2 research outputs found
Műszaki informatikai problémákhoz kapcsolódó diszkrét matematikai modellek vizsgálata = Discrete mathematical models related to problems in informatics
DiszkrĂ©t matematikai mĂłdszerekkel strukturális Ă©s kvantitatĂv összefĂĽggĂ©seket bizonyĂtottunk; algoritmusokat terveztĂĽnk, komplexitásukat elemeztĂĽk. Az eredmĂ©nyek a gráfok Ă©s hipergráfok elmĂ©letĂ©hez, valamint on-line ĂĽtemezĂ©shez kapcsolĂłdnak. NĂ©hány kiemelĂ©s: - Pontosan leĂrtuk azokat a szerkezeti feltĂ©teleket, amelyeknek teljesĂĽlni kell ahhoz, hogy egy kommunikáciĂłs hálĂłzatban Ă©s annak minden összefĂĽggĹ‘ rĂ©szĂ©ben legyen olyan, megadott tĂpusĂş összefĂĽggĹ‘ rĂ©szhálĂłzat, ahonnan az összes többi elem közvetlenĂĽl elĂ©rhetĹ‘. (A problĂ©ma kĂ©t Ă©vtizeden át megoldatlan volt.) - Aszimptotikusan pontos becslĂ©st adtunk egy n-elemű alaphalmaz olyan, k-asokbĂłl állĂł halmazrendszereinek minimális mĂ©retĂ©re, amelyekben minden k-osztályĂş partĂciĂłhoz van olyan halmaz, ami az összes partĂciĂł-osztályt metszi. (Nyitott problĂ©ma volt 1973 Ăłta, több szerzĹ‘ egymástĂłl fĂĽggetlenĂĽl is felvetette.) - Halmazrendszerek partĂciĂłira az eddigieknĂ©l általánosabb modellt vezettĂĽnk be, megvizsgáltuk rĂ©szosztályainak hierarchikus szerkezetĂ©t Ă©s hatĂ©kony algoritmusokat adtunk. (Sok alkalmazás várhatĂł az erĹ‘forrás-allokáciĂł terĂĽletĂ©n.) - Kidolgoztunk egy mĂłdszert, amellyel lokálisan vĂ©ges pozĂciĂłs játĂ©kok nyerĹ‘ stratĂ©giája megtalálhatĂł mindössze lineáris mĂ©retű memĂłria használatával. - FĂ©lig on-line ĂĽtemezĂ©si algoritmusokat terveztĂĽnk (kĂ©tgĂ©pes feladatra, nem azonos sebessĂ©gű processzorokra), amelyeknek versenykĂ©pessĂ©gi aránya bizonyĂtottan jobb, mint ami a legjobb teljesen on-line mĂłdszerekkel elĂ©rhetĹ‘. | Applying discrete mathematical methods, we proved structural and quantitative relations, designed algorithms and analyzed their complexity. The results deal with graph and hypergraph theory and on-line scheduling. Some selected ones are: - We described the exact structural conditions which have to hold in order that an intercommunication network and each of its connected parts contain a connected subnetwork of prescribed type, from which all the other nodes of the network can be reached via direct link. (This problem was open for two decades.) - We gave asymptotically tight estimates on the minimum size of set systems of k-element sets over an n-element set such that, for each k-partition of the set, the set system contains a k-set meeting all classes of the partition. (This was an open problem since 1973, raised by several authors independently.) - We introduced a new model, more general than the previous ones, for partitions of set systems. We studied the hierarchic structure of its subclasses, and designed efficient algorithms. (Many applications are expected in the area of resource allocation.) - We developed a method to learn winning strategies in locally finite positional games, which requires linear-size memory only. - We designed semi-online scheduling algorithms (for two uniform processors of unequal speed), whose competitive ratio provably beats the best possible one achievable in the purely on-line setting
Voloshin's conjecture for C-perfect hypertrees
In the “mixed hypergraph” model, proper coloring requires that vertex subsets of one type (called C-edges) should contain two vertices of the same color, while the other type (D-edges) should not be monochromatic. Voloshin [Australas. J. Combin. 11 (1995), 25–45] introduced the concept of C-perfectness, which can be viewed as a dual kind of graph perfectness in the classical sense, and proposed a characterization for C-perfect hypertrees without D-edges. (A hypergraph is called a hypertree if there exists a graph T which is a tree such that each hyperedge induces a subtree in T.) We prove that the structural characterization conjectured by Voloshin is valid indeed, and it can even be extended in a natural way to mixed hypertrees without (or, with only few) D-edges of size 2; but not to mixed hypertrees in general. The proof is constructive and leads to a fast coloring algorithm, too. In sharp contrast to perfect graphs which can be recognized in polynomial time, the recognition problem of C-perfect hypergraphs is pointed out to be co-NP-complete already on the class of C-hypertrees