Annual IEEE Symposium Research Articles

Special Issue on Foundations of Computer Science

This volume comprises the polished and fully refereed versions of a selection of papers presented at the Forty-Sixth Annual IEEE Symposium on Foundations of Computer Science (FOCS 2005), held in Pittsburgh, Pennsylvania, October 23–25, 2005. Unrefereed preliminary versions of the papers presented at the symposium appeared in the proceedings of the meeting, published by IEEE. The FOCS 2005 Program Committee consisted of Ziv Bar-Yossef, Paul Beame, Ran Canetti, Irit Dinur, Ashish Goel, Venkatesan Guruswami, Sariel Har-Peled, Michael Kearns, Richard Lipton, Frank McSherry, Satish Rao, Omer Reingold, Eva Tardos, Mikkel Thorup, Berthold Voecking, John Watrous, Mihalis Yannakakis, and David Zuckerman. Of 276 extended abstracts submitted to the FOCS 2005 Program Committee, 62 were selected for presentation at the symposium. Eight of those 62 papers are included in this volume. This collection encompasses a wide variety of topics and methods in theoretical computer science, often shedding new light on entire areas with a fresh approach. The topics include algorithms, learning, property testing, combinatorics, cryptography, and distributed and quantum computing. All papers were refereed in accordance with SICOMP's stringent standards, and most were substantially updated in the process. We take this opportunity to thank all the referees whose anonymous work has significantly contributed to the value of this volume. It was an honor to edit this special section in SIAM Journal on Computing.

Sums of squares based approximation algorithms for MAX-SAT

We investigate the Semidefinite Programming based sums of squares (SOS) decomposition method, designed for global optimization of polynomials, in the context of the (Maximum) Satisfiability problem. To be specific, we examine the potential of this theory for providing tests for unsatisfiability and providing MAX-SAT upper bounds. We compare the SOS approach with existing upper bound and rounding techniques for the MAX-2-SAT case of Goemans and Williamson [Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming, J. Assoc. Comput. Mach. 42(6) (1995) 1115–1145] and Feige and Goemans [Approximating the value of two prover proof systems, with applications to MAX2SAT and MAXDICUT, in: Proceedings of the Third Israel Symposium on Theory of Computing and Systems, 1995, pp. 182–189] and the MAX-3-SAT case of Karloff and Zwick [A 7/8-approximation algorithm for MAX 3SAT? in: Proceedings of the 38th Annual IEEE Symposium on Foundations of Computer Science, Miami Beach, FL, USA, IEEE Press, New York, 1997], which are based on Semidefinite Programming as well. We prove that for each of these algorithms there is an SOS-based counterpart which provides upper bounds at least as tight, but observably tighter in particular cases. Also, we propose a new randomized rounding technique based on the optimal solution of the SOS Semidefinite Program (SDP) which we experimentally compare with the appropriate existing rounding techniques. Further we investigate the implications to the decision variant SAT and compare experimental results with those yielded from the higher lifting approach of Anjos [On semidefinite programming relaxations for the satisfiability problem, Math. Methods Oper. Res. 60(3) (2004) 349–367; An improved semidefinite programming relaxation for the satisfiability problem, Math. Programming 102(3) (2005) 589–608; Semidefinite optimization approaches for satisfiability and maximum-satisfiability problems, J. Satisfiability Boolean Modeling Comput. 1 (2005) 1–47]. We give some impression of the fraction of the so-called unit constraints in the various SDP relaxations. From a mathematical viewpoint these constraints should be easily dealt within an algorithmic setting, but seem hard to be avoided as extra constraints in an SDP setting. Finally, we briefly indicate whether this work could have implications in finding counterexamples to uncovered cases in Hilbert's Positivstellensatz.

