Small Traveling Salesman Problems

Abstract

Abstract. This paper illustrates fundamental principles about small traveling salesman problems (TSPs) which are a current application for quantum annealing computers. The 2048 qubit, quantum annealing computer manufactured by D-Wave Systems is estimated to be able to solve all TSPs on 8 cities, which advances a recent 4-city result on a quantum simulator. Additionally, the D-Wave quantum computer is expected to find all optimal tours for each TSP. To prepare for this, we show the expected quantum output for 5,000 randomly generated TSPs on 6, 8 and 10 cities. The examples in the TSPLIB have 14 or more cities. These are too large for the current quantum annealing processors. We have included an annotated bibliography about solving the TSP on a quantum computer. Keywords:Traveling salesman, optimal tour, combinatorial analysis, discrete optimization, quantum annealing AMS Classification Codes: 68R05, 81P68, 90C27 IntroductionRecently first steps have been taken to solve the traveling salesman problem (TSP) using the computational advantages promised by quantum algorithms [13]. As with any emerging effort, the initial steps have been limited and focused on a small number of cities that fit on the hardware. The major contributions of this paper are for 6, 8 and 10 city TSPs. We present empirical evidence based on 15,000 random samples, 5,000 for each number of cities. The majority of evidence indicates that the number of optimal tours and the length of an optimal tour increase together. All of the data shows that as the number of cities increases, the number of optimal tours increases. In all cases, the distances between cities are random integers in the interval [1,21].Outside the quantum world, the traveling salesman problem continues to be a hot topic that attracts wide interest. The problem is easy to explain and has several diverse applications [9,18], as well as theoretical studies [2,20]. There are many fascinating ways to solve it, usually approximately. Given a set of cities and the directed distances between each pair of cities, the TSP asks for a shortest route that visits each city exactly once and returns to the starting city. There are excellent, non-quantum TSP surveys [5,7,10].Quantum computing, particularly quantum annealing [3], is a new paradigm for discrete optimization that could use TSP benchmarks for the hardware. Quantum computing opens the possibility of large speedup with high probability of optimal answers, but requires new techniques for solving the TSP [11,19].The paper is organized as follows: Section 2 has references for other reports that use quantum computing algorithms for solving the TSP. Section 3 contains the results from a study of the expected output for TSPs from current quantum annealing computers. The output is envisioned to be all optimal tours, since these are obtained on a quantum annealing simulator. We have classified this output for 6, 8 and 10-city TSPs which is the estimated size that can be solved currently. The last section points to future quantum work for the TS...

A QUBO Model for the Traveling Salesman Problem with Time Windows

“…Such works point out the fact that quantum techniques have already been proven to be useful when it comes to particular TSP variants, which motivated the endeavour of this work. Another work on quantum annealing and TSP was presented by Warren [37]. Warren studied small-scale instances of traveling salesman problems, showcasing how a D-Wave machine using quantum annealing would operate to solve these instance.…”

“…The authors introduced a different quantum annealing scheme based on path-integral Monte Carlo processes in order to address the symmetric version of the Traveling Salesman Problem (sTSP). In [36,37], the D-Wave platform was used as a test bed for evaluating and comparing the efficiency of quantum annealing against classical methods in solving the standard TSP. In [38,39], the well-known variation of the TSP, the (Capacitated) Vehicle Routing Problem is studied using, again, the D-Wave computer.…”

“…The existence of an Ising or QUBO formulation for a problem is the essential precondition for its solution on the current generation of D-Wave computers. For the vanilla TSP, there exists such a formulation, as presented in an elegant and comprehensive manner in [33], which has enabled the actual solution of TSP instances on the D-Wave platform (see [36][37][38]). In contrast, prior to this work, the TSPTW had not been cast in the QUBO framework.…”

“…We can now substitute Equations (31)- (36) into Equation (25) to finally arrive at the expanded form of the constraint, given by the following equation. W 1 = x 1 0,1 c 2 0,1 + x 1 0,2 c 2 0,2 + x 1 0,3 c 2 0,3 + x 1 0,4 c 2 0,4 + 2x 1 0,1 c 0,1 x 1 0,2 c 0,2 + 2x 1 0,1 c 0,1 x 1 0,3 c 0,3 + 2x 1 0,1 c 0,1 x 1 0,4 c 0,4 + 2x 1 0,2 c 0,2 x 1 0,3 c 0,3 + 2x 1 0,2 c 0,2 x 1 0,4 c 0,4 + 2x 1 0,3 c 0,3 x 1 0,4 c 0,4 + t 1,1 + 4t 2,1 + 4t 1,1 t 2,1 + x 1 0,1 l 2 1 + x 1 0,2 l 2 2 + x 1 0,3 l 2 3 + x 1 0,4 l 2 4 + 2x 1 0,1 l 1 x 1 0,2 l 2 + 2x 1 0,1 l 1 x 1 0,3 l 3 + 2x 1 0,1 l 1 x 1 0,4 l 4 + 2x 1 0,2 l 2 x 1 0,3 l 3 + 2x 1 0,2 l 2 x 1 0,4 l 4 + 2x 1 0,3 l 3 x 1 0,4 l 4 − 2x 1 0,1 c 0,1 t 1,1 − 2x 1 0,1 c 0,1 t 2,1 − 2x 1 0,2 c 0,2 t 1,1 − 2x 1 0,2 c 0,2 t 2,1 − 2x 1 0,3 c 0,3 t 1,1 − 2x 1 0,3 c 0,3 t 2,1 − 2x 1 0,4 c 0,4 t 1,1 − 2x 1 0,4 c 0,4 t 2,1 − 2x 1 0,1 c 0,1 l 1 − 2x 1 0,2 c 0,2 l 2 − 2x 1 0,3 c 0,3 l 3 − 2x 1 0,4 c 0,4 l 4 (37) − 2x 1 0,1 c 0,1 x 1 0,2 l 2 − 2x 1 0,1 c 0,1 x 1 0,3 l 3 − 2x 1 0,1 c 0,1 x 1 0,4 l 4 − 2x 1 0,2 c 0,2 x 1 0,1 l 1 − 2x 1 0,2 c 0,2 x 1 0,3 l 3 − 2x 1 0,2 c 0,2 x 1 0,4 l 4 − 2x 1 0,3 c 0,3 x 1 0,1 l 1 − 2x 1 0,3 c 0,3 x 1 0,2 l 2 − 2x 1 0,3 c 0,3 x 1 0,4 l 4 − 2x 1 0,4 c 0,4 x 1 0,1 l 1 − 2x 1 0,4 c 0,4 x 1 0,2 l 2 − 2x 1 0,4 c 0,4 x 1 0,3 l 3 − 2t 1,1 x 1 0,1 l 1 − 2t 1,1 x 1 0,2 l 2 − 2t 1,1 x 1 0,3 l 3 − 2t 1,1 x 1 0,4 l 4 − 4t 2,1 x 1 0,1 l 1 − 4t 2,1 x 1 0,2 l 2 − 4t 2,1 x 1 0,3 l 3 − 4t 2,1 x 1 0,4 l 4…”

Solving the traveling salesman problem on a quantum annealer

“…(B) What percent of 100 different TSPs with a fixed number of cities and random distances are solved optimally? See [12]. (C) Is the software designed for symmetric or asymmetric TSPs?…”