Randomness-optimal unique element isolation, with applications to perfect matching and related problems

In this paper, we precisely characterize the randomness complexity of the unique element isolation problem, a crucial step in the RNC algorithm for perfect matching due to Mulmuley, Vazirani \& Vazirani[21] and in several other applications. Given a set $S$ and an unknown family $\cal F \subseteq$ $2^{S}$ with $|\cal F| \leq$ $Z$, we present a scheme to assign polynomially bounded weights to the elements of $S$, using only $O(\log Z + \log |S|)$ ransom bits, such that the minimum weight set in $\cal F$ is unique with high probability. This generalizes and improves the results of Mulmuley, Vazirani \& Vazirani who give a scheme which uses $O(S \log S)$ random bits independent of $Z$. We also prove a matching lower bound for the randomness complexity of this problem. This new weight assignment scheme yields a randomness-efficient $RNC^{2}$ algorithm for perfect matching which uses $O(\log Z + \log n)$ random bits where $Z$ is any given upper bound on the number of perfect matchings in the input graph. This generalizes the result of Grigoriev \& Karpinski[11] who present an $NC^{3}$ algorithm when $Z$ is polynomially bounded and also gives an improvement on the running time in this case. The worst-case randomness complexity of our algorithm is $O(n \log (m/n))$ random bits, as opposed to the previous bound of $O(m \log n)$ bits. Our technique also gives randomness-efficient solutions for several problems in which the unique element isolation tool is used, such as $RNC$ algorithms for variants of matching and basic problems on linear matroids such as matroid intersection and matroid matching. We also obtain a randomness-efficient alternative to the random reduction from $SAT$ to $USAT$, the language of uniquely satisfiable formulas, due to Valiant and Vazirani[32]. This reduction can be derandomized in the case of languages in $F ew P$ to yield new proofs of the results $F ew P \subseteq \oplus P$ and $F ew P \subseteq C_{=} P$.