Randomized Search of Graphs in Log Space and Probabilistic Computation

Reingold has shown that L = SL, that s-t connectivity in a poly-mixing digraph is complete for promise-RL, and that s-t connectivity for a poly-mixing out-regular digraph with known stationary distribution is in L. Several properties that bound the mixing times of random walks on digraphs have been identified, including the digraph conductance and the digraph spectral expansion. However, rapidly mixing digraphs can still have exponential cover time, thus it is important to specifically identify structural properties of digraphs that effect cover times. We examine the complexity of random walks on a basic parameterized family of unbalanced digraphs called Strong Chains (which model weakly symmetric logspace computations), and a special family of Strong Chains called Harps. We show that the worst case hitting times of Strong Chain families vary smoothly with the number of asymmetric vertices and identify the necessary condition for non-polynomial cover time. This analysis also yields bounds on the cover times of general digraphs. Next we relate random walks on graphs to the random walks that arise in Monte Carlo methods applied to optimization problems. We introduce the notion of the asymmetric states of Markov chains and use this definition to obtain some results about Markov chains. We also obtain some results on the mixing times for Markov Chain Monte Carlo Methods. Finally, we consider the question of whether a single long random walk or many short walks is a better strategy for exploration. These are walks which reset to the start after a fixed number of steps. We exhibit digraph families for which a few short walks are far superior to a single long walk. We introduce an iterative deepening random search. We use this strategy estimate the cover time for poly-mixing subgraphs. Finally we discuss complexity theoretic implications and future work

Randomness in completeness and space-bounded computations

The study of computational complexity investigates the role of various computational resources such as processing time, memory requirements, nondeterminism, randomness, nonuniformity, etc. to solve different types of computational problems. In this dissertation, we study the role of randomness in two fundamental areas of computational complexity: NP-completeness and space-bounded computations. The concept of completeness plays an important role in defining the notion of \u27hard\u27 problems in Computer Science. Intuitively, an NP-complete problem captures the difficulty of solving any problem in NP. Polynomial-time reductions are at the heart of defining completeness. However, there is no single notion of reduction; researchers identified various polynomial-time reductions such as many-one reduction, truth-table reduction, Turing reduction, etc. Each such notion of reduction induces a notion of completeness. Finding the relationships among various NP-completeness notions is a significant open problem. Our first result is about the separation of two such polynomial-time completeness notions for NP, namely, Turing completeness and many-one completeness. This is the first result that separates completeness notions for NP under a worst-case hardness hypothesis. Our next result involves a conjecture by Even, Selman, and Yacobi [ESY84,SY82] which states that there do not exist disjoint NP-pairs all of whose separators are NP-hard via Turing reductions. If true, this conjecture implies that a certain kind of probabilistic public-key cryptosystems is not secure. The conjecture is open for 30 years. We provide evidence in support of a variant of this conjecture. We show that if there exist certain secure one-way functions, then the ESY conjecture for the bounded-truth-table reduction holds. Now we turn our attention to space-bounded computations. We investigate probabilistic space-bounded machines that are allowed to access their random bits {\em multiple times}. Our main conceptual contribution here is to establish an interesting connection between derandomization of such probabilistic space-bounded machines and the derandomization of probabilistic time-bounded machines. In particular, we show that if we can derandomize a multipass machine even with a small number of passes over random tape and only O(log^2 n) random bits to deterministic polynomial-time, then BPTIME(n) ⊆ DTIME(2^{o(n)}). Note that if we restrict the number of random bits to O(log n), then we can trivially derandomize the machine to polynomial time. Furthermore, it can be shown that if we restrict the number of passes to O(1), we can still derandomize the machine to polynomial time. Thus our result implies that any extension beyond these trivialities will lead to an unknown derandomization of BPTIME(n). Our final contribution is about the derandomization of probabilistic time-bounded machines under branching program lower bounds. The standard method of derandomizing time-bounded probabilistic machines depends on various circuit lower bounds, which are notoriously hard to prove. We show that the derandomization of low-degree polynomial identity testing, a well-known problem in co-RP, can be obtained under certain branching program lower bounds. Note that branching programs are considered weaker model of computation than the Boolean circuits

S-t connectivity on digraphs with a known stationary distribution

We present a deterministic logspace algorithm for solving S-T CONNECTIVITY on directed graphs if (i) we are given a stationary distribution for random walk on the graph and (ii) the random walk which starts at the source vertex s has polynomial mixing time. This result generalizes the recent deterministic logspace algorithm for S-T CONNECTIVITY on undirected graphs [15]. It identifies knowledge of the stationary distribution as the gap between the S-T CONNECTIVITY problems we know how to solve in logspace (L) and those that capture all of randomized logspace (RL)