Primal-Dual Approximation Algorithms for Feedback Problems in Planar Graphs

Received: August 22, 1996     

The Shannon Capacity of a Union

For an undirected graph , let denote the graph whose vertex set is in which two distinct vertices and are adjacent iff for all i between 1 and n either or . The Shannon capacity c(G) of G is the limit , where is the maximum size of an independent set of vertices in . We show that there are graphs G and H such that the Shannon capacity of their disjoint union is (much) bigger than the sum of their capacities. This disproves a conjecture of Shannon raised in 1956. Received: December 8, 1997     

OnQ-matrices

In a recent paper [1], Aganagic and Cottle have established a constructive characterization for aP ₀-matrix to be aQ-matrix. Among the principal results in this paper, we show that the same characterization holds for anL-matrix as well, and that the symmetric copositive-plusQ-matrices are precisely those which are strictly copositive.     

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.     

The Circular Flow Number of a 6-Edge Connected Graph is Less Than Four

We show that every 6-edge connected graph admits a circulation whose range lies in the interval [1,3). Received March 29, 2000 RID="*" ID="*" Supported by NATO-CNR Fellowship; this work was done while the author was visiting the Dept. of Mathematics and Statistics at Simon Fraser University, Canada. RID="†" ID="†" Supported by a National Sciences and Engineering Research Council Research Grant     

Perfect Graphs, Partitionable Graphs and Cutsets

We prove a theorem about cutsets in partitionable graphs that generalizes earlier results on amalgams, 2-amalgams and homogeneous pairs. Received December 13, 1999 RID="*" ID="*" This work was supported in part by the Fields Institute for Research in Mathematical Sciences, Toronto, Canada, and by NSF grants DMI-0098427 and DMI-9802773 and ONR grant N00014-97-1-0196.     

Convex Sets in the Plane with Three of Every Four Meeting

Dedicated to the memory of Paul Erdős Suppose we have a finite collection of closed convex sets in the plane, (which without loss of generality we can take to be polygons). Suppose further that among any four of them, some three have non-empty intersection. We show that 13 points are sufficient to meet every one of the convex sets. Received October 27, 1999/Revised April 11, 2000 RID="*" ID="*" Supported by grant OTKA-T-029074. RID="†" ID="†" Supported by NSF grant DMS-99-70071, OTKA-T-020914 and OTKA-F-22234.     

齐型空间上Littlewood-paleyg算子的加权BMO有界性

证明了齐型空间上由ε—算子族定义的Littlewood—paley g算子的加权BMO有界性。     

对社会主义初级阶段阶级、阶级矛盾的再认识

社会主义初级阶段非公有制经济的存在和长期发展，导致新的资产阶级的产生。由于对生产资料关系的不同和经济利益上的矛盾．工人阶级和新的资产阶级之间的矛盾已成为一个不争的事实。但其矛盾有着特殊的性质．即属于人民内部的非对抗性矛盾。对于这种新的矛盾我们应当给予高度的重视，以避免矛盾性质的转化。政府只有高度重视并能正确地解决这一矛盾，才能更好地促进社会主义和谐社会的发展。     

On the Single-Source Unsplittable Flow Problem

be a capacitated directed graph with a source s and k terminals with demands , . We would like to concurrently route every demand on a single path from s to the corresponding terminal without violating the capacities. There are several interesting and important variations of this unsplittable flow problem. If the necessary cut condition is satisfied, we show how to compute an unsplittable flow satisfying the demands such that the total flow through any edge exceeds its capacity by at most the maximum demand. For graphs in which all capacities are at least the maximum demand, we therefore obtain an unsplittable flow with congestion at most 2, and this result is best possible. Furthermore, we show that all demands can be routed unsplittably in 5 rounds, i.e., all demands can be collectively satisfied by the union of 5 unsplittable flows. Finally, we show that 22.6% of the total demand can be satisfied unsplittably. These results are extended to the case when the cut condition is not necessarily satisfied. We derive a 2-approximation algorithm for congestion, a 5-approximation algorithm for the number of rounds and a -approximation algorithm for the maximum routable demand. Received: July 12, 1998