A non-topological proof for the impossibility of $k$-set agreement

International audienceIn the \emph{$k$-set agreement} task each process proposes a value, and it is required that each correct process has to decide a value which was proposed and at most $k$ distinct values must be decided. Using topological arguments it has been proved that $k$-set agreement is unsolvable in the asynchronous \emph{wait-free} read/write shared memory model, when $k < n$, the number of processes. This paper presents a simple, non-topological impossibility proof of $k$-set agreement. The proof depends on two simple properties of the \emph{immediate snapshot executions}, a subset of all possible executions, and on the well known graph theory result stating that every graph has an even number of vertices with odd degree (the \emph{handshaking lemma})