Dependent Randomized Rounding via Exchange Properties of Combinatorial Structures

We consider the problem of randomly rounding a fractional solution $x$ in an integer polytope $P \subseteq [0,1]^n$ to a vertex $X$ of $P$, so that $\E[X] = x$. Our goal is to achieve {\em concentration properties} for linear and sub modular functions of the rounded solution. Such dependent rounding techniques, with concentration bounds for linear functions, have been developed in the past for two polytopes: the assignment polytope (that is, bipartite matchings and $b$-matchings)~\cite{S01, GKPS06, KMPS09}, and more recently for the spanning tree polytope~\cite{AGMGS10}. These schemes have led to a number of new algorithmic results. In this paper we describe a new {\em swap rounding} technique which can be applied in a variety of settings including {\em matroids} and {\em matroid intersection}, while providing Chernoff-type concentration bounds for linear and sub modular functions of the rounded solution. In addition to existing techniques based on negative correlation, we use a martingale argument to obtain an exponential tail estimate for monotone sub modular functions. The rounding scheme explicitly exploits {\em exchange properties} of the underlying combinatorial structures, and highlights these properties as the basis for concentration bounds. Matroids and matroid intersection provide a unifying framework for several known applications~\cite{GKPS06, KMPS09, CCPV09, KST09, AGMGS10} as well as new ones, and their generality allows a richer set of constraints to be incorporated easily. We give some illustrative examples, with a more comprehensive discussion deferred to a later version of the paper.