The Symmetric Travelling Salesman Problem (STSP) is to find a minimum weight Hamiltonian cycle in a weighted complete graph on n nodes. One direction which seems promising for finding improved solutions for the STSP is the study of a linear relaxation of this problem called the Subtour Elimination Problem (SEP). A well known conjecture in combinatorial optimization says that the integrality gap of the SEP is 4/3 in the metric case. Currently the best upper bound known for this integrality gap is 3/2. Finding the exact value for the integrality gap for the SEP is difficult even for small values of n due to the exponential size of the data involved. In this paper we describe how we were able to overcome such difficulties and obtain the exact integrality gap for all values of n up to 10. Our results give a verification of the 4/3 conjecture for small values of n, and also give rise to a new stronger form of the conjecture which is dependent on n.

Abstract-We consider parameter estimation in linear models when some of the parameters are known to be integers. Such problems arise, for example, in positioning using carrier phase measurements in the global positioning system (GPS), where the unknown integers enter the equations as the number of carrier signal cycles between the receiver and the satellites when the carrier signal is initially phase locked.Given a linear model, we address two problems: 1) the problem of estimating the parameters and 2) the problem of verifying the parameter estimates. We show that with additive Gaussian measurement noise• the maximum likelihood estimates of the parameters are given by solving an integer least-squares problem. Theoretically, this problem is very difficult computationally (NP-hard); • verifying the parameter estimates (computing the probability of estimating the integer parameters correctly) requires computing the integral of a Gaussian probability density function over the Voronoi cell of a lattice. This problem is also very difficult computationally. However, by using a polynomial-time algorithm due to Lenstra, Lenstra, and Lovász (LLL algorithm)• the integer least-squares problem associated with estimating the parameters can be solved efficiently in practice; • sharp upper and lower bounds can be found on the probability of correct integer parameter estimation. We conclude the paper with simulation results that are based on a synthetic GPS setup.

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Abstract. We study the Travelling Salesman Problem (TSP) on the metric completion of cubic and subcubic graphs, which is known to be NP-hard. The problem is of interest because of its relation to the famous 4/3 conjecture for metric TSP, which says that the integrality gap, i.e., the worst case ratio between the optimal values of the TSP and its linear programming relaxation, is 4/3. Using polyhedral techniques in an interesting way, we obtain a polynomial-time 4/3-approximation algorithm for this problem on cubic graphs, improving upon Christofides' 3/2-approximation, and upon the 3/2 − 5/389 ≈ 1.487-approximation ratio by Gamarnik, Lewenstein and Svirdenko for the case the graphs are also 3-edge connected. We also prove that, as an upper bound, the 4/3 conjecture is true for this problem on cubic graphs. For subcubic graphs we obtain a polynomial-time 7/5-approximation algorithm and a 7/5 bound on the integrality gap.

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