Lossless Condensers,Unbalanced Expanders,And Extractors

Trevisan showed that many pseudorandom generator constructions give rise to constructions of explicit extractors. We show how to use such constructions to obtain explicit lossless condensers. A lossless condenser is a probabilistic map using only O(logn) additional random bits that maps n bits strings to poly(logK) bit strings, such that any source with support size K is mapped almost injectively to the smaller domain. Our construction remains the best lossless condenser to date. By composing our condenser with previous extractors, we obtain new, improved extractors. For small enough min-entropies our extractors can output all of the randomness with only O(logn) bits. We also obtain a new disperser that works for every entropy loss, uses an O(logn) bit seed, and has only O(logn) entropy loss. This is the best disperser construction to date, and yields other applications. Finally, our lossless condenser can be viewed as an unbalanced bipartite graph with strong expansion properties. * Much of this work was done while the author was in the Computer Science Division, University of California, Berkeley, and supported in part by a David and Lucile Packard Fellowship for Science and Engineering and NSF NYI Grant No. CCR-9457799. The work was also supported in part by an Alon fellowship and by the Israel Science Foundation. † Much of this work was done while the author was a graduate student in the Computer Science Division, University of California, Berkeley. Supported in part by NSF Grants CCR-9820897, CCF-0346991, and an Alfred P. Sloan Research Fellowship. ‡ Much of this work was done while the author was on leave at the Computer Science Division, University of California, Berkeley. Supported in part by a David and Lucile Packard Fellowship for Science and Engineering, NSF Grants CCR-9912428 and CCR-0310960, NSF NYI Grant CCR-9457799, and an Alfred P. Sloan Research Fellowship.     

Density Conditions for Panchromatic Colourings of Hypergraphs

Let be a hypergraph. A panchromatic t-colouring of is a t-colouring of its vertices such that each edge has at least one vertex of each colour; and is panchromatically t-choosable if, whenever each vertex is given a list of t colours, the vertices can be coloured from their lists in such a way that each edge receives at least t different colours. The Hall ratio of is . Among other results, it is proved here that if every edge has at least t vertices and whenever , then is panchromatically t-choosable, and this condition is sharp; the minimum such that every t-uniform hypergraph with is panchromatically t-choosable satisfies ; and except possibly when t = 3 or 5, a t-uniform hypergraph is panchromatically t-colourable if whenever , and this condition is sharp. This last result dualizes to a sharp sufficient condition for the chromatic index of a hypergraph to equal its maximum degree. Received November 10, 1998 RID="*" ID="*" This work was carried out while the first author was visiting Nottingham, funded by Visiting Fellowship Research Grant GR/L54585 from the Engineering and Physical Sciences Research Council. The work of this author was also partly supported by grants 96-01-01614 and 97-01-01075 of the Russian Foundation for Fundamental Research.     

An Approximate Max-Steiner-Tree-Packing Min-Steiner-Cut Theorem*

Given an undirected multigraph G and a subset of vertices S ⊆ V (G), the STEINER TREE PACKING problem is to find a largest collection of edge-disjoint trees that each connects S. This problem and its generalizations have attracted considerable attention from researchers in different areas because of their wide applicability. This problem was shown to be APX-hard (no polynomial time approximation scheme unless P=NP). In fact, prior to this paper, not even an approximation algorithm with asymptotic ratio o(n) was known despite several attempts. In this work, we present the first polynomial time constant factor approximation algorithm for the STEINER TREE PACKING problem. The main theorem is an approximate min-max relation between the maximum number of edge-disjoint trees that each connects S (S-trees) and the minimum size of an edge-cut that disconnects some pair of vertices in S (S-cut). Specifically, we prove that if every S-cut in G has at least 26k edges, then G has at least k edge-disjoint S-trees; this answers Kriesells conjecture affirmatively up to a constant multiple. * A preliminary version appeared in the Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science (FOCS) 2004. † The author was supported by an Ontario Graduate Scholarship and a University of Toronto Fellowship.     

