Thursday, November 16, 2017 - 13:30 , Location: Skiles 005 , Vijay Vazirani , UC Irvine , Organizer: Robin Thomas
Is matching in NC, i.e., is there a deterministic fast parallel algorithm for it? This has been an outstanding open question in TCS for over three decades, ever since the discovery of Random NC matching algorithms. Within this question, the case of planar graphs has remained an enigma: On the one hand, counting the number of perfect matchings is far harder than finding one (the former is #P-complete and the latter is in P), and on the other, for planar graphs, counting has long been known to be in NC whereas finding one has resisted a solution!The case of bipartite planar graphs was solved by Miller and Naor in 1989 via a flow-based algorithm. In 2000, Mahajan and Varadarajan gave an elegant way of using counting matchings to finding one, hence giving a different NC algorithm.However, non-bipartite planar graphs still didn't yield: the stumbling block being odd tight cuts. Interestingly enough, these are also a key to the solution: a balanced odd tight cut leads to a straight-forward divide and conquer NC algorithm. The remaining task is to find such a cut in NC. This requires several algorithmic ideas, such as finding a point in the interior of the minimum weight face of the perfect matching polytope and uncrossing odd tight cuts.Joint work with Nima Anari.
Tuesday, January 19, 2016 - 13:05 , Location: Skiles 006 , Shuji Kijima , Kyushu University , Organizer: Prasad Tetali
The rotor-router model, also known as the Propp machine, is a deterministic process analogous to a simple random walk on a graph. In this talk, we are concerned with a generalized model, functional-router model, which imitates a Markov chain possibly containing irrational transition probabilities. We investigate the discrepancy of the number of tokens between the functional-router model and its corresponding Markov chain, and give some upper bounds in terms of the mixing time of the Markov chain.