorithmic results for the traveling salesman problem," Mathematics of Operations Research 21 (1996), 65--84.

....u j ) is an edge of G if and only if a ij = 1. To compute the permanent of a given 0 1 matrix is a #P complete problem and even to estimate per A seems to be difficult. Polynomial time algorithms for computing per A are known when A has some special structure, for example, when A has a small rank [5], or when A is a 0 1 matrix and per A is small (see [14] and Section 7.3 of [25] Since the exact computation is difficult, a natural question is how well one can approximate the permanent in polynomial time. In particular, is it true that for any ffl 0 there is a polynomial time (possibly ....

A.I. Barvinok, Two algorithmic results for the Traveling Salesman Problem, Mathematics of Operations Research, 21(1996), 65--84.

....of G if and only if a ij = 1. To compute the permanent of a given 0 1 matrix is a #P complete problem and even to estimate per A seems to be difficult. Polynomial time algorithms for computing per A are known when A has some special structure, for example, when A is sparse [11] or has small rank [4]. Polynomial time approximation schemes are known for dense 0 1 matrices [12] for almost all 0 1 matrices (see [12] 21] and [9] and for some special 0 1 matrices, such as corresponding to lattice graphs, see [13] for a survey on approximation algorithms) However, not much is known on how ....

A.I. Barvinok, Two algorithmic results for the Traveling Salesman Problem, Mathematics of Operations Research, 21(1996), 65-84.

....approximation scheme (PTAS) exists, i.e. a sequence of polynomial time algorithms A k , 1 k 1, where A k is guaranteed to find a tour whose length is within a ratio of 1 (1=k) of optimal. The situation for geometric versions of the Maximum TSP is less completely resolved. Barvinok [3] has shown that once again polynomial time approximation schemes exist for all fixed dimensions d and all L p or polyhedral norms (and in The Maximum TSP 3 a sense for any fixed norm; see [3] Until now, however, the complexity of the optimization problems themselves when d is fixed has ....

....optimal. The situation for geometric versions of the Maximum TSP is less completely resolved. Barvinok [3] has shown that once again polynomial time approximation schemes exist for all fixed dimensions d and all L p or polyhedral norms (and in The Maximum TSP 3 a sense for any fixed norm; see [3]) Until now, however, the complexity of the optimization problems themselves when d is fixed has remained open: For no fixed dimension d and L p or polyhedral norm was the problem of determining the maximum tour length known either to be NP hard or to be polynomial time solvable. In this paper, ....

[Article contains additional citation context not shown here]

Barvinok, A.I., "Two algorithmic results for the traveling salesman problem," Math. of Oper. Res. 21 (1996), 65--84.

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A. I. Barvinok. Two algorithmic results for the traveling salesman problem. Math. Oper. Res., 21:65--84, 1996.

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A.I. Barvinok. Two algorithmic results for the traveling salesman problem. Mathematics of Operations Research, 21:65--84, 1996.

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A.I. Barvinok. Two algorithmic results for the traveling salesman problem. Mathematics of Operations Research, 21:65--84, 1996.

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