Michal Koucky

Universal traversal sequences with backtracking

We introduce a new notion of traversal sequences that we call exploration sequences. Exploration sequences share many properties with the traversal sequences defined in (AKL+), but they also exhibit some new properties. In particular, they have an ability to backtrack, and their random properties are robust under choice of the probability distribution on labels. Further, we present extremely simple constructions of polynomial length universal exploration sequences for some previously studied classes of graphs (e.g. 2-regular graphs, cliques, expanders), and we also present universal exploration sequences for trees. Our constructions beat previously known lower-bounds on the length of universal traversal sequences.

Power from random strings

We show that sets consisting of strings of high Kolmogorov complexity provide examples of sets that are complete for several complexity classes under probabilistic and non-uniform reductions. These sets are provably not complete under the usual many-one reductions. Let R/sub K/, R/sub Kt/, R/sub KS/, R/sub KT/ be the sets of strings x having complexity at least |x|/2, according to the usual Kolmogorov complexity measure K, Levins time-bounded Kolmogorov complexity Kt [27], a space-bounded Kolmogorov measure KS, and the time-bounded Kolmogorov complexity measure KT that was introduced in [4], respectively. Our main results are: 1. R/sub KS/ and R/sub Kt/ are complete for PSPACE and EXP, respectively, under P/poly-truth-table reductions. 2. EXP = NP/sup R(Kt)/. 3. PSPACE = ZPP/sup R(KS)/ /spl sube/ P/sup R(K)/. 4. The Discrete Log, Factoring, and several lattice problems are solvable in BPP/sup R(KT)/.

Time-space tradeoffs in the counting hierarchy

Extends the lower-bound techniques of L. Fortnow (2000) to the unbounded-error probabilistic model. A key step in the argument is a generalization of V.A. Nepomnjas/spl caron/c/spl caron/ii/spl breve/s (1970) theorem from the Boolean setting to the arithmetic setting. This generalization is made possible due to the recent discovery of logspace-uniform TC/sup 0/ circuits for iterated multiplication (A. Chiu et al., 2000). As an example of the sort of lower bounds that we obtain, we show that MAJ-MAJSAT is not contained in PrTiSp(n/sup 1+o(1)/, n/sup /spl epsiv//) for any /spl epsiv/<1. We also extend one of Fortnows lower bounds, from showing that S~A~T~ does not have uniform NC/sup 1/ circuits of size n/sup 1+o(1)/, to a similar result for SAC/sup 1/ circuits.

Exact algorithms for solving stochastic games: extended abstract

Shapleys discounted stochastic games, Everetts recursive games and Gillettes undiscounted stochastic games are classical models of game theory describing two-player zero-sum games of potentially infinite duration. We describe algorithms for exactly solving these games. When the number of positions of the game isbconstant, our algorithms run in polynomial time.

Derandomization and distinguishing complexity

We continue an investigation of resource-bounded Kolmogorov complexity and derandomization techniques begun in [E. Allender (2001), E. Allender et al., (2002)]. We introduce nondeterministic time-bounded Kolmogorov complexity measures (KNt and KNT) and examine the properties of these measures using constructions of hitting set generators for nondeterministic circuits [P. B. Miltersen et al., (1999), R. Shaltiel et al., (2001)]. We observe that KNt bears many similarities to the nondeterministic distinguishing complexity CND of [H. Buhrman et al., (2002)]. This motivates the definition of a new notion of time-bounded distinguishing complexity KDt, as an intermediate notion with connections to the class FewEXP. The set of KDt-random strings is complete for EXP under P/poly reductions. Most of the notions of resource-bounded Kolmogorov complexity discussed here and in [E. Allender (2001), E. Allender et al., (2002)] have close connections to circuit size (on different types of circuits). We extend this framework to define notions of Kolmogorov complexity KB and KF that are related to branching program size and formula size, respectively. The sets of KB- and KF-random strings lie in coNP; we show that oracle access to these sets enables one to factor Blum integers. We obtain related intractability results for approximating minimum formula size, branching program size, and circuit size. The NEXP/spl sube/NC and NEXP/spl sube/L/poly questions are shown to be equivalent to conditions about the KF and KB complexity of sets in P.

The Hardness of Being Private

In 1989 Kushilevitz initiated the study of iinformation-theoretic privacy within the context of communication complexity. Unfortunately, it has been shown that most interesting functions are not privately computable. The unattainability of perfect privacy for many functions motivated the study of approximate privacy. Feigenbaum et al. define notions of worst-case as well as average-case approximate privacy, and present several interesting upper bounds, and some open problems for further study. In this paper, we obtain asymptotically tight bounds on the tradeoffs between both the worst-case and average-case approximate privacy of protocols and their communication cost for Vickrey-auctions. Further, we relate the notion of average-case approximate privacy to other measures based on information cost of protocols. This enables us to prove exponential lower bounds on the subjective approximate privacy of protocols for computing the Intersection function, independent of its communication cost. This proves a conjecture of Feigenbaum et al.

Amplifying Lower Bounds by Means of Self-Reducibility

We observe that many important computational problems in NC1 share a simple self-reducibility property. We then show that, for any problem A having this self-reducibility property, A has polynomial size TC0 circuits if and only if it has TC0 circuits of size n1+isin for every isin>0 (counting the number of wires in a circuit as the size of the circuit). As an example of what this observation yields, consider the Boolean formula evaluation problem (BFE), which is complete for NC1. It follows from a lower bound of Impagliazzo, Paturi, and Saks, that BFE requires depth d TC0 circuits of size n1+isin d. If one were able to improve this lower bound to show that there is some constant isin>0 such that every TC0 circuit family recognizing BFE has size n1+isin, then it would follow that TC0neNC1. We also show that problems with small uniform constant- depth circuits have algorithms that simultaneously have small space and time bounds. We then make use of known time-space tradeoff lower bounds to show that SAT requires uniform depth d TC0 and AC0 [6] circuits of size n1+c for some constant c depending on d.

Towards a Reverse Newman's Theorem in Interactive Information Complexity

Newman’s theorem states that we can take any public-coin communication protocol and convert it into one that uses only private randomness with but a little increase in communication complexity. We consider a reversed scenario in the context of information complexity: can we take a protocol that uses private randomness and convert it into one that only uses public randomness while preserving the information revealed to each player? We prove that the answer is yes, at least for protocols that use a bounded number of rounds. As an application, we prove new direct-sum theorems through the compression of interactive communication in the bounded-round setting. To obtain this application, we prove a new one-shot variant of the Slepian–Wolf coding theorem, interesting in its own right. Furthermore, we show that if a Reverse Newman’s Theorem can be proven in full generality, then full compression of interactive communication and fully-general direct-sum theorems will result.

Randomised Individual Communication Complexity

In this paper we study the individual communication complexity of the following problem. Alice receives an input string x and Bob an input string y, and Alice has to output y. For deterministic protocols it has been shown in Buhrman et al. (2004), that C(y) many bits need to be exchanged even if the actual amount of information C(y|x) is much smaller than C(y). It turns out that for randomised protocols the situation is very different. We establish randomised protocols whose communication complexity is close to the information theoretical lower bound. We furthermore initiate and obtain results about the randomised round complexity of this problem and show trade-offs between the amount of communication and the number of rounds. In order to do this we establish a general framework for studying these types of questions.