Variations of coloring problems related to scheduling and discrete tomography

The graph coloring problem is one of the most famous problems in graph theory and has a large range of applications. It consists in coloring the vertices of an undirected graph with a given number of colors such that two adjacent vertices get different colors. This thesis deals with some variations of this basic coloring problem which are related to scheduling and discrete tomography. These problems may also be considered as partitioning problems. In Chapter 1 basic definitions of computational complexity and graph theory are presented. An introduction to graph coloring and discrete tomography is given. In the next chapter we discuss two coloring problems in mixed graphs (i.e., graphs having edges and arcs) arising from scheduling. In the first one (strong mixed graph coloring problem) we have to cope with disjunctive constraints (some pairs of jobs cannot be processed simultaneously) as well as with precedence constraints (some pairs of jobs must be executed in a given order). It is known that this problem is NP-complete in mixed bipartite graphs. In this thesis we strengthen this result by proving that for k = 3 colors the strong mixed graph coloring problem is NP-complete even if the mixed graph is planar bipartite with maximum degree 4 and each vertex incident to at least one arc has maximum degree 2 or if the mixed graph is bipartite and has maximum degree 3. Furthermore we show that the problem is polynomially solvable in partial p-trees, for fixed p, as well as in general graphs with k = 2 colors. We also give upper bounds on the strong mixed chromatic number or even its exact value for some classes of graphs. In the second problem (weak mixed graph coloring problem), we allow jobs linked by precedence constraints to be executed at the same time. We show that for k = 3 colors this problem is NP-complete in mixed planar bipartite graphs of maximum degree 4 as well as in mixed bipartite graphs of maximum degree 3. Again, for partial p-trees, p fixed, and for general graphs with k = 2 colors, we prove that the weak mixed graph coloring problem is polynomially solvable. We consider in Chapter 3 the problem of characterizing in an undirected graph G = (V, E) a minimum set R of edges for which maximum matchings M can be found with specific values of p = |M ∩ R|. We obtain partial results for some classes of graphs and show in particular that for odd cacti with triangles only and for forests one can determine in polynomial time whether there exists a minimum set R for which there are maximum matchings M such that p= |R ∩ M|, for p= 0,1, ..., ν(G). The remaining chapters deal with some coloring (or partitioning) problems related to the basic image reconstruction problem in discrete tomography. In Chapter 4 we consider a generalization of the vertex coloring problem associated with the basic image reconstruction problem. We are given an undirected graph and a family of chains covering its vertices. For each chain the number of occurrences of each color is given. We then want to find a coloring respecting these occurrences. We are interested in both, arbitrary and proper colorings and give complexity results. In particular we show that for arbitrary colorings the problem is NP-complete with two colors even if the graph is a tree of maximum degree 3. We also consider the edge coloring version of both problems. Again we present some complexity results. We consider in Chapter 5 some generalized neighborhoods instead of chains. For each vertex x we are given the number of occurrences of each color in its open neighborhood Nd(x) (resp. closed neighborhood Nd+(x)), representing the set of vertices which are at distance d from x (resp. at distance at most d from x). We are interested in arbitrary colorings as well as proper colorings. We present some complexity results and we show in particular that for d = 1 the problems are polynomially solvable in trees using a dynamic programming approach. For the open neighborhood and d = 2 we obtain a polynomial time algorithm for quatrees (i.e. trees where all internal vertices have degree at least 4). We also examine the bounded version of these problems, i.e., instead of the exact number of occurrences of each color we are given upper bounds on these occurrences. In particular we show that the problem for proper colorings is NP-complete in bipartite graphs of maximum degree 3 with four colors and each color appearing at most once in the neighborhood N(x) of each vertex x. This result implies that the L(1,1)-labelling problem is NP-complete in this class of graphs for four colors. Finally in Chapter 6 we consider the edge partitioning version of the basic image reconstruction problem, i.e., we have to partition the edge set of a complete bipartite graph into k subsets such that for each vertex there must be a given number of edges of each set of the partition incident to this vertex. For k = 3 the complexity status is still open. Here we present a new solvable case for k = 3. Then we examine some variations where the union of two subsets E1, E2 has to satisfy some additional constraints as for example it must form a tree or a collection of disjoint chains. In both cases we give necessary and sufficient conditions for a solution to exist. We also consider the case where we have a complete graph instead of a complete bipartite graph. We show that the edge partitioning problem in a complete graph is at least as difficult as in a complete bipartite graph. We also give necessary and sufficient conditions for a solution to exist if E1 ∪ E2 form a tree or if they form a Hamiltonian cycle in the case of a complete graph. Finally we examine for both, complete and complete bipartite graphs, the case where each one of the sets E1 and E2 is structured (two disjoint Hamiltonian chains, two edge disjoint cycles) and present necessary and sufficient conditions

