Correcting the Foundational Analysis of Karp--Vazirani--Vazirani (STOC 1990): A Rigorous Revision of the $1-1/e$ Upper Bound

We revisit the classical analysis of Karp, Vazirani, and Vazirani (KVV, STOC~1990), which established the well-known upper bound of $1 - 1/e$ as the limiting proportion of vertices that can be matched by any online procedure in a canonical bipartite structure. Although foundational, the original analysis contains several inaccuracies, including a fundamental technical gap in the treatment of the underlying discrete process. We give a transparent and fully rigorous reconstruction of the KVV argument by reformulating the evolution of available neighbors as a discrete-time death process and deriving a sharp upper bound via a simple factor-revealing linear program that captures the correct recurrence structure. This yields a precise bound $\lceil n(1 - 1/e) + 2 - 1/e \rceil$ on the expected number of matched vertices, refining the classical claim $n(1 - 1/e) + o(n)$. Our goal is not to optimize this upper bound, but to provide a mathematically sound and conceptually clean correction of the classical KVV analysis, while remaining faithful to its original combinatorial framework.