Approximating Buy-at-Bulk and Shallow-Light k-Steiner Trees

We study two related network design problems with two cost functions. In the buy-at-bulk k-Steiner tree problem we are given a graph G(V,E) with a set of terminals T⊆V including a particular vertex s called the root, and an integer k≤|T|. There are two cost functions on the edges of G, a buy cost b:E→ℝ+ and a distance cost r:E→ℝ+. The goal is to find a subtree H of G rooted at s with at least k terminals so that the cost ∑e∈H b(e)+∑t∈T−s dist(t,s) is minimized, where dist(t,s) is the distance from t to s in H with respect to the r cost. We present an O(log 4 n)-approximation algorithm for the buy-at-bulk k-Steiner tree problem. The second and closely related one is bicriteria approximation algorithm for Shallow-light k-Steiner trees. In the shallow-light k-Steiner tree problem we are given a graph G with edge costs b(e) and distance costs r(e), and an integer k. Our goal is to find a minimum cost (under b-cost) k-Steiner tree such that the diameter under r-cost is at most some given bound D. We develop an (O(log n),O(log 3 n))-approximation algorithm for a relaxed version of Shallow-light k-Steiner tree where the solution has at least $\frac{k}{8}$terminals. Using this we obtain an (O(log 2 n),O(log 4 n))-approximation algorithm for the shallow-light k-Steiner tree and an O(log 4 n)-approximation algorithm for the buy-at-bulk k-Steiner tree problem. Our results are recently used to give the first polylogarithmic approximation algorithm for the non-uniform multicommodity buy-at-bulk problem (Chekuri, C., et al. in Proceedings of 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS’06), pp. 677–686, 2006).

From first principles to the Burrows and Wheeler transform and beyond, via combinatorial optimization

We introduce a combinatorial optimization framework that naturally induces a class of optimal word permutations with respect to a suitably defined cost function taking into account various measures of relatedness between words. The Burrows and Wheeler transform ( bwt) (cf. [M. Burrows, D. Wheeler, A block sorting lossless data compression algorithm, Technical Report 124, Digital Equipment Corporation, 1994]), and its analog for labelled trees (cf. [P. Ferragina, F. Luccio, G. Manzini, S. Muthukrishnan, Structuring labeled trees for optimal succinctness, and beyond, in: Proc. of the 45th Annual IEEE Symposium on Foundations of Computer Science, 2005, pp. 198–207]), are special cases in the class. We also show that the class of optimal word permutations defined here is identical to the one identified by Ferragina et al. for compression boosting [P. Ferragina, R. Giancarlo, G. Manzini, M. Sciortino, Boosting textual compression in optimal linear time, Journal of the ACM 52 (2005) 688–713]. Therefore, they are all highly compressible. We also provide, by using techniques from Combinatorics on Words, a fast method to compute bwt without using any end-of-string symbol. We also investigate more general classes of optimal word permutations, where relatedness of symbols may be measured by functions more complex than context length. For this general problem we provide an instance that is MAX-SNP hard, and therefore unlikely to be solved or approximated efficiently. The results presented here indicate that a key feature of the Burrows and Wheeler transform seems to be, besides compressibility, the existence of efficient algorithms for its computation and inversion.

Learning a hidden graph using [formula omitted] queries per edge

We consider the problem of learning a general graph using edge-detecting queries. In this model, the learner may query whether a set of vertices induces an edge of the hidden graph. This model has been studied for particular classes of graphs by Grebinski and Kucherov [V. Grebinski, G. Kucherov, Optimal query bounds for reconstructing a Hamiltonian cycle in complete graphs, in: Fifth Israel Symposium on the Theory of Computing Systems, 1997, pp. 166–173] and Alon et al. [N. Alon, R. Beigel, S. Kasif, S. Rudich, B. Sudakov, Learning a hidden matching, in: The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002, pp. 197–206], motivated by problems arising in genome sequencing. We give an adaptive deterministic algorithm that learns a general graph with n vertices and m edges using O ( m log n ) queries, which is tight up to a constant factor for classes of non-dense graphs. Allowing randomness, we give a 5-round Las Vegas algorithm using O ( m log n + m log 2 n ) queries in expectation. We give a lower bound of Ω ( ( 2 m / r ) r / 2 ) for learning the class of non-uniform hypergraphs of dimension r with m edges.