An analysis of a Monte Carlo algorithm for estimating the permanent

Karmarkar, Karp, Lipton, Lovász, and Luby proposed a Monte Carlo algorithm for approximating the permanent of a non-negativen×n matrix, which is based on an easily computed, unbiased estimator. It is not difficult to construct 0,1-matrices for which the variance of this estimator is very large, so that an exponential number of trials is necessary to obtain a reliable approximation that is within a constant factor of the correct value. Nevertheless, the same authors conjectured that for a random 0,1-matrix the variance of the estimator is typically small. The conjecture is shown to be true; indeed, for almost every 0,1-matrixA, just O(nw(n)e ^-2) trials suffice to obtain a reliable approximation to the permanent ofA within a factor 1±ɛ of the correct value. Here ω(n) is any function tending to infinity asn→∞. This result extends to random 0,1-matrices with density at leastn ^−1/2ω(n). It is also shown that polynomially many trials suffice to approximate the permanent of any dense 0,1-matrix, i.e., one in which every row- and column-sum is at least (1/2+α)n, for some constant α>0. The degree of the polynomial bounding the number of trials is a function of α, and increases as α→0. Supported by NSF grant CCR-9225008. The work described here was partly carried out while the author was visiting Princeton University as a guest of DIMACS (Center for Discrete Mathematics and Computer Science).     

On the Knowledge Complexity of

We show that if a language has an interactive proof of logarithmic statistical knowledge-complexity, then it belongs to the class . Thus, if the polynomial time hierarchy does not collapse, then -complete languages do not have logarithmic knowledge complexity. Prior to this work, there was no indication that would contradict languages being proven with even one bit of knowledge. Our result is a common generalization of two previous results: The first asserts that statistical zero knowledge is contained in [11, 2], while the second asserts that the languages recognizable in logarithmic statistical knowledge complexity are in [19]. Next, we consider the relation between the error probability and the knowledge complexity of an interactive proof. Note that reducing the error probability via repetition is not free: it may increase the knowledge complexity. We show that if the negligible error probability is less than (where k(n) is the knowledge complexity) then the language proven is in the third level of the polynomial time hierarchy (specifically, it is in ). In the standard setting of negligible error probability, there exist PSPACE-complete languages which have sub-linear knowledge complexity. However, if we insist, for example, that the error probability is less than , then PSPACE-complete languages do not have sub-quadratic knowledge complexity, unless . In order to prove our main result, we develop an AM protocol for checking that a samplable distribution D has a given entropy h. For any fractions , the verifier runs in time polynomial in and and fails with probability at most to detect an additive error in the entropy. We believe that this protocol is of independent interest. Subsequent to our work Goldreich and Vadhan [16] established that the problem of comparing the entropies of two samplable distributions if they are noticeably different is a natural complete promise problem for the class of statistical zero knowledge (). Received January 6, 2000 RID=" " ID=" " This research was performed while the authors were visiting the Computer Science Department at the University of Toronto, preliminary version of this paper appeared in [27] RID="*" ID="*" Partially supported by Technion V. P. R. Found––N. Haar and R. Zinn Research Fund. RID="†" ID="†" Partially supported by the grants OTKA T-029255, T-030059, FKFP 0607/1999, and AKP 2000-78 2.1.     

Ramsey-type Theorems with Forbidden Subgraphs

Dedicated to the memory of Paul Erdős   A graph is called H-free if it contains no induced copy of H. We discuss the following question raised by Erdős and Hajnal. Is it true that for every graph H, there exists an such that any H-free graph with n vertices contains either a complete or an empty subgraph of size at least ? We answer this question in the affirmative for a special class of graphs, and give an equivalent reformulation for tournaments. In order to prove the equivalence, we establish several Ramsey type results for tournaments. Received August 22, 1999 RID="*" ID="*" Supported by a USA Israeli BSF grant, by a grant from the Israel Science Foundation and by the Hermann Minkowski Minerva Center for Geometry at Tel Aviv University. RID="†" ID="†" Supported by NSF grant CR-9732101, PSC-CUNY Research Award 663472, and OTKA-T-020914. RID="‡" ID="‡" Supported by TKI grant Stochastics@TUB, and OTKA-T-026203.     

An Approximation Scheme For Cake Division With A Linear Number Of Cuts

In the cake cutting problem, n≥2 players want to cut a cake into n pieces so that every player gets a ‘fair’ share of the cake by his own measure. We prove the following result: For every ε>0, there exists a cake division scheme for n players that uses at most c_εn cuts, and in which each player can enforce to get a share of at least (1-ε)/n of the cake according to his own private measure. * Partially supported by Institute for Theoretical Computer Science, Prague (project LN00A056 of MŠMT ČR) and grant IAA1019401 of GA AV ČR.     

