1 Traveling salesman problem in unweighted graphs
The traveling salesman problem (TSP
) is one of the most studied topics in both computer science and combinatorial optimization. Given a set of points and the distance between every pair of them, the task is to find the shortest Hamiltonian circuit.Graphic TSP is a special case of TSP: we are given an unweighted graph and want to find a shortest closed walk that contains all vertices of the graph. Such walk is called a traveling salesman tour or a TSP tour. We will denote its length by .
Recently, significant progress has been achieved in approximating graphic TSP. A series of incremental improvements culminated in a -approximation algorithm of Sebö and Vygen , providing the best ratio for general graphs known at present. We refer the reader to the survey of Vygen  for more information on approximation algorithms for TSP.
Together with general graphic TSP, several special instances of graphic TSP have been intensively studied. In particular, we are interested in cubic TSP and subcubic TSP, where the graph in question is cubic and subcubic, respectively. (A cubic graph is a graph with all vertices of degree three and a subcubic graph is a graph with all vertices of degree at most three.) Note that every bridge (i.e., an edge whose removal increases the number of components of a graph) is used exactly twice in any TSP tour: it must be used at least twice and if it was used more, it would be possible to construct a shorter TSP tour. After removing a bridge from a graph and finding TSP tours and of the resulting two components, we can create a TSP tour of by starting with , then using , continuing with , and closing the tour with . Thus to solve the problem of finding a TSP tour in cubic graphs with bridges, it suffices to solve the problem of finding a TSP tour in bridgeless subcubic graphs. Since a graph must be connected to have a TSP tour, we are interested in -connected subcubic graphs.
All present approximation algorithms for TSP on -connected cubic and subcubic graphs only compare the length of the found tour with the number of vertices, that is, they basically do not care about the optimal TSP tour and just aim to find a TSP tour no longer than a constant multiple of the order of the given graph. We find the latter problem interesting in its own right.
Mömke and Svensson  proved that every -connected subcubic graph has a TSP tour of length at most . This bound is tight for both -connected subcubic graphs and -connected cubic multigraphs. To improve the bounds further we have to examine a smaller class of graphs. Simple -connected cubic graphs seem to be the least restrictive class of graphs where improvement is possible.
Boyd et al.  proved that a simple -connected cubic graph on vertices has a TSP tour of length at most (assuming ). The result was subsequently improved by Correa, Larré, and Soto  to , by van Zuylen  to , by Candráková and Lukoťka  to , and very recently by Dvořák, Kráľ and Mohar  to .
For the class of subcubic graphs, graphs with no short TSP tours are easy to find. Indeed, it is sufficient to take two vertices and connect them by three equally long paths; one of the paths has to be used twice in any TSP tour. This subcubic graph can be turned into a cubic graph by replacing vertices of degree by digons (cycles of length ); this replacement does not change the ratio of because whenever a TSP tour passes through a digon, it has to pass through all its vertices. This would fail, however, if we try to replace the vertices of degree with larger structures in order to avoid parallel edges. Thus the situation is much less clear for the class of simple -connected cubic graphs. Boyd et al.  found an infinite family of -connected cubic graphs with no TSP tour shorter than . We improve this lower bound by constructing, for any positive , a -connected cubic graph such that (see Theorem 1). We believe that the value is the best possible.
Every simple -connected cubic graph with vertices has a traveling salesman tour of length at most .
A similar construction and conjecture were recently independently proposed by Dvořák, Kráľ and Mohar . For cyclically -connected cubic graphs, Conjecture 1 is implied by the dominating cycle conjecture of Fleischner : a dominating cycle in a cubic graph must contain at least of the vertices of and every other vertex has a neighbour on , thus we can construct a walk with length at most visiting all the vertices.
TSP was also studied for cubic bipartite graphs. The best present result (proved by van Zuylen in ) guarantees that a connected bipartite cubic graph different from has a TSP tour of length at most . We employ our method from Theorem 1 to show that for any positive , there exists a -connected bipartite cubic graph such that (see Theorem 2).
Considering the rapid recent development in research on TSP for simple -connected cubic graphs, it is quite possible that Conjecture 1 will be soon proved. After that, -edge-connected graphs constitute a natural subclass of cubic graphs to study if we want to improve the bound on TSP length below . In Theorem 3 we prove the existence of a family of -edge-connected graphs that have TSP tours of lengths asymptotically approaching .
