Past exam of the computer science course of the University of Oxford / 2023 / Combinatorial optimisation / 3 / d

Assume $G$ has no isolated vertex; otherwise an edge cover does not exist. In particular, a one-vertex connected graph is an excluded degenerate case of the statement.

Let $M$ be a maximum cardinality matching, of size $m(G)$, and let $U$ be its unmatched vertices in a matching. There is no edge joining two vertices of $U$, since adding it would enlarge $M$. Choose one incident edge for each $u\in U$. These chosen edges are distinct and outside $M$, so together they form an edge cover of sizeConsequently $e(G)\le |V|-m(G)$.

For the opposite bound, let $F$ be a minimum-cardinality edge cover. It is inclusion-minimal. Every selected edge has an endpoint of vertex degree one in $(V,F)$: otherwise deleting that edge would leave both endpoints covered. Each connected component of a graph of $(V,F)$ is therefore a star graph. To see this, a component can have at most one vertex of vertex degree greater than one; a shortest graph path between two such vertices would contain an edge whose endpoints both have vertex degree greater than one. A component with no such vertex is a single edge, also a star graph.

If there are $t$ such star graphs, each has one fewer edge than vertices, so $|F|=|V|-t$. Select one edge from each component to get a matching in a graph of size $t$. Hence $m(G)\ge t=|V|-e(G)$, proving the edge cover and matching identity

First test for an isolated vertex, reporting infeasibility if one exists. Otherwise compute a maximum cardinality matching $M$ using a general-graph polynomial-time algorithm, then add one arbitrary incident edge for each unmatched vertex in a matching. The proof in (i) shows that the resulting edge cover has minimum cardinality $|V|-|M|$.

The final scan takes $O(|V|+|E|)$ time with adjacency lists; the maximum cardinality matching computation is polynomial. One may obtain it from a general-graph maximum-weight matching routine with every weight equal to one.^[ref]

Use minimum-weight edge cover by matching savings. Reject inputs with an isolated vertex. For every vertex $v$, choose a cheapest incident edge $e_v$ and write $c_v=w(e_v)$. Define a baseline and a saving for each edge $uv$ byCompute a maximum-weight matching $M$ using $p$ as its weights, allowing the empty matching in a graph and imposing no maximum-cardinality condition. Negative savings may be ignored. ReturnThis is an edge cover. Since original weights are nonnegative, removing duplicate appearances of an edge cannot increase the total, and henceTo prove optimality, take a minimum-weight edge cover and remove redundant edges until it is inclusion-minimal. Nonnegative weights ensure that its weight does not increase, so it remains optimal. As proved in (i), its components are star graphs. Choose one edge from each star graph, forming a matching in a graph $N$.

In a star graph with center $a$, selected leaf $b$, and other leaves $L$, this choice contributes $w(ab)+{\sum }_{v\in L}{c}_v$ to $B-p(N)$. Each other leaf $v$ has its edge $av$ available, so $c_v\le w(av)$. Thus this contribution is at most the weight of that star graph. Summing over components yieldsSince $M$ maximizes saving, $p(M)\ge p(N)$, and consequentlyAll inequalities are equalities. Therefore $F_M$ is a minimum-weight edge cover. This proof also handles zero weights and duplicate cheapest-edge choices.

Finding all $c_v$ and constructing $F_M$ takes $O(|V|+|E|)$ time. A general-graph maximum-weight matching polynomial-time algorithm supplies the only substantial step; the documented routine runs in $O(|V{|}^3)$ arithmetic time. Exact rational or integer weights avoid floating-point comparisons affecting optimality.^[ref]