Symposium On Theory Of Computing Research Articles

Minimum 2SAT-DELETION: Inapproximability results and relations to Minimum Vertex Cover

The M INIMUM 2SAT-D ELETION problem is to delete the minimum number of clauses in a 2SAT instance to make it satisfiable. It is one of the prototypes in the approximability hierarchy of minimization problems Khanna et al. [Constraint satisfaction: the approximability of minimization problems, Proceedings of the 12th Annual IEEE Conference on Computational Complexity, Ulm, Germany, 24–27 June, 1997, pp. 282–296], and its approximability is largely open. We prove a lower approximation bound of 8 5 - 15 ≈ 2.88854 , improving the previous bound of 10 5 - 21 ≈ 1.36067 by Dinur and Safra [The importance of being biased, Proceedings of the 34th Annual ACM Symposium on Theory of Computing (STOC), May 2002, pp. 33–42, also ECCC Report TR01-104, 2001]. For highly restricted instances with exactly four occurrences of every variable we provide a lower bound of 3 2 . Both inapproximability results apply to instances with no mixed clauses (the literals in every clause are both either negated, or unnegated). We further prove that any k-approximation algorithm for the M INIMUM 2SAT-D ELETION problem polynomially reduces to a ( 2 - 2 / ( k + 1 ) ) -approximation algorithm for the M INIMUM V ERTEX C OVER problem. One ingredient of these improvements is our proof that the M INIMUM V ERTEX C OVER problem is hardest to approximate on graphs with perfect matching. More precisely, the problem to design a ρ -approximation algorithm for the M INIMUM V ERTEX C OVER on general graphs polynomially reduces to the same problem on graphs with perfect matching. This improves also on the results by Chen and Kanj [On approximating minimum vertex cover for graphs with perfect matching, Proceedings of the 11st ISAAC, Taipei, Taiwan, Lecture Notes in Computer Science, vol. 1969, Springer, Berlin, 2000, pp. 132–143].

An information statistics approach to data stream and communication complexity

We present a new method for proving strong lower bounds in communication complexity. This method is based on the notion of the conditional information complexity of a function which is the minimum amount of information about the inputs that has to be revealed by a communication protocol for the function. While conditional information complexity is a lower bound on communication complexity, we show that it also admits a direct sum theorem . Direct sum decomposition reduces our task to that of proving conditional information complexity lower bounds for simple problems (such as the AND of two bits). For the latter, we develop novel techniques based on Hellinger distance and its generalizations. Our paradigm leads to two main results: (1) An improved lower bound for the multi-party set-disjointness problem in the general communication complexity model, and a nearly optimal lower bound in the one-way communication model. As a consequence, we show that for any real k >2, approximating the k th frequency moment in the data stream model requires essentially Ω(n 1−2/k ) space; this resolves a conjecture of Alon et al. (J. Comput. System Sci. 58(1) (1999) 137). (2) A lower bound for the L p approximation problem in the general communication model; this solves an open problem of Saks and Sun (in: Proceedings of the 34th Annual ACM Symposium on Theory of Computing (STOC), 2002, pp. 360–369). As a consequence, we show that for p >2, approximating the L p norm to within a factor of n ε in the data stream model with constant number of passes requires Ω(n 1−4ε−2/p ) space.

An information statistics approach to data stream and communication complexity

We present a new method for proving strong lower bounds in communication complexity. This method is based on the notion of the conditional information complexity of a function which is the minimum amount of information about the inputs that has to be revealed by a communication protocol for the function. While conditional information complexity is a lower bound on communication complexity, we show that it also admits a direct sum theorem. Direct sum decomposition reduces our task to that of proving conditional information complexity lower bounds for simple problems (such as the AND of two bits). For the latter, we develop novel techniques based on Hellinger distance and its generalizations. Our paradigm leads to two main results: (1) An improved lower bound for the multi-party set-disjointness problem in the general communication complexity model, and a nearly optimal lower bound in the one-way communication model. As a consequence, we show that for any real k>2, approximating the kth frequency moment in the data stream model requires essentially Ω(n 1−2/k) space; this resolves a conjecture of Alon et al. (J. Comput. System Sci. 58(1) (1999) 137). (2) A lower bound for the L p approximation problem in the general communication model; this solves an open problem of Saks and Sun (in: Proceedings of the 34th Annual ACM Symposium on Theory of Computing (STOC), 2002, pp. 360–369). As a consequence, we show that for p>2, approximating the L p norm to within a factor of n ε in the data stream model with constant number of passes requires Ω(n 1−4ε−2/p) space.