2 Construction of graphs with no short tours
Consider a TSP tour in a simple cubic graph . No edge in is used three or more times, otherwise could be shortened. Thus the edges in are used either once or twice. The set of all the edges of that are used exactly once in induces a subgraph which is even (that is, all its vertices have even degree). Actually, since is a cubic graph, all vertices of have degree , and so is a collection of disjoint circuits. The extension of by all the isolated vertices of yields an even factor of , that is, a subgraph which contains all vertices of and has even degrees of all vertices. Since is cubic, we can view its even factors as collections of disjoint circuits and isolated vertices.
Assume that the even factor corresponding to the tour contains circuits and isolated vertices. We can easily express the length of in terms of , , and the order of , denoted by . First, the edges of in (we contract all the circuits of ) constitute a spanning tree of , otherwise could be shortened. Each edge of this spanning tree is used twice in , thus they contribute to . The edges in circuits of contribute . Altogether, . Our aim is to find graphs where the excess is large for every even factor. For future reference, we denote by the excess of an even factor in a graph .
Our construction is based on suitable construction blocks. Consider a gadget obtained from a -edge-connected cubic graph by cutting an edge into a pair of dangling edges. Such structures are called cubic -poles (see [8, 5]). The structure obtained from a -edge-connected cubic graph by removing a vertex and replacing the three edges incident to by dangling edges is a -pole. The notion of excess can be naturally extended to -poles (we confine ourselves to in the rest of this article). Even factors of an -pole obtained from a graph are simply even factors of with the exception that whenever the original even factor contains an edge of that is not present in , we replace by corresponding dangling edges. In particular, if is a -pole, we replace by adding both dangling edges of ; if is a -pole, we replace by the dangling edge sharing an endvertex with . In both cases, two dangling edges are added and one circuit of is replaced by a path starting and ending with a dangling edge. This path contributes nothing to .
Every -pole can be assigned a triple , where is the minimum of over all even factors of containing no dangling edges, is the minimum of over all even factors of containing two dangling edges, and is the number of vertices in . (The number of dangling edges belonging to an even factor is always even.) For instance, the -pole arising from is assigned the triple .
Given a -pole , we construct a -pole as indicated in Fig. 1: one dangling edge of each copy of is attached to a new vertex , the other is attached to a new vertex ; in addition, contains a path , one dangling edge incident to , and one dangling edge incident to . The relationship between and is captured in the following lemma.
Let be a -pole such that . Then
Let be an even factor in . First, we determine . If contains a path containing the dangling edges, there are two cases: either passes through , , one copy of , , and , or it passes just through and . In the first case, the copy of with nonempty intersection with contributes at least to , and the other copy contributes at least , so is . In the second case, either and are isolated vertices in , so contribute , and each copy of contributes at least to , or there is a circuit passing through , and both copies of , which contributes at least to . If we take the minimum, we get .
Next, we consider that does not contain the dangling edges of and determine . There are two cases: either and belong to a circuit (which must include , , and a copy of ), or and are isolated vertices in . In the first case, the contribution to is from the circuit and at least from the copies of . In the second case, we have contribution of from and and at least from the rest of (the situation was analysed in the previous paragraph). Altogether, .
Finally, is obviously . ∎
We are ready to state and prove the main theorem.
For any , there exists a simple planar cubic graph with no traveling salesman tour of length shorter than .
We construct a sequence of -poles with increasing ratio of excess to the number of vertices.
Let be the -pole that arises from by cutting an edge. We have . For every integer , let . Note that if is planar, then is planar, thus all the constructed -poles are planar (since is planar). According to Lemma 1, the sequence of triples satisfies
Solving the last two recurrence relations using the standard machinery of generating functions (see ) yields and for every .
By inserting into an edge of a planar cubic graph isomorphic to , we obtain a simple planar cubic graph with the ratio of the length of a traveling salesman tour to the number of vertices arbitrarily close to . Indeed, consider an even factor of . If contains the edges arising from , its excess is at least (we add for the circuit passing through the vertices of ). If does not contain those edges, it is composed of an even factor of with excess at least and an even factor of with excess at least . Altogether, the length of a TSP tour is at least , and thus the ratio is equal to , which belongs to for a sufficiently large . ∎
For any , there exists a simple bipartite cubic graph with no traveling salesman tour of length shorter than .
We construct a sequence of -poles with increasing ratio of excess to the number of vertices. All the constructed 2-poles are truly bipartite
in the following sense: they do not contain any odd circuit and every path both starting and ending with a dangling edge contains an even number of vertices.
Let be the truly bipartite -pole that arises from by cutting an edge. We have . For every integer , let . Note that if is truly bipartite, then is truly bipartite, thus all the constructed -poles are truly bipartite. According to Lemma 1, the sequence of triples satisfies
thus and for every .
