Gráfok és algoritmusok = Graphs and algorithms

A kutatás az elvárt eredménnyel zárult: tekintélyes nemzetközi konferenciákon és pubikációkban hoztuk nyilvánosságra az eredményéket, ideértve a STOC, SIAM és IEEE kiadványokat is, valamint egy könyvet is. A publikációk száma a matematikában elég magas (74). Ez nemzetközi összehasonlításban is kiemelkedő mutató a támogatás összegére vetítve. A projektben megmutattuk, hogy a gráfelmelet és a diszkrét matematika eszköztára számos helyen jól alkalmazható, ilyen terület a nagysebességű kommunikációs hálózatok tervezése, ezekben igen gyors routerek létrehozása. Egy másik terület a biológiai nagymolekulákon definiált gráfok és geometriai struktúrák. | The research concluded with the awaited results: in good international conferences and journals we published 74 works, including STOC conference, SIAM conferences and journals and one of the best IEEE journal. This number is high above average in mathematics research. We showed in the project that the tools of graph theory and discrete mathematics can be well applied in the high-speed communication network design, where we proposed fast and secure routing solutions. Additionally we also found applications in biological macromolecules

Superselectors: Efficient Constructions and Applications

We introduce a new combinatorial structure: the superselector. We show that superselectors subsume several important combinatorial structures used in the past few years to solve problems in group testing, compressed sensing, multi-channel conflict resolution and data security. We prove close upper and lower bounds on the size of superselectors and we provide efficient algorithms for their constructions. Albeit our bounds are very general, when they are instantiated on the combinatorial structures that are particular cases of superselectors (e.g., (p,k,n)-selectors, (d,\ell)-list-disjunct matrices, MUT_k(r)-families, FUT(k, a)-families, etc.) they match the best known bounds in terms of size of the structures (the relevant parameter in the applications). For appropriate values of parameters, our results also provide the first efficient deterministic algorithms for the construction of such structures

Kombinatorikus informatika = Combinatorial Computer Science

A [1, 3, 4, 7-9, 13, 14] eredmények közvetlenül kódokhoz kapcsolódnak. A [7, 9, 3] dolgozatokban bevezettük az egy felhasználót meghatározó superimposed kódokat, majd Alon és Körner eredményeit felhasználva sikerült kód korlátokat kapni. Vizsgáltuk az identifying kódókat véletlen gráfokban [1,8]. Itt (1 valószínüséggel) éles korlátokat kaptunk a minimális kód méretére. A kódokhoz kapcsolódó Turán rendszereket írtunk le a The CRC Handbook of Combinatorial Designs-ba [13]. Ezen kívül könyvet írunk a többszörös hozzáférésű csatornák kódolásáról, 250 oldal elkészült. A kódok korrelációs tulajdonságaihoz kapcsolódó véletlen metsző halmaz-rendszerek tulajdonságait (1 valószínüséggel) leírtuk adott részhalmaz méreten belül [5,6]. Az elméleti informatikában nagyon fontos Ramsey problémákat vizsgáltunk [2,10-12]. A Szemerédi lemma segítségével néhány régi Ramsey típusú nyitott problémát sikerült megoldani. Meghatároztuk az utak Ramsey számát pontosan három szín esetén (több, mint 25 éven át volt nyitott probléma), a körfedési számot pontosítottuk és teljes páros gráfok Ramsey szinezéseit is vizsgáltuk. | Our results in [1, 3, 4, 7-9, 13, 14] are related to codes. In [7, 9, 3] we introduced the single user tracing superimposed codes and using results of Alon, Körner we gave bounds on their minimum length. We investigated identifying codes in random graphs [1, 8]. We obtained (with probability 1) tight bounds on the minimum size of the code. We described the Turán systems related to codes [13]. We are writing a book on coding of multiple access channels, 250 pages are ready. The related to correlation properties of codes, random intersecting systems were described (with probability 1) in [5, 6] up to a certain subset size. We investigated some Ramsey problems [2,10-12] which in general are very important in theoretical computer science. Using the Szemerédi lemma we managed to solve some long standing open Ramsey problems. We determined exactly the Ramsey number of paths in case of three colors (this problem was open for more than 25 years), narrowed the bounds on cycle partition number and we also investigated the Ramsey colorings of bipartite graphs

Asymptotics of Fingerprinting and Group Testing: Capacity-Achieving Log-Likelihood Decoders

We study the large-coalition asymptotics of fingerprinting and group testing, and derive explicit decoders that provably achieve capacity for many of the considered models. We do this both for simple decoders (fast but suboptimal) and for joint decoders (slow but optimal), and both for informed and uninformed settings. For fingerprinting, we show that if the pirate strategy is known, the Neyman-Pearson-based log-likelihood decoders provably achieve capacity, regardless of the strategy. The decoder built against the interleaving attack is further shown to be a universal decoder, able to deal with arbitrary attacks and achieving the uninformed capacity. This universal decoder is shown to be closely related to the Lagrange-optimized decoder of Oosterwijk et al. and the empirical mutual information decoder of Moulin. Joint decoders are also proposed, and we conjecture that these also achieve the corresponding joint capacities. For group testing, the simple decoder for the classical model is shown to be more efficient than the one of Chan et al. and it provably achieves the simple group testing capacity. For generalizations of this model such as noisy group testing, the resulting simple decoders also achieve the corresponding simple capacities.Comment: 14 pages, 2 figure

放送型暗号の組合せ的構造及びマルチメディア指紋符号に関する進展

筑波大学 (University of Tsukuba)201

Efficient Probabilistic Group Testing Based on Traitor Tracing

Inspired by recent results from collusion-resistant traitor tracing, we provide a framework for constructing efficient probabilistic group testing schemes. In the traditional group testing model, our scheme asymptotically requires T ~ 2 K ln N tests to find (with high probability) the correct set of K defectives out of N items. The framework is also applied to several noisy group testing and threshold group testing models, often leading to improvements over previously known results, but we emphasize that this framework can be applied to other variants of the classical model as well, both in adaptive and in non-adaptive settings.Comment: 8 pages, 3 figures, 1 tabl

A comparison of three Algorithms for Tracing Nonlinear Equilibrium Paths of Structural Systems

The relative efficiencies of the Riks/Wempner, Crisfield, and normal flow solution algorithms for tracking nonlinear equilibrium paths of structural systems are compared. It is argued that the normal flow algorithm maybe both more computationally efficient and more robust compared to the other two algorithms when tracing the path through severe nonlinearities such as those associated with structural collapse. This is demonstrated qualitatively by comparing the relative behaviors of each algorithm in the vicinity of a severe nonlinearity. Quantitative results are presented for the collapse a blade stiffened panel