Alphabet-independent linear-time construction of compressed suffix arrays using [formula omitted]-bit working space

The suffix array is a fundamental index data structure in string algorithms and bioinformatics, and the compressed suffix array (CSA) and the FM-index are its compressed versions. Many algorithms for constructing these index data structures have been developed. Recently, Hon et al. [W.K. Hon, K. Sadakane, W.K. Sung, Breaking a time-and-space barrier in constructing full-text indices, in: Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science, 2003, pp. 251–260] proposed a construction algorithm using O ( n log log | Σ | ) time and O ( n log | Σ | ) -bit working space, which is the fastest algorithm using O ( n log | Σ | ) -bit working space. In this paper we give an efficient algorithm to construct the index data structures. Our algorithm constructs the suffix array, the CSA, the FM-index, and Burrows–Wheeler transform using alphabet-independent O ( n ) time and O ( n log | Σ | log | Σ | α n ) -bit working space, where α = log 3 2 . Our algorithm takes less time and more space than Hon et al.’s algorithm. Our algorithm uses least working space among alphabet-independent linear-time algorithms.

Event Domains, Stable Functions and Proof-Nets

We pursue the program of exposing the intrinsic mathematical structure of the “space of proofs” of a logical system [S. Abramsky and R. Jagadeesan. Games and Full Completeness for Multiplicative Linear Logic, Journal of Symbolic Logic, (1994), vol. 59, no. 2, 543–574]. We study the case of Multiplicative-Additive Linear Logic (MALL). We use tools from Domain theory to develop a semantic notion of proof net for MALL, and prove a Sequentialization Theorem. This work forms part of a continuation of previous joint work with Radha Jagadeesan [S. Abramsky and R. Jagadeesan. Games and Full Completeness for Multiplicative Linear Logic, Journal of Symbolic Logic, (1994), vol. 59, no. 2, 543–574] and Paul-André Melliès [S. Abramsky and P.-A. Melliès. Concurrent Games and Full Completeness. Proceedings of the 14th Annual IEEE Symposium on Logic in Computer Science, 431–44, 1999].

De-linearizing Linearity: Projective Quantum Axiomatics From Strong Compact Closure

Elaborating on our joint work with Abramsky in [S. Abramsky, B. Coecke, B. (2004) A categorical semantics of quantum protocols. Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science (LiCS'04), IEEE Computer Science Press. Extended version including proofs at arXiv:quant-ph/0402130, S. Abramsky, B. Coecke, (2005) Abstract physical traces. Theory and Applications of Categories 14, 111–124] we further unravel the linear structure of Hilbert spaces into several constituents. Some prove to be very crucial for particular features of quantum theory while others obstruct the passage to a formalism which is not saturated with physically insignificant global phases.First we show that the bulk of the required linear structure is purely multiplicative, and arises from the strongly compact closed tensor which, besides providing a variety of notions such as scalars, trace, unitarity, self-adjointness and bipartite projectors [S. Abramsky, B. Coecke, B. (2004) A categorical semantics of quantum protocols. Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science (LiCS'04), IEEE Computer Science Press. Extended version including proofs at arXiv:quant-ph/0402130, S. Abramsky, B. Coecke, (2005) Abstract physical traces. Theory and Applications of Categories 14, 111–124], also provides Hilbert-Schmidt norm, Hilbert-Schmidt inner-product, and in particular, the preparation-state agreement axiom which enables the passage from a formalism of the vector space kind to a rather projective one, as it was intended in the (in)famous Birkhoff & von Neumann paper [G. Birkhoff, J. von Neumann, (1936) The logic of quantum mechanics. Annals of Mathematics 37, 823–843].Next we consider additive types which distribute over the tensor, from which measurements can be build, and the correctness proofs of the protocols discussed in [S. Abramsky, B. Coecke, (2004) A categorical semantics of quantum protocols. Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science (LiCS'04), IEEE Computer Science Press. Extended version including proofs at arXiv:quant-ph/0402130] carry over to the resulting weaker setting. A full probabilistic calculus is obtained when the trace is moreover linear and satisfies the diagonal axiom, which brings us to a second main result, characterization of the necessary and sufficient additive structure of a both qualitatively and quantitatively effective categorical quantum formalism without redundant global phases. Along the way we show that if in a category a (additive) monoidal tensor distributes over a strongly compact closed tensor, then this category is always enriched in commutative monoids.

