🤖 AI Summary
This paper addresses the problem of selecting one state from each of multiple state types to construct an *m*-state Markov chain that minimizes the average reward (i.e., “gain”) under its stationary distribution—a problem arising from optimal binary AIFV-*m* lossless coding design. Conventional approaches require exponential-time enumeration. We introduce the novel concept of the “Markov chain polytope,” modeling state selection as searching for extreme points of a convex envelope lying beneath a specific hyperplane in convex geometry. Leveraging a separation oracle and the ellipsoid method, we devise the first polynomial-time algorithm for exact optimization. Our framework unifies the construction of optimal Markov chains across several lossless compression paradigms, reducing computational complexity from exponential to polynomial time. This yields the first efficient, exact solver for AIFV-*m* code design and related problems.
📝 Abstract
This paper addresses the problem of finding a minimum-cost $m$-state Markov chain $(S_0,ldots,S_{m-1})$ in a large set of chains. The chains studied have a reward associated with each state. The cost of a chain is its"gain", i.e., its average reward under its stationary distribution. Specifically, for each $k=0,ldots,m-1$ there is a known set ${mathbb S}_k$ of type-$k$ states. A permissible Markov chain contains exactly one state of each type; the problem is to find a minimum-cost permissible chain. The original motivation was to find a cheapest binary AIFV-$m$ lossless code on a source alphabet of size $n$. Such a code is an $m$-tuple of trees, in which each tree can be viewed as a Markov Chain state. This formulation was then used to address other problems in lossless compression. The known solution techniques for finding minimum-cost Markov chains were iterative and ran in exponential time. This paper shows how to map every possible type-$k$ state into a type-$k$ hyperplane and then define a"Markov Chain Polytope"as the lower envelope of all such hyperplanes. Finding a minimum-cost Markov chain can then be shown to be equivalent to finding a"highest"point on this polytope. The local optimization procedures used in the previous iterative algorithms are shown to be separation oracles for this polytope. Since these were often polynomial time, an application of the Ellipsoid method immediately leads to polynomial time algorithms for these problems.