Results

**1 - 6**of**6**### Algorithmic and Probabilistic Aspects of the Bipartite Traveling Salesman Problem

, 2001

"... Contents Introduction 7 1 Preliminaries 11 1.1 It is not only salesmen who travel . . . . . . . . . . . . . . . . . . 11 1.2 The alternating TSP . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.3 Basic denitions and notations . . . . . . . . . . . . . . . . . . . . 12 2 Euclidean bipartite T ..."

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Contents Introduction 7 1 Preliminaries 11 1.1 It is not only salesmen who travel . . . . . . . . . . . . . . . . . . 11 1.2 The alternating TSP . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.3 Basic denitions and notations . . . . . . . . . . . . . . . . . . . . 12 2 Euclidean bipartite TSP 15 2.1 Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.2 The trouble with good TSP approximations . . . . . . . . . . . . 16 2.3 The matching method . . . . . . . . . . . . . . . . . . . . . . . . 17 2.4 The spanning tree strategy . . . . . . . . . . . . . . . . . . . . . . 20 2.5 Cycle covers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3 Computational results 33 3.1 Uniformly distributed points . . . . . . . . . . . . . . . . . . . . . 33 3.2 Two non-uniform point distributions . . . . . . . . . . . . . . . . 34 4 Random points in the unit square 37 4.1 An optimal algorithm .

### On the b-partite Random Asymmetric Traveling Salesman Problem and its Assignment Relaxation

"... We study the relationship between the value of optimal solutions to the random asymmetric b-partite traveling salesman problem and its assignment relaxation. In particular we prove that given a bn × bn weight matrix W = (w ij ) such that each finite entry has probability p n of being zero, ..."

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We study the relationship between the value of optimal solutions to the random asymmetric b-partite traveling salesman problem and its assignment relaxation. In particular we prove that given a bn &times; bn weight matrix W = (w ij ) such that each finite entry has probability p n of being zero, the optimal values bATSP (W ) and AP (W ) are equal (almost surely), whenever np n tends to infinity with n. On the other hand, if np n tends to some constant c then P[bATSP (W ) 6= AP (W )] > > 0, and for np n ! 0, P[bATSP (W ) 6= AP (W )] ! 1 (a.s.). This generalizes results of Frieze, Karp and Reed (1995) for the ordinary asymmetric TSP.

### Alan Frieze ∗

, 2007

"... An n-lift of a digraph K, is a digraph with vertex set V (K) × [n] and for each directed edge (i,j) ∈ E(K) there is a perfect matching between fibers {i} × [n] and {j} × [n], with edges directed from fiber i to fiber j. If these matchings are chosen independently and uniformly at random then we ..."

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An n-lift of a digraph K, is a digraph with vertex set V (K) × [n] and for each directed edge (i,j) ∈ E(K) there is a perfect matching between fibers {i} × [n] and {j} × [n], with edges directed from fiber i to fiber j. If these matchings are chosen independently and uniformly at random then we say that we have a random n-lift. We show that if h is sufficiently large then a random n-lift of the complete digraph �Kh is hamiltonian whp. 1

### One-in-Two-Matching Problem is NP-complete

, 2006

"... Abstract. 2-dimensional Matching Problem, which requires to find a matching of left- to rightvertices in a balanced 2n-vertex bipartite graph, is a well-known polynomial problem, while various variants, like the 3-dimensional analogoue (3DM, with triangles on a tripartite graph), or the Hamiltonian ..."

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Abstract. 2-dimensional Matching Problem, which requires to find a matching of left- to rightvertices in a balanced 2n-vertex bipartite graph, is a well-known polynomial problem, while various variants, like the 3-dimensional analogoue (3DM, with triangles on a tripartite graph), or the Hamiltonian Circuit Problem (HC, a restriction to “unicyclic ” matchings) are among the main examples of NP-hard problems, since the first Karp reduction series of 1972. The same holds for the weighted variants of these problems, the Linear Assignment Problem being polynomial, and the Numerical 3-Dimensional Matching and Travelling Salesman Problem being NP-complete. In this paper we show that a small modification of the 2-dimensional Matching and Assignment Problems in which for each i ≤ n/2 it is required that either π(2i − 1) = 2i − 1 or π(2i) = 2i, is a NP-complete problem. The proof is by linear reduction from SAT (or NAE-SAT), with the size n of the Matching Problem being four times the number of edges in the factor graph representation of the boolean problem. As a corollary, in combination with the simple linear reduction of Onein-Two Matching to 3-Dimensional Matching, we show that SAT can be linearly reduced to 3DM, while the original Karp reduction was only cubic.

### Hamilton cycles in random lifts of . . .

, 2007

"... An n-lift of a digraph K, is a digraph with vertex set V (K) × [n] and for each directed edge (i, j) ∈ E(K) there is a perfect matching between fibers {i} × [n] and {j} × [n], with edges directed from fiber i to fiber j. If these matchings are chosen independently and uniformly at random then we ..."

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An n-lift of a digraph K, is a digraph with vertex set V (K) × [n] and for each directed edge (i, j) ∈ E(K) there is a perfect matching between fibers {i} × [n] and {j} × [n], with edges directed from fiber i to fiber j. If these matchings are chosen independently and uniformly at random then we say that we have a random n-lift. We show that if h is sufficiently large then a random n-lift of the complete digraph ⃗ Kh is hamiltonian whp.

### On the b-partite Directed Random Traveling Salesman Problem and its Assignment Relaxation

"... We study the relationship between the value of optimal solutions to the b-partite directed random traveling salesman problem and its assignment relaxation. In particular we prove that given a bn bn weight matrix W = (w ij ) such that each nite entry has probability p n of being zero, the optimal ..."

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We study the relationship between the value of optimal solutions to the b-partite directed random traveling salesman problem and its assignment relaxation. In particular we prove that given a bn bn weight matrix W = (w ij ) such that each nite entry has probability p n of being zero, the optimal values b-TSP(W ) and AP(W ) are equal (almost surely), whenever np n tends to innity with n. On the other hand, if np n tends to some constant c then P[b-TSP(W ) 6= AP (W )] > > 0, and for np n ! 0, P[b-TSP(W ) 6= AP(W )] ! 1 (a.s.). This generalizes results of Frieze, Karp and Reed (1995) for the asymmetric TSP.