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Compute a minimum spanning tree, then a minimum-weight perfect matching on its odd-degree vertices. Their multiset union is connected with even degrees, so it has an Euler circuit. Shortcut repeated vertices to obtain a metric tour. The tree costs at most the optimal tour, and shortcutting the optimal tour to its odd-degree subset gives a cyclic order whose alternating matchings show that the added matching costs at most half the optimum. The result is a polynomial-time $3/2$-approximation algorithm for symmetric metric instances.

An Euler circuit is a closed edge trail that traverses every edge exactly once. Vertices may repeat, and parallel edges are counted separately. A connected finite undirected multigraph has such a circuit exactly when every vertex has even degree. Necessity follows by pairing arrivals and departures; sufficiency follows by forming closed unused-edge trails and splicing them until every edge is used, as in the Euler circuit criterion.

Let $C^{\ast }$ be an optimal metric tour. Removing any one of its edges gives a spanning tree, so the minimum spanning tree $T$ satisfies $c(T)\le c(C^{\ast })$. The handshaking lemma shows that its odd-degree set $U$ has even size.

Follow $C^{\ast }$ and retain only vertices of $U$, shortcutting between consecutive retained vertices. This produces a cyclic order on $U$ of total cost at most $c(C^{\ast })$. If $U$ is nonempty, the alternating edges of that cyclic order form two perfect matchings; their costs sum to the cycle cost. One has cost at most $c(C^{\ast })/2$, so a minimum-weight perfect matching $M$ hasFor $|U|=2$, count the same undirected edge twice in the cyclic order; each alternating matching consists of one copy. For $U=\varnothing$, use the empty matching.

The multigraph with edges $T\uplus M$ is connected because it contains $T$. Each odd-degree vertex receives exactly one matching edge, and the other degrees are unchanged, so every degree is even. It therefore has an Euler circuit. Constructively, follow unused edges until returning to the starting vertex; parity prevents getting stuck elsewhere. If unused edges remain, connectivity supplies a vertex of the current circuit incident with one, and its additional closed trail can be spliced into the circuit. Repetition gives a circuit using every edge exactly once, including parallel copies.

Shortcut repeated vertices of this Euler circuit to obtain a metric tour. Its cost is at mostComputing a minimum spanning tree, a minimum-weight perfect matching, an Euler circuit and the shortcut tour is polynomial-time; weighted perfect matching is a standard polynomial-time graph optimization problem. Thus the Christofides algorithm is a $3/2$-approximation algorithm for symmetric metric tours.