Partial DNA Assembly: A Rate-Distortion Perspective

Earlier formulations of the DNA assembly problem were all in the context of perfect assembly; i.e., given a set of reads from a long genome sequence, is it possible to perfectly reconstruct the original sequence? In practice, however, it is very often the case that the read data is not sufficiently rich to permit unambiguous reconstruction of the original sequence. While a natural generalization of the perfect assembly formulation to these cases would be to consider a rate-distortion framework, partial assemblies are usually represented in terms of an assembly graph, making the definition of a distortion measure challenging. In this work, we introduce a distortion function for assembly graphs that can be understood as the logarithm of the number of Eulerian cycles in the assembly graph, each of which correspond to a candidate assembly that could have generated the observed reads. We also introduce an algorithm for the construction of an assembly graph and analyze its performance on real genomes.Comment: To be published at ISIT-2016. 11 pages, 10 figure

On Linkedness of Cartesian Product of Graphs

We study linkedness of Cartesian product of graphs and prove that the product of an $a$-linked and a $b$-linked graphs is $(a+b-1)$-linked if the graphs are sufficiently large. Further bounds in terms of connectivity are shown. We determine linkedness of product of paths and product of cycles

Pairs of disjoint matchings and related classes of graphs

For a finite graph $G$, we study the maximum $2$-edge colorable subgraph problem and a related ratio $\frac{\mu(G)}{\nu(G)}$, where $\nu(G)$ is the matching number of $G$, and $\mu(G)$ is the size of the largest matching in any pair $(H,H')$ of disjoint matchings maximizing $|H| + |H'|$ (equivalently, forming a maximum $2$-edge colorable subgraph). Previously, it was shown that $\frac{4}{5} \le \frac{\mu(G)}{\nu(G)} \le 1$, and the class of graphs achieving $\frac{4}{5}$ was completely characterized. We show here that any rational number between $\frac{4}{5}$ and $1$ can be achieved by a connected graph. Furthermore, we prove that every graph with ratio less than $1$ must admit special subgraphs

Percolation on an infinitely generated group

We give an example of a long range Bernoulli percolation process on a group non-quasi-isometric with $\mathbb{Z}$, in which clusters are almost surely finite for all values of the parameter. This random graph admits diverse equivalent definitions, and we study their ramifications. We also study its expected size and point out certain phase transitions.Comment: 23 page

When the Cut Condition is Enough: A Complete Characterization for Multiflow Problems in Series-Parallel Networks

Let $G=(V,E)$ be a supply graph and $H=(V,F)$ a demand graph defined on the same set of vertices. An assignment of capacities to the edges of $G$ and demands to the edges of $H$ is said to satisfy the \emph{cut condition} if for any cut in the graph, the total demand crossing the cut is no more than the total capacity crossing it. The pair $(G,H)$ is called \emph{cut-sufficient} if for any assignment of capacities and demands that satisfy the cut condition, there is a multiflow routing the demands defined on $H$ within the network with capacities defined on $G$. We prove a previous conjecture, which states that when the supply graph $G$ is series-parallel, the pair $(G,H)$ is cut-sufficient if and only if $(G,H)$ does not contain an \emph{odd spindle} as a minor; that is, if it is impossible to contract edges of $G$ and delete edges of $G$ and $H$ so that $G$ becomes the complete bipartite graph $K_{2,p}$, with $p\geq 3$ odd, and $H$ is composed of a cycle connecting the $p$ vertices of degree 2, and an edge connecting the two vertices of degree $p$. We further prove that if the instance is \emph{Eulerian} --- that is, the demands and capacities are integers and the total of demands and capacities incident to each vertex is even --- then the multiflow problem has an integral solution. We provide a polynomial-time algorithm to find an integral solution in this case. In order to prove these results, we formulate properties of tight cuts (cuts for which the cut condition inequality is tight) in cut-sufficient pairs. We believe these properties might be useful in extending our results to planar graphs.Comment: An extended abstract of this paper will be published at the 44th Symposium on Theory of Computing (STOC 2012