Efficient construction of a small hitting set for combinatorial rectangles in high dimension

We describe a deterministic algorithm which, on input integersd, m and real number (0,1), produces a subset S of [m] ^d ={1,2,3,...,m} ^d that hits every combinatorial rectangle in [m] ^d of volume at least , i.e., every subset of [m] ^d the formR ₁×R ₂×...×R _d of size at least m ^d . The cardinality of S is polynomial inm(logd)/, and the time to construct it is polynomial inmd/. The construction of such sets has applications in derandomization methods based on small sample spaces for general multivalued random variables.A preliminary version of this paper appeared in Proceedings of the 25th Annual ACM Symposium on Theory of Computing, 1993.Research partially done while visiting the International Computer Science Institute. Research supported in part by a grant from the Israel-USA Binational Science Foundation.A large portion of this research was done while still at the International Computer Science Institute in Berkeley, California. Research supported in part by National Science Foundation operating grants CCR-9304722 and NCR-9416101, and United States-Israel Binational Science Foundation grant No. 92-00226.Supported in part by NSF under grants CCR-8911388 and CCR-9215293 and by AFOSR grants AFOSR-89-0512 AFOSR-90-0008, and by DIMACS, which is supported by NSF grant STC-91-19999 and by the New Jersey Commission on Science and Technology. Research partially done while visiting the International Computer Science Institute.Partially supported by NSF NYI Grant No. CCR-9457799. Most of this research was done while the author was at MIT, partially supported by an NSF Postdoctoral Fellowship. Research partially done while visiting the International Computer Science Institute.     

Crossing families

Given a set of points in the plane, a crossing family is a collection of line segments, each joining two of the points, such that any two line segments intersect internally. Two setsA andB of points in the plane are mutually avoiding if no line subtended by a pair of points inA intersects the convex hull ofB, and vice versa. We show that any set ofn points in general position contains a pair of mutually avoiding subsets each of size at least . As a consequence we show that such a set possesses a crossing family of size at least , and describe a fast algorithm for finding such a family.Research supported in part by DARPA grant N00014-89-J-1988, Air Force AFOSR-89-0271, NSF grant DMS-8606225, and an ONR graduate fellowship. Further, part of this work was conducted at and supported by DIMACS (Center for Discrete Mathematics and Theoretical Computer Science), a National Science Foundation Science and Technology Center-NSF-STC8809648.     

Existence of Polyhedral Embeddings of Graphs

It is proved that the decision problem about the existence of an embedding of face-width 3 of a given graph is NP-complete. A similar result is proved for some related decision problems. This solves a problem raised by Neil Robertson. Received July 6, 1998 RID="*" ID="*" Supported in part by the Ministry of Science and Technology of Slovenia, Research Project J1–0502–0101–98.     

On the spanning trees of weighted graphs

Given a weighted graph, letW ₁,W ₂,W ₃,... denote the increasing sequence of all possible distinct spanning tree weights. Settling a conjecture due to Kano, we prove that every spanning tree of weightW ₁ is at mostk–1 edge swaps away from some spanning tree of weightW _k . Three other conjectures posed by Kano are proven for two special classes of graphs. Finally, we consider the algorithmic complexity of generating a spanning tree of weightW _k .This work was supported in part by a grant from the AT&T foundation and NSF grant DCR-8351757.Primarily supported by a 1967 Science and Engineering Scholarship from the Natural Sciences and Engineering Research Council of Canada.     

Computational complexity of stable partitions with B-preferences

Consider a special stable partition problem in which the player's preferences over sets to which she could belong are identical with her preferences over the most attractive member of a set and in case of indifference the set of smaller cardinality is preferred. If the preferences of all players over other (individual) players are strict, a strongly stable and a stable partition always exists. However, if ties are present, we show that both the existence problems are NP-complete. These results are very similar to what is known for the stable roommates problem. Received: July 2000/Revised: October 2002 RID="*" ID="*"  This work was supported by the Slovak Agency for Science, contract #1/7465/20 “Combinatorial Structures and Complexity of Algorithms”.     