Fast matrix multiplication is stable

We perform forward error analysis for a large class of recursive matrix multiplication algorithms in the spirit of Bini and Lotti [Numer. Math. 36:63–72, 1980]. As a consequence of our analysis, we show that the exponent of matrix multiplication (the optimal running time) can be achieved by numerically stable algorithms. We also show that new group-theoretic algorithms proposed in Cohn and Umans [Foundations of Computer Science, 44th Annual IEEE Symposium, pp. 438–449, 2003] and Cohn et al. [Foundations of Computer Science, 46th Annual IEEE Symposium, pp. 379–388, 2005] are all included in the class of algorithms to which our analysis applies, and are therefore numerically stable. We perform detailed error analysis for three specific fast group-theoretic algorithms.

Localized Client-Server Load Balancing without Global Information

We consider distributed algorithms for maximizing throughput in a network of clients and servers, modeled as a bipartite graph. We seek algorithms and lower bounds for decentralized algorithms in which each participant has only local knowledge about the state of itself and its neighbors. Our problem is analogous to recent work on oblivious routing [M. Bienkowski, M. Korzeniowski, and H. Räcke, Proceedings of the $15$th Annual ACM Symposium on Parallel Algorithms and Architectures, 2003, pp. 24–33, C. Harrelson, K. Hildrum, and S. Rao, Proceedings of the $15$th Annual ACM Symposium on Parallel Algorithms and Architectures, 2003, pp. 34–43, H. Räcke, Proceedings of the $43$rd Annual IEEE Symposium on Foundations of Computer Science, 2002, pp. 43–52] but with the objective of maximizing throughput rather than minimizing congestion. In contrast to that work, we prove a strong lower bound (polynomial in n, the size of the graph) on the competitive ratio of any oblivious algorithm. This is accompanied by simple algorithms achieving upper bounds which are tight in terms of $\OPT$, the maximum throughput achievable by an omniscient algorithm, and are also tight in terms of m, the number of servers. Finally, we investigate an online version of the problem, in a restricted model which requires that clients, upon becoming active, must remain so for at least $log(n)$ time steps. In contrast to our primarily negative results in the oblivious case, here we present an algorithm which is constant-competitive. Our lower bounds justify the intuition, implicit in earlier work on the subject [B. Awerbuch and Y. Azar, Proceedings of the $35$th Annual IEEE Symposium on Foundations of Computer Science, 1994, pp. 240–249], that some such restriction (i.e., requiring some stability in the demand pattern over time) is necessary in order to achieve a constant—or even polylogarithmic—competitive ratio.

Special Section on Foundations of Computer Science

This volume comprises the polished and fully refereed versions of a selection of papers presented at the Forty‐Fifth Annual IEEE Symposium on Foundations of Computer Science (FOCS 2004), held in Rome, Italy, October 17–19, 2004. Unrefereed preliminary versions of the papers presented at the symposium appeared in the proceedings of the meeting, published by IEEE. The FOCS 2004 Program Committee consisted of Dimitris Achlioptas, Micah Adler, Eli Ben‐Sasson, Faith Fich, Oded Goldreich, Martin Grohe, Sean Hallgren, Johan Håstad, Giuseppe F. Italiano, Vladlen Koltun, Yuval Rabani, Miklos Santha, Leonard Schulman, Rocco Servedio, D. Sivakumar, Eli Upfal, and David Williamson. Out of 272 “Extended Abstracts” submitted to the FOCS 2004 Program Committee, 64 were selected for presentation at the symposium. Eight of those 64 papers are included in this volume. This collection encompasses a wide variety of questions and methods in theoretical computer science, often shedding new light on entire areas with a fresh approach. The topics include fundamental questions of complexity theory and algorithms as well as foundational mathematical problems. All papers were refereed in accordance with SICOMP’s stringent standards, and most were substantially updated in the process. We take this opportunity to thank all the referees whose anonymous work has significantly contributed to the value of this volume. It was an honor to edit this special section in SIAM Journal on Computing.

