Let M be a class of 0/1-matrices. A 0/1/?-matrix A where the ?s induce a submatrix is a probe matrix of M if the ?s in A can be replaced by 0s and 1s such that A becomes a member of M. We show that for M being the class of totally balanced matrices, it can be decided in polynomial time whether A is a probe totally balanced matrix. On our route toward proving this main result, we also prove that...

The Balanced Network Search (BNS) is an algorithm which finds a maximum balanced flow in a balanced network N. This algorithm is a way of using network flows to solve a number of standard problems, including maximum matchings, the factor problem, maximum capacitated b-matchings, etc., in general graphs. The value of a maximum balanced flow equals the capacity of a minimum balanced edge-cut. Flo...

A hole in a graph is an induced cycle on at least four vertices. A graph is Berge if it has no odd hole and if its complement has no odd hole. In 2002, Chudnovsky, Robertson, Seymour and Thomas proved a decomposition theorem for Berge graphs saying that every Berge graph either is in a well understood basic class or has some kind of decomposition. Then, Chudnovsky proved stronger theorems. One ...

Consider the balanced Ramsey game, in which a player has r colors and where in each round r random edges of an initially empty graph on n vertices are presented. The player has to immediately assign a different color to each edge and her goal is to avoid creating a monochromatic copy of some fixed graph F for as long as possible. The Achlioptas game is similar, but the player only loses when sh...

Previous studies on social networks are often focused on networks with only positive relations between individual nodes. As a significant extension, we conduct the spectral analysis on graphs with both positive and negative edges. Specifically, we investigate the impacts of introducing negative edges and examine patterns in the spectral space of the graph’s adjacency matrix. Our theoretical res...

This article discusses algorithms for scheduling task graphs G V E to LogP machines These algorithms depend on the granularity of G i e on the ratio of computation v and communication times in the LogP cost model and on the structure of G We de ne a class of coarse grained task graphs that can be scheduled with a performance guarantee of Topt G where Topt G is the time required for the optimal ...

A graph G is called a complete k-partite (k ≥ 2) graph if its vertices can be partitioned into k independent sets V1, . . . , Vk such that each vertex in Vi is adjacent to all the other vertices in Vj for 1 ≤ i < j ≤ k. A complete k-partite graph G is a complete balanced kpartite graph if |V1| = |V2| = · · · = |Vk|. An edge-coloring of a graph G with colors 1, . . . , t is an interval t-colorin...

For a fixed graph H, and a (larger) graph G, an H-factor of G is a vertex disjoint collection of copies of H, that cover all the vertices of G. We are interested in the threshold functions for the existence of an H-factor in the Erdős-Rényi random graph. The case where H is a single edge (a perfect matching) has been known since 1964 but even the case where H is a triangle is far more difficult...

Bruce Reed asks the following question: Can we determine whether a bipartite graph contains a chordless cycle whose length is a multiple of 4? We show that the two following more general questions are equivalent and we provide an answer. Given a bipartite graph G where each edge is assigned a weight + 1 or I, l determine whether G contains a cycle whose weight is a multiple of 4, l determine wh...

I would like to thank my advisor, Gerard Cornuéjols, for his support and his precious guidance. I also want to express my gratitude towards Michele Conforti, who introduced me to the field of combinatorics when I has an undergraduate student in Padova, and encouraged me to continue. Several parts of this dissertation resulted from working jointly with Michele Con-forti, Gerard Cornuéjols, and K...