Greedy linear extensions for minimizing bumps

Let P be a finite poset and let L={x ₁<...n} be a linear extension of P. A bump in L is an ordered pair (x _i , x _i+1) where x _ii+1 in P. The bump number of P is the least integer b(P), such that there exists a linear extension of P with b(P) bumps. We call L optimal if the number of bumps of L is b(P). We call L greedy if x _{i j} for every j>i, whenever (x _i, x _i+1) is a bump. A poset P is called greedy if every greedy linear extension of P is optimal. Our main result is that in a greedy poset every optimal linear extension is greedy. As a consequence, we prove that every greedy poset of bump number k is the linear sum of k+1 greedy posets, each of bump number zero.This research (Math/1406/31) was supported by the Research Center, College of Science, King Saud University, Riyadh, Saudi Arabia.     

On Heilbronn’s Problem in Higher Dimension

Heilbronn conjectured that given arbitrary n points in the 2-dimensional unit square [0, 1]², there must be three points which form a triangle of area at most O(1/n²). This conjecture was disproved by a nonconstructive argument of Komlós, Pintz and Szemerédi [10] who showed that for every n there is a configuration of n points in the unit square [0, 1]² where all triangles have area at least (log n/n²). Considering a generalization of this problem to dimensions d3, Barequet [3] showed for every n the existence of n points in the d-dimensional unit cube [0, 1]^d such that the minimum volume of every simplex spanned by any (d+1) of these n points is at least (1/n^d). We improve on this lower bound by a logarithmic factor (log n).     

On Dwork's accessory parameter problem

We study the question which ordinary second order linear differential equation allows power series solutions whose p-adic radius of convergence is at least one, a question raised by B.Dwork. In particular we shall consider the case of Fuchsian equations with four singularities and local exponent differences 0. Received: 28 August 2000; final form: 20 November 2001/ Published online: 17 June 2002 RID="*" ID="*" Part of this work was supported by EPSRC grant L99920     

On Bipartite Graphs with Linear Ramsey Numbers

Dedicated to the memory of Paul Erdős We provide an elementary proof of the fact that the ramsey number of every bipartite graph H with maximum degree at most is less than . This improves an old upper bound on the ramsey number of the n-cube due to Beck, and brings us closer toward the bound conjectured by Burr and Erdős. Applying the probabilistic method we also show that for all and there exists a bipartite graph with n vertices and maximum degree at most whose ramsey number is greater than for some absolute constant c>1. Received December 1, 1999 RID="*" ID="*" Supported by NSF grant DMS-9704114 RID="**" ID="**" Supported by KBN grant 2 P03A 032 16     

On the power of interaction

LetIP[f(n)] be the class of languages recognized by interactive proofs withf(¦x¦) interactions. Babai [2] showed that all languages recognized by interactive proofs with a bounded number of interactions can be recognized by interactive proofs with only two interactions; i.e., for every constantk, IP[k] collapses toIP[2].In this paper, we give evidence that interactive proofs with an unbounded number of interactions may be more powerful than interactive proofs with a bounded number of interactions. We show that for any polynomially bounded polynomial time computable functionf(n) and anyg(n)=o(f(n)) there exists an oracleB such thatIP ^B [f(n)] ₌ IP ^B [g(n)].The techniques employed are extensions of the techniques for proving lower bounds on small depth circuits used in [6], [14] and [10].Research done while in the Department of Mathematics at M. I. T. and supported by an ONR graduate fellowship.Supported in part by NSF Grant DCR MCS8509905.Research done while at the Laboratory for Computer Science at M. I. T. and Supported by an IBM fellowship.     

The Erdős–Pósa Property for Odd Cycles in Highly Connected Graphs

Dedicated to the memory of Paul Erdős In [9] Thomassen proved that a -connected graph either contains k vertex disjoint odd cycles or an odd cycle cover containing at most 2k-2 vertices, i.e. he showed that the Erdős–Pósa property holds for odd cycles in highly connected graphs. In this paper, we will show that the above statement is still valid for 576k-connected graphs which is essentially best possible. Received November 17, 1999 RID="*" ID="*" This work was supported by a post-doctoral DONET grant. RID="†" ID="†" This work was supported by an NSF-CNRS collaborative research grant. RID="‡" ID="‡" This work was performed while both authors were visiting the LIRMM, Université de Montpellier II, France.