Improved approximations for cubic bipartite and cubic TSP

Abstract

We show improved approximation guarantees for the traveling salesman problem on cubic bipartite graphs and cubic graphs. For connected cubic bipartite graphs with n nodes, we improve on recent results of Karp and Ravi by giving a “local improvement” algorithm that finds a tour of length at most $5/4n-2$. For 2-connected cubic graphs, we show that the techniques of Mömke and Svensson can be combined with the techniques of Correa, Larré and Soto, to obtain a tour of length at most $(4/3-1/8754)n$.

A Tightness of the analysis of Algorithm 1

We give an example of a cubic bipartite graph $G=(V, E)$ for which both the 2-factors $F_1$ and $F_2$ that result from Algorithm 1 have |V| / 8 components.

The instance has 48 nodes, numbered 1 through 48, and $(V, F_1)$ contains six cycles, four cycles of size 6, one cycle of size 10 and one cycle of size 14. For brevity, we denote the cycles by only giving an ordered listing of their nodes; an edge between consecutive nodes and between the last and first node is implicit. The cycles in $(V, F_1)$ are:

$$\begin{aligned}&C_1=(1,2,3,4,5,6), \\&C_2=(7,8,9,10,11,12), \\&C_3=(13,14,15,16,17,18), \\&C_4=(19,20,21,22,23,24), \\&C_5=(25,26,27,28,29,30,31,32,33,34), \\&C_6=(35,36,37,38,39,40,41,42,43,44,45,46,47,48). \end{aligned}$$

The second 2-factor $(V,F_2)$ has six cycles as well, namely, three cycles of size 6 and three cycles of size 10. We again denote the cycles by giving an ordered listing of the nodes, but now a semicolon between subsequent nodes indicates that the nodes are connected by an edge in $F_2{\setminus } F_1$, and a comma denotes that they are connected by an edge in $F_2\cap F_1$. The cycles in $(V, F_2)$ are

$$\begin{aligned}&D_1 =(6,1;30,31;42,43;28,29;40,41;),&D_4=(18,13;38,39;44,45;), \\&D_2=(48,35;10,11;4,5;8,9;2,3;),&D_5=(24,19;26,27;32,33;), \\&D_3=(34,25;16,17;22,23;14,15;20,21;),&D_6=(12,7;36,37;46,47;). \end{aligned}$$

It is straightforward to verify that every node occurs in exactly one cycle in $(V, F_1)$ and exactly one cycle in $(V, F_2)$, and that each cycle in $(V, F_1)$ alternates edges in $F_1{\setminus } F_2$ and edges in $F_1\cap F_2$, and that each cycle in $(V, F_2)$ alternates edges in $F_2\cap F_1$ and edges in $F_2{\setminus } F_1$. Furthermore, each cycle $C_i$ in $(V, F_1)$ has exactly two edges in a cycle D in $(V, F_2)$ of size exactly 10.

Local Optimum

Figure 3 depicts each of the six cycles $C_i$ in $(V, F_1)$, together with the cycles $D_j$ in $(V,F_2)$ that intersect the given cycle $C_i$. For any of the cycles $C_i$, replacing $F_2$ by $F_2\triangle E(C_i)$ does not decrease the number of components of $(V, F_2)$ for any cycle $C_i$, nor does the modification of $F_2$ for a chorded cycle described in Algorithm 1.

Hence, if, rather than following Algorithm 1, we would execute one of the two possible modifications of $F_2$ suggested by the algorithm, as long as this reduced the number of components of $(V, F_2)$, then the 2-factor $F_2$ is in fact a local optimum with respect to this process since none of the possible modifications reduces the number of components.

We note that the instance does have a Hamilton cycle. This cycle contains subpaths of more than two adjacent edges in $E{\setminus } F_1$, so it can never be found from $F_1$ and $F_2$ using the “moves” we defined, even if we allow “moves” that do not reduce the number of components of $(V, F_2)$. We give the Hamilton cycle by again giving an ordered listing of the nodes, where a semicolon between subsequent nodes indicates that the nodes are connected by an edge in $E{\setminus } F_1$, and a comma denotes that they are connected by an edge in $F_1$.

$$\begin{aligned}&(1;30, 29 ; 40,39,38 ; 13,14,15,16;25,26;19,20,21;34,33;24,23,22;17,18;45,\\&44,43;28,27;32,31;42,41;6,5;8,7,12, 11; 4,3;48,47,46;37,36,35;10,9;2,1). \end{aligned}$$

Fig. 3

Depicted are six figures, one for each cycle $C_i$ in $(V, F_1)$ for $i=1,\ldots , 6$. The white nodes are the nodes in the cycle $C_i$, and the dashed edges are the edges in $F_1{\setminus } E(C_i)$. The non-dashed and thick dashed edges are edges in $F_2$. For each $C_i$, the number of components of $(V, F_2)$ does not decrease by replacing $F_2$ by $F_2\triangle E(C_i)$