Optimal Inapproximability Results for MAX‐CUT and Other 2‐Variable CSPs?

In this paper we show a reduction from the Unique Games problem to the problem of approximating MAX‐CUT to within a factor of $\alpha_{\text{\tiny{GW}}} + \epsilon$ for all $\epsilon > 0$; here $\alpha_{\text{\tiny{GW}}} \approx .878567$ denotes the approximation ratio achieved by the algorithm of Goemans and Williamson in [J. Assoc. Comput. Mach., 42 (1995), pp. 1115–1145]. This implies that if the Unique Games Conjecture of Khot in [Proceedings of the 34th Annual ACM Symposium on Theory of Computing, 2002, pp. 767–775] holds, then the Goemans–Williamson approximation algorithm is optimal. Our result indicates that the geometric nature of the Goemans–Williamson algorithm might be intrinsic to the MAX‐CUT problem. Our reduction relies on a theorem we call Majority Is Stablest. This was introduced as a conjecture in the original version of this paper, and was subsequently confirmed in [E. Mossel, R. O’Donnell, and K. Oleszkiewicz, Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science, 2005, pp. 21–30]. A stronger version of this conjecture called Plurality Is Stablest is still open, although [E. Mossel, R. O’Donnell, and K. Oleszkiewicz, Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science, 2005, pp. 21–30] contains a proof of an asymptotic version of it. Our techniques extend to several other two‐variable constraint satisfaction problems. In particular, subject to the Unique Games Conjecture, we show tight or nearly tight hardness results for MAX‐2SAT, MAX‐q‐CUT, and MAX‐2LIN(q). For MAX‐2SAT we show approximation hardness up to a factor of roughly $.943$. This nearly matches the $.940$ approximation algorithm of Lewin, Livnat, and Zwick in [Proceedings of the 9th Annual Conference on Integer Programming and Combinatorial Optimization, Springer‐Verlag, Berlin, 2002, pp. 67–82]. Furthermore, we show that our .943... factor is actually tight for a slightly restricted version of MAX‐2SAT. For MAX‐q‐CUT we show a hardness factor which asymptotically (for large q) matches the approximation factor achieved by Frieze and Jerrum [Improved approximation algorithms for MAX k‐CUT and MAX BISECTION, in Integer Programming and Combinatorial Optimization, Springer‐Verlag, Berlin, pp. 1–13], namely $1 - 1/q + 2({\rm ln}\,q)/q^2$. For MAX‐2LIN(q) we show hardness of distinguishing between instances which are $(1-\epsilon)$‐satisfiable and those which are not even, roughly, $(q^{-\epsilon/2})$‐satisfiable. These parameters almost match those achieved by the recent algorithm of Charikar, Makarychev, and Makarychev [Proceedings of the 38th Annual ACM Symposium on Theory of Computing, 2006, pp. 205–214]. The hardness result holds even for instances in which all equations are of the form $x_i - x_j = c$. At a more qualitative level, this result also implies that $1-\epsilon$ vs. ε hardness for MAX‐2LIN(q) is equivalent to the Unique Games Conjecture.

