Gábor Tardos

A constructive proof of the general lovász local lemma

The Lovász Local Lemma discovered by Erdős and Lovász in 1975 is a powerful tool to non-constructively prove the existence of combinatorial objects meeting a prescribed collection of criteria. In 1991, József Beck was the first to demonstrate that a constructive variant can be given under certain more restrictive conditions, starting a whole line of research aimed at improving his algorithms performance and relaxing its restrictions. In the present article, we improve upon recent findings so as to provide a method for making almost all known applications of the general Local Lemma algorithmic.

On the power of randomization in online algorithms

Against in adaptive adversary, we show that the power of randomization in on-line algorithms is severely limited! We prove the existence of an efficient “simulation” of randomized on-line algorithms by deterministic ones, which is best possible in general. The proof of the upper bound is existential. We deal with the issue of computing the efficient deterministic algorithm, and show that this is possible in very general cases.

Excluded permutation matrices and the Stanley-Wilf conjecture

This paper examines the extremal problem of how many 1-entries an n × n 0-1 matrix can have that avoids a certain fixed submatrix P. For any permutation matrix P we prove a linear bound, settling a conjecture of Zoltan Furedi and Peter Hajnal (Discrete Math. 103(1992) 233). Due to the work of Martin Klazar (D. Krob, A.A. Mikhalev, A.V. Mikhalev (Eds.), Formal Power Series and Algebraics Combinatorics, Springer, Berlin, 2000, pp. 250-255), this also settles the conjecture of Stanley and Wilf on the number of n -permutations avoiding a fixed permutation and a related conjecture of Alon and Friedgut (J. Combin Theory Ser A 89(2000) 133).

On the power of randomization in on-line algorithms

Against in adaptive adversary, we show that the power of randomization in on-line algorithms is severely limited! We prove the existence of an efficient “simulation” of randomized on-line algorithms by deterministic ones, which is best possible in general. The proof of the upper bound is existential. We deal with the issue of computing the efficient deterministic algorithm, and show that this is possible in very general cases.

Local Chromatic Number, KY Fan's Theorem, And Circular Colorings

The local chromatic number of a graph was introduced in [14]. It is in between the chromatic and fractional chromatic numbers. This motivates the study of the local chromatic number of graphs for which these quantities are far apart. Such graphs include Kneser graphs, their vertex color-critical subgraphs, the Schrijver (or stable Kneser) graphs; Mycielski graphs, and their generalizations; and Borsuk graphs. We give more or less tight bounds for the local chromatic number of many of these graphs.We use an old topological result of Ky Fan [17] which generalizes the Borsuk–Ulam theorem. It implies the existence of a multicolored copy of the complete bipartite graph K⌈t/2⌉,⌊t/2⌋ in every proper coloring of many graphs whose chromatic number t is determined via a topological argument. (This was in particular noted for Kneser graphs by Ky Fan [18].) This yields a lower bound of ⌈t/2⌉ + 1 for the local chromatic number of these graphs. We show this bound to be tight or almost tight in many cases.As another consequence of the above we prove that the graphs considered here have equal circular and ordinary chromatic numbers if the latter is even. This partially proves a conjecture of Johnson, Holroyd, and Stahl and was independently attained by F. Meunier [42]. We also show that odd chromatic Schrijver graphs behave differently, their circular chromatic number can be arbitrarily close to the other extreme.

Tight bounds for Lp samplers, finding duplicates in streams, and related problems

