Michal Koucký

How to Explore a Fast-Changing World (Cover Time of a Simple Random Walk on Evolving Graphs)

Motivated by real world networks and use of algorithms based on random walks on these networks we study the simple random walks on dynamicundirected graphs with fixed underlying vertex set, i.e., graphs which are modified by inserting or deleting edges at every step of the walk. We are interested in the expected time needed to visit all the vertices of such a dynamic graph, the cover time, under the assumption that the graph is being modified by an oblivious adversary. It is well known that on connected staticundirected graphs the cover time is polynomial in the size of the graph. On the contrary and somewhat counter-intuitively, we show that there are adversary strategies which force the expected cover time of a simple random walk on connected dynamic graphs to be exponential. We relate this result to the cover time of static directed graphs. In addition we provide a simple strategy, the lazyrandom walk, that guarantees polynomial cover time regardless of the changes made by the adversary.

Many random walks are faster than one

We pose a new and intriguing question motivated by distributed computing regarding random walks on graphs: How long does it take for several independent random walks, starting from the same vertex, to cover an entire graph? We study the cover time-the expected time required to visit every node in a graph at least once-and we show that for a large collection of interesting graphs, running many random walks in parallel yields a speed-up in the cover time that is linear in the number of parallel walks. We demonstrate that an exponential speed-up is sometimes possible, but that some natural graphs allow only a logarithmic speed-up. A problem related to ours (in which the walks start from some probabilistic distribution on vertices) was previously studied in the context of space efficient algorithms for undirected s-t connectivity and our results yield, in certain cases, an improvement upon some of the earlier bounds.

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 n^1+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 n^1+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 n^1+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 n^1+c for some constant c depending on d.

What can be efficiently reduced to the kolmogorov-random strings?

Abstract We investigate the question of whether one can characterize complexity classes (such as PSPACE or NEXP) in terms of efficient reducibility to the set of Kolmogorov-random strings R C . This question arises because PSPACE ⊆ P R C and NEXP ⊆ NP R C , and no larger complexity classes are known to be reducible to R C in this way. We show that this question cannot be posed without explicitly dealing with issues raised by the choice of universal machine in the definition of Kolmogorov complexity. What follows is a list of some of our main results. • Although Kummer showed that, for every universal machine U there is a time bound t such that the halting problem is disjunctive truth-table reducible to R C U in time t , there is no such time bound t that suffices for every universal machine U . We also show that, for some machines U , the disjunctive reduction can be computed in as little as doubly-exponential time. • Although for every universal machine U , there are very complex sets that are ≤ dtt P -reducible to R C U , it is nonetheless true that P = REC ∩ ⋂ U { A : A ≤ dtt P R C U } . (A similar statement holds for parity-truth-table reductions.) • Any decidable set that is polynomial-time monotone-truth-table reducible to R C is in P / poly . • Any decidable set that is polynomial-time truth-table reducible to R C via a reduction that asks at most f ( n ) queries on inputs of size n lies in P / ( f ( n ) 2 f ( n ) 3 log f ( n ) ) .

Pseudorandom generators for group products: extended abstract

We prove that the pseudorandom generator introduced by Impagliazzo et al. (1994) with proper choice of parameters fools group products of a given finite group G. The seed length is O((|G|O(1) + log 1/δ)log n), where n is the length of the word and δ is the allowed error. The result implies that the pseudorandom generator with seed length O((2O(w log w) + log 1/δ)log n) fools read-once permutation branching programs of width w. As an application of the pseudorandom generator one obtains small-bias spaces for products over all finite groups Meka and Zuckerman (2009).

Bounded-depth circuits: separating wires from gates

We develop a new method to analyze the flow of communication in constant-depth circuits. This point of view allows usto prove new lower bounds on the number of wires required to recognize certain languages. We are able to provide explicit languages that can be recognized by <i>AC</i>⁰ circuits with <i>O</i>(<i>n</i>) gates but not with <i>O</i>(<i>n</i>) wires, and similarly for <i>ACC</i>⁰ circuits. We are also able to characterize exactly the regular languages that can be recognized with <i>O</i>(<i>n</i>) wires, both in <i>AC</i>⁰ and <i>ACC</i>⁰ framework.

Detection of feto-maternal infection/inflammation by the soluble receptor for advanced glycation end products (sRAGE): results of a pilot study

Abstract Objective: The receptor for advanced glycation end products, RAGE, plays an important role in the pathogenesis of several diseases. sRAGE, soluble receptor for advanced glycation end products, is an inhibitor of the pathological effect mediated via RAGE. The aim of this study was to assess the usefulness of measuring sRAGE concentration in pregnant women with threatening preterm labor. Methods: Serum levels of sRAGE, interleukin-6 (IL-6) and routine markers of inflammation were determined in 46 pregnant women with threatening preterm labor, 35 healthy pregnant women and 15 non-pregnant controls. Results: Serum levels of sRAGE in healthy pregnant women were significantly lower than in non-pregnant controls (669±296 vs. 1929±727 pg/mL, P<0.05). Women with threatening preterm birth had a significantly higher concentration of serum sRAGE in comparison with healthy pregnant women (819±329 pg/mL vs. 669±296 pg/mL, P<0.05). Conversely, patients with PPROM had significantly lower levels of sRAGE compared with patients with threatening premature labor (600±324 pg/mL, P<0.05). sRAGE correlated negatively with leukocyte counts (r=−0.325, P<0.05). Conclusions: sRAGE might be a new and promising marker of premature labor, especially with the symptoms of PPROM.

Derandomizing from Random Strings

In this paper we show that BPP is truth-table reducible to the set of Kolmogorov random strings R_K. It was previously known that PSPACE, and hence BPP is Turing-reducible to R_K. The earlier proof relied on the adaptivity of the Turing-reduction to find a Kolmogorov-random string of polynomial length using the set R_K as oracle. Our new non-adaptive result relies on a new fundamental fact about the set R_K, namely each initial segment of the characteristic sequence of R_K has high Kolmogorov complexity. As a partial converse to our claim we show that strings of very high Kolmogorov-complexity when used as advice are not much more useful than randomly chosen strings.

Computing with a full memory: catalytic space

We define the notion of a catalytic-space computation. This is a computation that has a small amount of clean space available and is equipped with additional auxiliary space, with the caveat that the additional space is initially in an arbitrary, possibly incompressible, state and must be returned to this state when the computation is finished. We show that the extra space can be used in a nontrivial way, to compute uniform TC1-circuits with just a logarithmic amount of clean space. The extra space thus works analogously to a catalyst in a chemical reaction. TC1-circuits can compute for example the determinant of a matrix, which is not known to be computable in logspace. In order to obtain our results we study an algebraic model of computation, a variant of straight-line programs. We employ register machines with input registers x1,..., xn and work registers r1,..., rm. The instructions available are of the form ri ← ri±u×v, with u, v registers (distinct from ri) or constants. We wish to compute a function f(x1,..., xn) through a sequence of such instructions. The working registers have some arbitrary initial value ri = τi, and they may be altered throughout the computation, but by the end all registers must be returned to their initial value τi, except for, say, r1 which must hold τ1 + f(x1,..., xn). We show that all of Valiants class VP, and more, can be computed in this model. This significantly extends the framework and techniques of Ben-Or and Cleve [6]. Upper bounding the power of catalytic computation we show that catalytic logspace is contained in ZPP. We further construct an oracle world where catalytic logpace is equal to PSPACE, and show that under the exponential time hypothesis (ETH), SAT can not be computed in catalytic sub-linear space.

The Hardness of Being Private

In 1989 Kushilevitz initiated the study of information-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.