Learning Monotone Decision Trees in Polynomial Time

We give an algorithm that learns any monotone Boolean function $\fisafunc$ to any constant accuracy, under the uniform distribution, in time polynomial in n and in the decision tree size of $f.$ This is the first algorithm that can learn arbitrary monotone Boolean functions to high accuracy, using random examples only, in time polynomial in a reasonable measure of the complexity of $f.$ A key ingredient of the result is a new bound showing that the average sensitivity of any monotone function computed by a decision tree of size s must be at most $\sqrt{\log s}$. This bound has proved to be of independent utility in the study of decision tree complexity [O. Schramm, R. O'Donnell, M. Saks, and R. Servedio, Every decision tree has an influential variable, in Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science, IEEE Computer Society, Los Alamitos, CA, 2005, pp. 31–39]. We generalize the basic inequality and learning result described above in various ways—specifically, to partition size (a stronger complexity measure than decision tree size), p-biased measures over the Boolean cube (rather than just the uniform distribution), and real-valued (rather than just Boolean-valued) functions.

Integrality Ratio for Group Steiner Trees and Directed Steiner Trees

The natural relaxation for the group Steiner tree problem, as well as for its generalization, the directed Steiner tree problem, is a flow-based linear programming relaxation. We prove new lower bounds on the integrality ratio of this relaxation. For the group Steiner tree problem, we show that the integrality ratio is $\Omega(\log^2 k)$, where $k$ denotes the number of groups; this holds even for input graphs that are hierarchically well-separated trees, introduced by Bartal [in Proceedings of the 37th Annual IEEE Symposium on Foundations of Computer Science, 1996, pp. 184-193], in which case this lower bound is tight. This also applies for the directed Steiner tree problem. In terms of the number $n$ of vertices, our results for the directed Steiner problem imply an $\Omega(\frac{\log^2 n}{(\log \log n)^2})$ integrality ratio. For both problems, these are the first lower bounds on the integrality ratio that are superlogarithmic in the input size. This exhibits, for the first time, a relaxation of a natural optimization problem whose integrality ratio is known to be superlogarithmic but subpolynomial. Our results and techniques have been used by Halperin and Krauthgamer [in Proceedings of the 35th Annual ACM Symposium on Theory of Computing, 2003, pp. 585-594] to show comparable inapproximability results, assuming that NP has no quasi-polynomial Las Vegas algorithms. We also show algorithmically that the integrality ratio for the group Steiner tree problem is much better for certain families of instances, which helps pinpoint the types of instances (parametrized by optimal solutions to their flow-based relaxations) that appear to be most difficult to approximate.

Strongly Compact Closed Semantics

These lecture notes survey some joint work with Samson Abramsky as it was presented by me at Mathematical Foundations of Programming Semantics XXI in Birmingham (2005).22I only present part of that talk since some material is already available elsewhere [15]. It concerns a categorical semantics for quantum mechanics introduced in [Abramsky, S. and Coecke, B. (2004) A categorical semantics of quantum protocols. In: Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science (LiCS'04), IEEE Computer Science Press. An extended & improved version is available at http://arXiv.org/quant-ph/0402130; Abramsky, S. and Coecke, B. (2005) Abstract physical traces. Theory and Applications of Categories 14, 111–124. Available at www.tac.mta.ca/tac/volumes/14/6/14-06abs.html]. We present this semantics with a particular focus on the connection between categorical diagrams and certain pictures involving typed squares, triangles, diamonds and lines. Along the way we unravel the structural components which come with strong compact closure. We provide pointers to related literature. Keywords: Strong compact closure, semantics, quantum mechanics, quantum logic, quantum informatics.