In this paper, we present near-optimal space bounds for <i>L_p</i>-samplers. Given a stream of updates (additions and subtraction) to the coordinates of an underlying vector <i>x</i> in <i>Rⁿ</i>, a perfect <i>L_p</i> sampler outputs the <i>i</i>-th coordinate with probability <i>x_i</i><i>^p</i><i>x</i><i>pp</i>. In SODA 2010, Monemizadeh and Woodruff showed polylog space upper bounds for approximate <i>L_p</i>-samplers and demonstrated various applications of them. Very recently, Andoni, Krauthgamer and Onak improved the upper bounds and gave a <i>O</i>(ε<i>^-p</i>log³<i>n</i>) space ε relative error and constant failure rate <i>L_p</i>-sampler for <i>p</i> є [1,2]. In this work, we give another such algorithm requiring only O(ε<i>^-p</i>log²<i>n</i>) space for <i>p</i> є (1,2). For <i>p</i> є (0,1), our space bound is O(ε^-1log²<i>n</i>), while for the <i>p</i>=1 case we have an <i>O</i>(log(1/ε)ε^-log²<i>n</i>) space algorithm. We also give a <i>O</i>(log²<i>n</i>) bits zero relative error <i>L</i>₀-sampler, improving the <i>O</i>(log³<i>n</i>) bits algorithm due to Frahling, Indyk and Sohler. As an application of our samplers, we give better upper bounds for the problem of finding duplicates in data streams. In case the length of the stream is longer than the alphabet size, <i>L</i>₁ sampling gives us an <i>O</i>(log²<i>n</i>) space algorithm, thus improving the previous <i>O</i>(log³<i>n</i>) bound due to Gopalan and Radhakrishnan. In the second part of our work, we prove an Ω (log²<i>n</i>) lower bound for sampling from 0, ± 1 vectors (in this special case, the parameter <i>p</i> is not relevant for <i>L_p</i> sampling). This matches the space of our sampling algorithms for constant ε>0. We also prove tight space lower bounds for the finding duplicates and heavy hitters problems. We obtain these lower bounds using reductions from the communication complexity problem augmented indexing.

On the maximum number of edges in quasi-planar graphs

A topological graph is quasi-planar, if it does not contain three pairwise crossing edges. Agarwal et al. [P.K. Agarwal, B. Aronov, J. Pach, R. Pollack, M. Sharir, Quasi-planar graphs have a linear number of edges, Combinatorica 17 (1) (1997) 1-9] proved that these graphs have a linear number of edges. We give a simple proof for this fact that yields the better upper bound of 8n edges for n vertices. Our best construction with 7n-O(1) edges comes very close to this bound. Moreover, we show matching upper and lower bounds for several relaxations and restrictions of this problem. In particular, we show that the maximum number of edges of a simple quasi-planar topological graph (i.e., every pair of edges have at most one point in common) is 6.5n-O(1), thereby exhibiting that non-simple quasi-planar graphs may have many more edges than simple ones.

Query complexity, or why is it difficult to separateNP A ∩coNP A fromP A by random oraclesA?

By thequery-time complexity of a relativized algorithm we mean the total length of oracle queries made; thequery-space complexity is the maximum length of the queries made. With respect to these cost measures one can define polynomially time- or space-bounded deterministic, nondeterministic, alternating, etc. Turing machines and the corresponding complexity classes. It turns out that all known relativized separation results operate essentially with this cost measure. Therefore, if certain classes do not separate in the query complexity model, this can be taken as an indication that their relativized separation in the classical cost model will require entirely new principles.A notable unresolved question in relativized complexity theory is the separation of NPA∩ ∩ co NPA fromPA under random oraclesA. We conjecture that the analogues of these classes actually coincide in the query complexity model, thus indicating an answer to the question in the title. As a first step in the direction of establishing the conjecture, we prove the following result, where polynomial bounds refer to query complexity.If two polynomially query-time-bounded nondeterministic oracle Turing machines accept precisely complementary (oracle dependent) languages LA and {0, 1}*∖LA under every oracle A then there exists a deterministic polynomially query-time-bounded oracle Turing machine that accept LA. The proof involves a sort of greedy strategy to selecting deterministically, from the large set of prospective queries of the two nondeterministic machines, a small subset that suffices to perform an accepting computation in one of the nondeterministic machines. We describe additional algorithmic strategies that may resolve the same problem when the condition holds for a (1−ε) fraction of the oracles A, a step that would bring us to a non-uniform version of the conjecture. Thereby we reduce the question to a combinatorial problem on certain pairs of sets of partial functions on finite sets.

Polynomial bound for a chip firing game on graphs

Bjorner, Lvasz, and Shor have introduced a chip firing game on graphs. This paper proves a polynomial bound on the length of the game in terms of the number of vertices of the graph provided the length is finite. The obtained bound is best possible within a constant factor.

A lower bound on the mod 6 degree of the OR function

Abstract. We examine the computational power of modular counting, where the modulus m is not a prime power, in the setting of polynomials in Boolean variables over Zm. In particular, we say that a polynomial P weakly represents a Boolean function f (both have n variables) if for any inputs x and y in {0,1}n, we have