298 research outputs found
Enumerating Colorings, Tensions and Flows in Cell Complexes
We study quasipolynomials enumerating proper colorings, nowhere-zero
tensions, and nowhere-zero flows in an arbitrary CW-complex , generalizing
the chromatic, tension and flow polynomials of a graph. Our colorings, tensions
and flows may be either modular (with values in for
some ) or integral (with values in ). We obtain
deletion-contraction recurrences and closed formulas for the chromatic, tension
and flow quasipolynomials, assuming certain unimodularity conditions. We use
geometric methods, specifically Ehrhart theory and inside-out polytopes, to
obtain reciprocity theorems for all of the aforementioned quasipolynomials,
giving combinatorial interpretations of their values at negative integers as
well as formulas for the numbers of acyclic and totally cyclic orientations of
.Comment: 28 pages, 3 figures. Final version, to appear in J. Combin. Theory
Series
House Markets with Matroid and Knapsack Constraints
Classical online bipartite matching problem and its generalizations are central algorithmic optimization problems. The second related line of research is in the area of algorithmic mechanism design, referring to the broad class of house allocation or assignment problems. We introduce a single framework that unifies and generalizes these two streams of models. Our generalizations allow for arbitrary matroid constraints or knapsack constraints at every object in the allocation problem. We design and analyze approximation algorithms and truthful mechanisms for this framework. Our algorithms have best possible approximation guarantees for most of the special instantiations of this framework, and are strong generalizations of the previous known results
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