A Novel Stochastic Game Via the Quantitative μ-calculus

The quantitative μ-calculus qMμ extends the applicability of Kozen's standard μ-calculus [D. Kozen, Results on the propositional μ-calculus, Theoretical Computer Science 27 (1983) 333–354] to probabilistic systems. Subsequent to its introduction [C. Morgan, and A. McIver, A probabilistic temporal calculus based on expectations, in: L. Groves and S. Reeves, editors, Proc. Formal Methods Pacific '97 (1997), available at [PSG, Probabilistic Systems Group: Collected reports, http://web.comlab.ox.ac.uk/oucl/research/areas/probs/bibliography.html]; also appears at [A. McIver, and C. Morgan, “Abstraction, Refinement and Proof for Probabilistic Systems,” Technical Monographs in Computer Science, Springer, New York, 2005, Chap. 9], M. Huth, and M. Kwiatkowska, Quantitative analysis and model checking, in: Proceedings of 12th annual IEEE Symposium on Logic in Computer Science, 1997] it has been developed by us [A. McIver, and C. Morgan, Games, probability and the quantitative μ-calculus qMu, in: Proc. LPAR, LNAI 2514 (2002), pp. 292–310, revised and expanded at [A. McIver, and C. Morgan, Results on the quantitative μ-calculus qMμ (2005), to appear in ACM TOCL]; also appears at [A. McIver, and C. Morgan, “Abstraction, Refinement and Proof for Probabilistic Systems,” Technical Monographs in Computer Science, Springer, New York, 2005, Chap. 11], A. McIver, and C. Morgan, “Abstraction, Refinement and Proof for Probabilistic Systems,” Technical Monographs in Computer Science, Springer, New York, 2005, A. McIver, and C. Morgan, Results on the quantitative μ-calculus qMμ (2005), to appear in ACM TOCL] and by others [L. de Alfaro, and R. Majumdar, Quantitative solution of omega-regular games, Journal of Computer and System Sciences 68 (2004) 374–397]. Beyond its natural application to define probabilistic temporal logic [C. Morgan, and A. McIver, An expectation-based model for probabilistic temporal logic, Logic Journal of the IGPL 7 (1999), pp. 779–804, also appears at [A. McIver, and C. Morgan, “Abstraction, Refinement and Proof for Probabilistic Systems,” Technical Monographs in Computer Science, Springer, New York, 2005, Chap.10]], there are a number of other areas that benefit from its use.One application is stochastic two-player games, and the contribution of this paper is to depart from the usual notion of “absolute winning conditions” and to introduce a novel game in which players can “draw”.The extension is motivated by examples based on economic games: we propose an extension to qMμ so that they can be specified; we show that the extension can be expressed via a reduction to the original logic; and, via that reduction, we prove that the players can play optimally in the extended game using memoryless strategies.

Cryptography in $NC^0$

We study the parallel time‐complexity of basic cryptographic primitives such as one‐way functions (OWFs) and pseudorandom generators (PRGs). Specifically, we study the possibility of implementing instances of these primitives by $NC^0$ functions, namely, by functions in which each output bit depends on a constant number of input bits. Despite previous efforts in this direction, there has been no convincing theoretical evidence supporting this possibility, which was posed as an open question in several previous works. We essentially settle this question by providing strong positive evidence for the possibility of cryptography in $NC^0$. Our main result is that every “moderately easy” OWF (resp., PRG), say computable in $NC^1$, can be compiled into a corresponding OWF (resp., “low‐stretch” PRG) in which each output bit depends on at most 4 input bits. The existence of OWFs and PRGs in $NC^1$ is a relatively mild assumption, implied by most number‐theoretic or algebraic intractability assumptions commonly used in cryptography. A similar compiler can also be obtained for other cryptographic primitives such as one‐way permutations, encryption, signatures, commitment, and collision‐resistant hashing. Our techniques can also be applied to obtain (unconditional) constructions of “noncryptographic” PRGs. In particular, we obtain ε‐biased generators and a PRG for space‐bounded computation in which each output bit depends on only 3 input bits. Our results make use of the machinery of randomizing polynomials [Y. Ishai and E. Kushilevitz, Proceedings of the 41st Annual IEEE Symposium on Foundations of Computer Science (FOCS), 2000, pp. 294–304], which was originally motivated by questions in the domain of information‐theoretic secure multiparty computation.

