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We consider the classic problem of envy-free division of a heterogeneous good("cake") among several agents. It is known that, when the allotted pieces mustbe connected, the problem cannot be solved by a finite algorithm for 3 or moreagents. The impossibility result, however, assumes that the entire cake must beallocated. In this paper we replace the entire-allocation requirement with aweaker \emph{partial-proportionality} requirement: the piece given to eachagent must be worth for it at least a certain positive fraction of the entirecake value. We prove that this version of the problem is solvable in boundedtime even when the pieces must be connected. We present simple, bounded-timeenvy-free cake-cutting algorithms for: (1) giving each of $n$ agents aconnected piece with a positive value; (2) giving each of 3 agents a connectedpiece worth at least 1/3; (3) giving each of 4 agents a connected piece worthat least 1/7; (4) giving each of 4 agents a disconnected piece worth at least1/4; (5) giving each of $n$ agents a disconnected piece worth at least$(1-\epsilon)/n$ for any positive $\epsilon$.

Waste Makes Haste: Bounded Time Protocols for Envy-Free Cake Cutting with Free Disposal