Polynomial Time Solvable Subclass of the Generalized Traveling Salesman Problem on Grid Clusters

Abstract

The Generalized Traveling Salesman Problem on Grid Clusters (GTSP-GC) is the geometric setting of the famous Generalized Traveling Salesman Problem, where the nodes of a given graph are points on the Euclidean plane and the clusters are defined implicitly by the cells of a unit grid. The problem in question is strongly NP-hard but can be approximated in polynomial time with a fixed ratio. In this paper we describe a new non-trivial polynomially solvable subclass of GTSP-GC. Providing new min-max guarantee for the optimal clustering loss in one-dimensional 2-medians problem, we show that any instance of this subclass has a quasi-pyramidal optimal route, which can be found by dynamic programming in polynomial time.

A Proof of Lemma 1

Indeed, consider the following two-gamer zero-sum antagonistic game. The first player choose an n-length sample $\xi =(p_1,\ldots ,p_n)$ from [0, 1]. The second player proposes a 2-partition $C_1\cup C_2=[1,n]$. Payoff function

$$\begin{aligned} F(\xi ,(C_1,C_2))=\sum _{i\in C_1}|p_i-m_1|+\sum _{i\in C_2}|p_i-m_2|=\sum _{i=1}^n\min \{|p_i-m_1|,|p_i-m_2|\}, \end{aligned}$$

where $m_1$ and $m_2$ are medians of subsamples $\xi _1=(p_i:i\in C_1)$ and $\xi _2=(p_i:i\in C_2)$, respectively.

It is easy to verify that this game has no value. To obtain its lower value, notice that equation

$$\begin{aligned} \mathop {\sum }\nolimits _{i\in C}|p_i-m|={\left\{ \begin{array}{ll}\mathop {\sum }\nolimits _{i=k+1}^{2k}p_i-\sum _{i=1}^{k}p_i,&{} \text{ if } |C|=2k,\\[1ex] \mathop {\sum }\nolimits _{i=k+2}^{2k+1}p_i-\mathop {\sum }\nolimits _{i=1}^{k}p_i,&{} \text{ if } |C|=2k+1\end{array}\right. } \end{aligned}$$

is valid for any non-emplty index set C, median m, and sample

$$\begin{aligned} p_1\le p_2\le \ldots \le p_{|C|}. \end{aligned}$$

Therefore $\sup _{\xi }\inf _{C_1,C_2}F(\xi ,(C_1,C_2))$ is an optimum of the appropriate linear program

$$\begin{aligned} \begin{array}{lll} \alpha =&{}\max &{} u\\[1ex] &{}\text{ s.t. }&{} \sum \limits _{i=\lceil |C_1|/2\rceil +1}^{|C_1|}p_i-\sum \limits _{i=1}^{\lfloor |C_1|/2\rfloor }p_i+ \sum \limits _{i=\lceil |C_2|/2\rceil +1}^{|C_2|}p_{i+|C_1|}-\sum \limits _{i=1}^{\lfloor |C_2|/2\rfloor }p_{i+|C_1|}\ge u\\[3ex] &{}&{} \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad (C_1\cup C_2=[1,n])\\[1ex] &{}&{} 0\le p_1,\ldots ,p_n\le 1 \end{array} \end{aligned}$$

(9)

Using one of common linear programming techniques, e.g. variable elimination, it is easy to show that $\alpha =n/6$. Lemma is proved.