Short Combinatorial Proof that the DFJ Polytope is contained in the MTZ Polytope for the Asymmetric Traveling Salesman Problem

Mark Velednitsky
UC Berkeley
marvel@berkeley.edu Fellow, National Physical Science Consortium

Abstract

For the Asymmetric Traveling Salesman Problem (ATSP), it is known that the Dantzig-Fulkerson-Johnson (DFJ) polytope is contained in the Miller-Tucker-Zemlin (MTZ) polytope. The analytic proofs of this fact are quite long. Here, we present a proof which is combinatorial and significantly shorter by relating the formulation to distances in a modified graph.

Keywords— salesman, polytope, mtz, subtour, combinatorial

1 Introduction

The Asymmetric Traveling Salesman Problem (ATSP) on the graph $G=(V,A)$ is typically formulated as an Integer Program (IP) by assigning each arc $(i,j)$ , of weight $c_{ij}$ , a binary variable $x_{ij}$ indicating whether or not it participates in the tour:

minimize	$\displaystyle\sum_{(i,j)\in A}{c_{ij}x_{ij}}$
subject to	$\displaystyle\sum_{j}{x_{ij}}=1\ \$	$\displaystyle\forall i\in V$
	$\displaystyle\sum_{i}{x_{ij}}=1\ \$	$\displaystyle\forall j\in V$
	$\displaystyle\text{no sub-tours in}\ \{(i,j)\|x_{ij}=1\}\ \$		(1)
	$\displaystyle x_{ij}\in\{0,1\}\ \$	$\displaystyle\forall(i,j)\in A$

Several variants of the sub-tour elimination constraint (1) have been proposed. The DFJ constraints are:

\sum_{i\in Q}{\sum_{j\in Q}{x_{ij}}}\leq|Q|-1

(2)

for any $Q\subseteq\{2,3,\ldots,n\}$ . The MTZ constraints introduce a new variable $u_{i}$ at each node $i\in V$ such that [5]:

u_{i}-u_{j}+nx_{ij}\leq n-1\ \forall(i,j)\in A

(3)

The $u_{i}$ are meant to enumerate the order in which nodes appear in the tour. That is, $u_{i}=1$ for the first node, $u_{i}=2$ for the second, and so on.

The DFJ and MTZ polytopes are the feasible regions of the respective LP relaxations. It is known that the MTZ formulation produces a weaker LP relaxation. However, rigorous proofs of this fact are quite involved [2, 3, 4, 6].

Even though they are weaker, MTZ-like constraints have been applied to Vehicle Routing Problems and are popular for solving small instances of ATSP [1]. Having a concise proof of their weakness could be instructive for understanding the constraints, teaching them, and applying them elsewhere [7].

2 Proof

Theorem 1.

The DFJ polytope is contained in the MTZ polytope.

Proof.

Let $x_{ij}$ be feasible for formulation DFJ. We define a new graph $G$ where the arc weights are $(n-1)-nx_{ij}.$ We let $-u_{j}$ be the length of the shortest path from $1$ to $j$ in $G$ . We claim that these $u_{j}$ are well-defined and make the $u_{j}$ and $x_{ij}$ together satisfy formulation MTZ. To check that MTZ is satisfied, we write the shortest path condition in $G$ :

-u_{j}\leq-u_{i}+(n-1)-nx_{ij}\implies u_{i}-u_{j}+nx_{ij}\leq(n-1).

To confirm that the $u_{j}$ are well-defined, we need to prove there are no negative-cost cycles in $G$ . Assume there is a negative cost cycle with edge set $C$ with node set $Q$ :

\sum_{(i,j)\in C}{((n-1)-nx_{ij})}<0\implies|Q|(n-1)-n\sum_{(i,j)\in C}{x_{ij}}<0\implies|Q|\frac{n-1}{n}<\sum_{(i,j)\in C}{x_{ij}}.

But the conditions of formulation DFJ give us

\sum_{(i,j)\in C}{x_{ij}}\leq|Q|-1.

This is a contradiction (since $|Q|=|C|\leq n$ ), so there are no negative cost cycles. ∎

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