By inserting into an edge of the bipartite cubic graph , we obtain a simple bipartite cubic graph with the ratio of the length of a TSP tour to the number of vertices being , which approaches for large values of . ∎
Next, we describe an analogous construction which yields -connected simple cubic graphs with no short tours. A -pole is symmetric if for each permutation of the three dangling edges of there exists an automorphism of that permutes the dangling edges according to . Given a symmetric -pole , we construct a symmetric -pole as indicated in Fig. 2: we take the Petersen graph, remove a vertex, and replace each of the remaining vertices by a copy of . Note that since is symmetric, the resulting is uniquely determined and also symmetric. The relationship between and is captured in the following lemma.
Let be a symmetric -pole such that . Then
First, we determine . If there is a circuit of an even factor passing through at least copies of , then contributes and each copy of contributes at least to the value of . If there is no such circuit, then each copy of contributes at least to . Consequently, . Since is symmetric, for every choice of two dangling edges of , there is a -factor containing the chosen dangling edges with . It is, therefore, possible to get for a -factor of containing a circuit passing through all the nine copies of in , hence .
Next, we determine . Any path containing two dangling edges can visit at most copies of (otherwise the Petersen graph would be Hamiltonian; equality is attainable since is symmetric). There is thus at least one copy not visited by , which contributes at least , and each of the remaining copies contributes at least to . Altogether, .
Finally, has obviously vertices. ∎
For any , there exists a -connected simple cubic graph with no traveling salesman tour of length shorter than .
We construct a sequence of symmetric -poles with increasing ratio of excess to number of vertices.
Let be a vertex incident to three dangling edges; we have . For every integer , let . According to Lemma 2, the sequence of triples satisfies
Solving the last two recurrence relations by standard methods yields and for every .
By attaching a new vertex to the three dangling edges of , we obtain a cubic graph with the ratio of the length of a TSP tour to the number of vertices arbitrarily close to . Indeed, consider an even factor of . If contains two of the edges incident to , its excess is at least (we add for the circuit passing through ). If contains as an isolated vertex, its excess is . Put together, . Consequently, the length of a TSP tour is at least , and thus the ratio is equal to , which belongs to for a sufficiently large .
Acknowledgements. We are thankful to the anonymous referees for useful suggestions for improving this paper, especially for pointing out that our construction principle is also applicable to bipartite graphs. We acknowledge support from the research grants VEGA 1/0474/15, VEGA 1/0876/16, and APVV-15-0220.
-  S. Boyd, R. Sitters, S. van der Ster, L. Stougie, The traveling salesman problem on cubic and subcubic graphs, Mathematical Programming 144 (2014), 227–245.
-  B. Candráková, R. Lukoťka, Cubic TSP—a 1.3-approximation, arXiv:1506.06369 [cs.DM] (2015).
-  J. R. Correa, O. Larré, J. A. Soto, TSP tours in cubic graphs: Beyond , SIAM J. Discrete Math 29 (2015), 915–939.
-  Z. Dvořák, D. Kráľ, B. Mohar, Graphic TSP in cubic graphs, Proceedings of 34th International Symposium on Theoretical Aspects of Computer Science (STACS’17), Leibniz International Proceedings in Informatics (LIPIcs) vol. 66, article no. 27, Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2017.
-  M. A. Fiol, J. Vilaltella, Some results on the structure of multipoles in the study of snarks, Electron. J. Combin. 22 (1) (2015), #P1.45.
-  H. Fleischner, Some blood, sweat, but no tears in Eulerian graph theory, Congr. Numer. 63 (1988), 8–48.
-  T. Mömke, O. Svensson, Approximating graphic tsp by matchings. In Proceedings of the 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS ’11, 560–569, Washington, DC, USA, 2011. IEEE Computer Society.
-  R. Nedela, M. Škoviera, Decompositions and reductions of snarks, J. Graph Theory 22 (3) (1996), 253–279.
-  A. Sebö, J. Vygen, Shorter tours by nicer ears: 7/5-approximation for the graph-TSP, 3/2 for the path version, and 4/3 for two-edge-connected subgraphs, Combinatorica (2014), 1–34.
-  A. van Zuylen, Improved Approximations for Cubic Bipartite and Cubic TSP, in: Q. Louveaux, M. Skutella (eds) Integer Programming and Combinatorial Optimization, IPCO 2016, Lecture Notes in Computer Science, vol 9682, Springer, Cham, 250–261.
-  J. Vygen, New approximation algorithms for the TSP, OPTIMA 90 (2012), 1–12.
-  H. S. Wilf, Generatingfunctionology, Third Edition, A K Peters / CRC Press (2005), ISBN 9781439864395.