Approximating [formula omitted] to within almost-polynomial factors is NP-hard

We show SVP ∞ and CVP ∞ to be NP-hard to approximate to within n c/ log log n for some constant c>0. We show a direct reduction from SAT to these problems, that combines ideas from Arora et al. (Proc. 34th IEEE Symp. on Foundations of Computer Science, 1993, p. 724) and Dinur et al. (Approximating-CVP to within almost-polynomial factors is NP-hard, manuscript, 1999), along with some modifications. Our result is obtained without relying on the PCP characterization of NP, although some of our techniques are derived from the proof of the PCP characterization itself (STOC: ACM Symposium on Theory of Computing (STOC), 1999).

Review of Introduction to Distributed Algorithms

This book presents a view of several important topics in the modern field of distributed algorithms. The book is intended to provide useful background and reference information for professional engineers dealing with distributed systems, and for entering researchers who wish to develop a sufficient background in distributed algorithms. In addition, the book is suitable for providing a textbook-style introduction to distributed algorithms for advanced undergraduates or early graduate students who are taking a relevant course or consider pursuing research in the field.The reference sources for this book incluude mainly research papers on the modern theory of Distributed Computing that routinely appear in conferences such as the ACM Symposium on Principles of Distributed Computing (PODC), or the International Symposium on DIStributed Computing (DISC). Fundamental milestones of this theory include also research papers from other major conferences of Theoretical Computer Science that are of wider interest, such as the ACM Symposium on Theory of Computing (STOC), or from leading theory journals, such as the Journal of the ACM or SIAM Journal on Computing.

Logic activities in Europe

During Fall 1992, thanks to ONR, I had an opportunity to visit a fair number of West European centers of logic research. I tried to learn more about logic investigations and applications in Europe with the hope that my experience may be useful to American researchers. This report is concerned only with logic activities related to computer science, and Europe here means usually Western Europe (one can learn only so much in one semester). The idea of such a visit may seem ridiculous to some. The modern world is quickly growing into a global village. There is plenty of communication between Europe and the US. Many European researchers visit the US, and many American researchers visit Europe. Neither Americans nor Europeans make secret of their logic research. Quite the opposite is true. They advertise their research. From ESPRIT reports, the Bulletin of European Association for Theoretical Computer Science, the Newsletter of European Association for Computer Science Logics, publications of European Foundation for Logic, Language and Information, publications of particular European universities, etc., one can get a good idea of what is going on in Europe and who is doing what. Some European colleagues asked me jokingly if I was on a reconnaissance mission. Well, sometimes a cow wants to suckle more than the calf wants to suck (a Hebrew proverb). It is amazing, however, how di erent computer science is, especially theoretical computer science, in Europe and the US. American theoretical computer science centers heavily around complexity theory. The two prime American theoretical conferences | ACM Symposium on Theory of Computing (STOC) and IEEE Foundation of Computer Science Conference (FOCS) | are largely devoted to complexity theory (in a wider sense of the term). That does not exclude logic. As a matter of fact, important logic results have been published in those conferences. However, STOC and FOCS logic papers belong, as a rule, to branches of logic intimately related to complexity. Finite model theory is a good example of that; see [Fagin 1990, Gurevich 1988] and especially [Immerman 1989]. The di erence between theoretical computer science and the rest of computer science (and computer engineering) is relatively well delineated in this country. In Europe, the line between theoretical computer science and the rest of computer science and engineering is much more blurred, partially because computer science and engineering in general are more theoretical. A much greater proportion of European research goes into programming language theory, semantics, speci cation languages, proof systems, veri cation methods, etc. For brevity and the lack of a better term, I will use the term \formal methods for all of these areas. Europeans put much more faith and e orts into