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An augmenting path alternates between edges outside and inside a matching in a graph and joins two unmatched endpoints. Flipping its edges increases the matching size by one. In an assignment equality graph it moves toward a perfect matching while preserving zero reduced costs on matched edges.

The Hungarian algorithm maintains feasible assignment dual potentials and a matching of equality edges. If the alternating search cannot augment, adjust reachable job and agent potentials by the smallest reduced cost leaving the search set. This preserves feasibility and existing matching equalities while exposing a new equality edge. A resulting perfect matching attains the dual lower bound.

For a linear program with unchanged cost and constraint matrix, an optimal simplex basis remains optimal while its basic solution remains feasible. Its reduced costs and dual certificate do not change. The new basic variables are $A_B^{-1}(b+ϵ)$, and the objective is the displayed affine function, with $y$ the basis's dual solution. This gives an exact piecewise-affine sensitivity region, determined by nonnegativity of the basic variables, rather than merely a formal first derivative.

The network simplex algorithm specializes the simplex algorithm to network flows. A basis is represented by a spanning tree with nonbasic flows at their bounds; potentials give reduced costs, and entering arcs produce cycle pivots. Degeneracy in linear programming is particularly common in assignment instances.

Use a directed bipartite graph with one vertex for each job and one for each agent. Job vertices have supply $1$ and agent vertices demand $1$. An arc from job $i$ to agent $j$ has cost $c_{ij}$ and nonnegative flow $x_{ij}$; an upper bound $1$ may be added, but is already implied by the balances. The minimum-cost flow formulation of the assignment problem isInteger supplies and demands give an integral optimum by integrality of the transportation problem. Its $n$ positive arcs form a perfect matching, hence give one job per agent.

A network simplex algorithm basis for this connected $2n$-vertex network has $2n-1$ tree arcs, whereas an integral assignment has only $n$ positive flows. For $n>1$, at least $n-1$ basic tree arcs therefore carry zero flow. This unavoidable degeneracy in linear programming can cause zero-step pivots, extra bookkeeping and cycling unless an appropriate pivot rule is used. The Hungarian algorithm exploits the assignment structure directly rather than maintaining many zero-flow basic arcs.

Suppose the assignment dual potentials satisfy ${\lambda }_i-{\mu }_j\le c_{ij}$. For every feasible assignment $y$,If the selected assignment $x$ uses only arcs with ${\lambda }_i-{\mu }_j=c_{ij}$, its cost equals this lower bound. Weak duality and complementary slackness certify optimality. The PDF correctly denotes the feasible assignment by $x_{ij}$; the TeX aid mistakenly replaces it with ${\lambda }_{ij}$.

The Hungarian algorithm maintains feasible assignment dual potentials and a matching in a graph of zero reduced cost edges, where $r_{ij}=c_{ij}-{\lambda }_i+{\mu }_j\ge 0$. Search for an augmenting path in a matching in this equality graph. If there is none, let $S$ be the reachable jobs and $T$ the reachable agents in the alternating search from unmatched jobs. Every agent in $T$ is matched to a job in $S$. SetReduced costs from $S$ to its complement decrease by $\Delta$, those from outside $S$ into $T$ increase, and the remaining ones do not change. Thus feasibility and current matching equalities are preserved, while at least one new equality edge appears. Continue until an augmenting path in a matching increases the matching size. A perfect matching in the equality graph then has the same value as the dual linear program and is optimal.

For the example, start with the row minima $\lambda =(10,50,50)$ and $\mu =(0,0,0)$. The reduced cost matrix isMatch job $1$ to agent $1$. Searching from both unmatched jobs gives $S=\{1,2,3\}$ and $T=\{1\}$; the minimum cost to an agent outside $T$ is $\Delta =10$. Update to $\lambda =(20,60,60)$ and $\mu =(10,0,0)$:The new equality edge $(2,3)$ augments the matching. From unmatched job $3$, the next alternating search reaches agent $1$ and its matched job $1$, giving $S=\{1,3\}$ and $T=\{1\}$. Now $\Delta =15$. The updated feasible assignment dual potentials and reduced costs areThe augmenting path in a matching $J_3\to A_1\to J_1\to A_2$ uses job $3$ to agent $1$, the reversed matched edge to job $1$, then the new edge to agent $2$. It gives the optimal assignmentThe dual linear program value is $35+60+75-25=145$, an independent optimality certificate.