Deterministic Extractors for Bit‐Fixing Sources by Obtaining an Independent Seed

An $(n,k)$‐bit‐fixing source is a distribution X over $\{0,1\}^n$ such that there is a subset of k variables in $X_1,\ldots,X_n$ which are uniformly distributed and independent of each other, and the remaining $n-k$ variables are fixed. A deterministic bit‐fixing source extractor is a function $E:\{0,1\}^n \rightarrow \{0,1\}^m$ which on an arbitrary $(n,k)$‐bit‐fixing source outputs m bits that are statistically close to uniform. Recently, Kamp and Zuckerman [Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science, 2003, pp. 92–101] gave a construction of a deterministic bit‐fixing source extractor that extracts $\Omega(k^2/n)$ bits and requires $k>\sqrt{n}$. In this paper we give constructions of deterministic bit‐fixing source extractors that extract $(1-o(1))k$ bits whenever $k>(\log n)^c$ for some universal constant $c>0$. Thus, our constructions extract almost all the randomness from bit‐fixing sources and work even when k is small. For $k \gg \sqrt{n}$ the extracted bits have statistical distance $2^{-n^{\Omega(1)}}$ from uniform, and for $k \le \sqrt{n}$ the extracted bits have statistical distance $k^{-\Omega(1)}$ from uniform. Our technique gives a general method to transform deterministic bit‐fixing source extractors that extract few bits into extractors which extract almost all the bits.

A Combinatorial Logarithmic Approximation Algorithm for the Directed Telephone Broadcast Problem

Consider a synchronous network of processors, modeled by directed or undirected graph $G = (V,E)$, in which in each round every processor is allowed to choose one of its neighbors and to send a message to this neighbor. Given a processor $s \in V$ and a subset $T \subseteq V$ of processors, the telephone multicast problem requires computing the shortest schedule (in terms of the number of rounds) that delivers a message from $s$ to all the processors of $T$. The particular case $T = V$ is called the telephone broadcast problem. These problems have multiple applications in distributed computing. Several approximation algorithms with polylogarithmic ratio, including one with logarithmic ratio, for the undirected variants of these problems are known. However, all these algorithms involve solving large linear programs. Devising a polylogarithmic approximation algorithm for the directed variants of these problems is an open problem, posed by Ravi in [Proceedings of the 35th Annual IEEE Symposium on Foundations of Computer Science, (FOCS '94), 1994, pp. 202-213]. We devise a combinatorial logarithmic approximation algorithm for these problems that applies also for the directed broadcast problem. Our algorithm has significantly smaller running time and seems to reveal more information about the combinatorial structure of the solution than the previous algorithms that are based on linear programming. We also improve the lower bounds on the approximation threshold of these problems. Both problems are known to be 3/2-inapproximable. For the undirected (resp., directed) broadcast problem we show that it is NP-hard (resp., impossible unless $NP \subseteq DTIME(n^{O(\log n)})$) to approximate it within a ratio of $3 - \epsilon$ for any $\epsilon > 0$ (resp., $\Omega(\sqrt{\log n})$).

Approximation Algorithms for the 0-Extension Problem

In the 0-extension problem, we are given a weighted graph with some nodes marked as terminals and a semimetric on the set of terminals. Our goal is to assign the rest of the nodes to terminals so as to minimize the sum, over all edges, of the product of the edge's weight and the distance between the terminals to which its endpoints are assigned. This problem generalizes the multiway cut problem of Dahlhaus et al. [SIAM J. Comput.}, 23 (1994), pp. 864--894] and is closely related to the metric labeling problem introduced by Kleinberg and Tardos [Proceedings of the 40th IEEE Annual Symposium on Foundations of Computer Science, New York, 1999, pp. 14--23]. We present approximation algorithms for {\sc 0-Extension}. In arbitrary graphs, we present a O(log k)-approximation algorithm, k being the number of terminals. We also give O(1)-approximation guarantees for weighted planar graphs. Our results are based on a natural metric relaxation of the problem previously considered by Karzanov [European J. Combin., 19 (1998), pp. 71--101]. It is similar in flavor to the linear programming relaxation of Garg, Vazirani, and Yannakakis [SIAM J. Comput.}, 25 (1996), pp. 235--251] for the multicut problem, and similar to relaxations for other graph partitioning problems. We prove that the integrality ratio of the metric relaxation is at least $c \sqrt{\lg k}$ for a positive c for infinitely many k. Our results improve some of the results of Kleinberg and Tardos, and they further our understanding on how to use metric relaxations.