In the present paper, we study algorithmic questions for the arc-intersection graph of directed paths on a tree. Such graphs are known to be perfect (proved by Monma and Wei in 1986). We present faster algorithms than all previously known algorithms for solving the minimum coloring and the minimum clique cover problems. They both run in $O(np)$ time, where $n$ is the number of vertices of the tree and $p$ the number of paths. Another result is a polynomial algorithm computing a kernel in the intersection graph, when its edges are oriented in a clique-acyclic way. Indeed, such a kernel exists for any perfect graph by a theorem of Boros and Gurvich. Such algorithms computing kernels are known only for few classes of perfect graphs.
In the minimum Multicut problem, the input is an edge-weighted supply graph $G=(V,E)$ and a simple demand graph $H=(V,F)$. Either $G$ and $H$ are directed (DMulC) or both are undirected (UMulC). The goal is to remove a minimum weight set of edges in $G$ such that there is no path from $s$ to $t$ in the remaining graph for any $(s,t) \in F$. UMulC admits an $O(\log k)$-approximation where $k$ is the vertex cover size of $H$ while the best known approximation for DMulC is $\min\{k, \tilde{O}(n^{11/23})\}$. These approximations are obtained by proving corresponding results on the multicommodity f…
We introduce the fractional version of oriented coloring and initiate its study. We prove some basic results and study the parameter for directed cycles and sparse planar graphs. In particular, we show that for every $ε> 0$, there exists an integer $g_ε \geq 12$ such that any oriented planar graph having girth at least $g_ε$ has fractional oriented chromatic number at most $4+ε$. Whereas, it is known that there exists an oriented planar graph having girth at least $g_ε$ with oriented chromatic number equal to $5$. We also study the fractional oriented chromatic number of directed cycles and pr…
In this paper, we investigate the manufacturing of vias in integrated circuits with a new technology combining lithography and Directed Self Assembly (DSA). Optimizing the production time and costs in this new process entails minimizing the number of lithography steps, which constitutes a generalization of graph coloring. We develop integer programming formulations for several variants of interest in the industry, and then study the computational performance of our formulations on true industrial instances. We show that the best integer programming formulation achieves good computational perfo…
We investigate structural and algorithmic advantages of a directed version of the well-researched class of distance-hereditary graphs. Since the previously defined distance-hereditary digraphs do not permit a recursive structure, we define directed twin-distance-hereditary graphs, which can be constructed by several twin and pendant vertex operations analogously to undirected distance-hereditary graphs and which still preserves the distance hereditary property. We give a characterization by forbidden induced subdigraphs and place the class in the hierarchy, comparing it to related classes. We …
The packing chromatic number $χ$ $ρ$ (G) of an undirected (resp. oriented) graph G is the smallest integer k such that its set of vertices V (G) can be partitioned into k disjoint subsets V 1,..., V k, in such a way that every two distinct vertices in V i are at distance (resp. directed distance) greater than i in G for every i, 1 $\le$ i $\le$ k. The generalized theta graph $Θ$ {\ell} 1,...,{\ell}p consists in two end-vertices joined by p $\ge$ 2 internally vertex-disjoint paths with respective lengths 1 $\le$ {\ell} 1 $\le$ . . . $\le$ {\ell} p. We prove that the packing chromatic number of …
We show a flow-augmentation algorithm in directed graphs: There exists a randomized polynomial-time algorithm that, given a directed graph $G$, two vertices $s,t \in V(G)$, and an integer $k$, adds (randomly) to $G$ a number of arcs such that for every minimal $st$-cut $Z$ in $G$ of size at most $k$, with probability $2^{-\mathrm{poly}(k)}$ the set $Z$ becomes a minimum $st$-cut in the resulting graph. We also provide a deterministic counterpart of this procedure. The directed flow-augmentation tool allows us to prove fixed-parameter tractability of a number of problems parameterized by the ca…
In this report, we describe a novel graph invariant for computational graphs (colored directed acylic graphs) and how we used it to generate all distinct computational graphs up to isomorphism for small graphs. The algorithm iteratively applies isomorphism-invariant operations, which take into account the graph structure and coloring, and outputs a fixed-length hash that is identical for all isomorphic computational graphs. While the algorithm cannot perfectly distinguish all pairs of non-isomorphic computational graphs, we suggest that it may be useful as a heuristic for comparing graphs.
The reversible pebble game is a combinatorial game played on rooted DAGs. This game was introduced by Bennett (1989) motivated by applications in designing space efficient reversible algorithms. Recently, Chan (2013) showed that the reversible pebble game number of any DAG is the same as its Dymond-Tompa pebble number and Raz-Mckenzie pebble number. We show, as our main result, that for any rooted directed tree T, its reversible pebble game number is always just one more than the edge rank coloring number of the underlying undirected tree U of T. It is known that given a DAG G as input, determ…
Motivated by applications in cancer genomics and following the work of Hajirasouliha and Raphael (WABI 2014), Hujdurović et al. (IEEE TCBB, to appear) introduced the minimum conflict-free row split (MCRS) problem: split each row of a given binary matrix into a bitwise OR of a set of rows so that the resulting matrix corresponds to a perfect phylogeny and has the minimum possible number of rows among all matrices with this property. Hajirasouliha and Raphael also proposed the study of a similar problem, in which the task is to minimize the number of distinct rows of the resulting matrix. Hujdur…
