Disjoint cycles in directed graphs on the torus and the Klein bottle

We give necessary and sufficient conditions for a directed graph embedded on the torus or the Klein bottle to contain pairwise disjoint circuits, each of a given orientation and homotopy, and in a given order. For the Klein bottle, the theorem is new. For the torus, the theorem was proved before by P. D. Seymour. This paper gives a shorter proof of that result. © 1993 by Academic Press, Inc

Generalized Interlinked Cycle Cover for Index Coding

A source coding problem over a noiseless broadcast channel where the source is pre-informed about the contents of the cache of all receivers, is an index coding problem. Furthermore, if each message is requested by one receiver, then we call this an index coding problem with a unicast message setting. This problem can be represented by a directed graph. In this paper, we first define a structure (we call generalized interlinked cycles (GIC)) in directed graphs. A GIC consists of cycles which are interlinked in some manner (i.e., not disjoint), and it turns out that the GIC is a generalization of cliques and cycles. We then propose a simple scalar linear encoding scheme with linear time encoding complexity. This scheme exploits GICs in the digraph. We prove that our scheme is optimal for a class of digraphs with message packets of any length. Moreover, we show that our scheme can outperform existing techniques, e.g., partial clique cover, local chromatic number, composite-coding, and interlinked cycle cover.Comment: Extended version of the paper which is to be presented at the IEEE Information Theory Workshop (ITW), 2015 Jej

Packing Directed Cycles Quarter- and Half-Integrally

The celebrated Erd\H{o}s-P\'osa theorem states that every undirected graph that does not admit a family of $k$ vertex-disjoint cycles contains a feedback vertex set (a set of vertices hitting all cycles in the graph) of size $O(k \log k)$. After being known for long as Younger's conjecture, a similar statement for directed graphs has been proven in 1996 by Reed, Robertson, Seymour, and Thomas. However, in their proof, the dependency of the size of the feedback vertex set on the size of vertex-disjoint cycle packing is not elementary. We show that if we compare the size of a minimum feedback vertex set in a directed graph with the quarter-integral cycle packing number, we obtain a polynomial bound. More precisely, we show that if in a directed graph $G$ there is no family of $k$ cycles such that every vertex of $G$ is in at most four of the cycles, then there exists a feedback vertex set in $G$ of size $O(k^4)$. Furthermore, a variant of our proof shows that if in a directed graph $G$ there is no family of $k$ cycles such that every vertex of $G$ is in at most two of the cycles, then there exists a feedback vertex set in $G$ of size $O(k^6)$. On the way there we prove a more general result about quarter-integral packing of subgraphs of high directed treewidth: for every pair of positive integers $a$ and $b$, if a directed graph $G$ has directed treewidth $\Omega(a^6 b^8 \log^2(ab))$, then one can find in $G$ a family of $a$ subgraphs, each of directed treewidth at least $b$, such that every vertex of $G$ is in at most four subgraphs.Comment: Accepted to European Symposium on Algorithms (ESA '19

Packing Arc-Disjoint 4-Cycles in Oriented Graphs

Given a directed graph G and a positive integer k, the Arc Disjoint r-Cycle Packing problem asks whether G has k arc-disjoint r-cycles. We show that, for each integer r ? 3, Arc Disjoint r-Cycle Packing is NP-complete on oriented graphs with girth r. When r is even, the same result holds even when the input class is further restricted to be bipartite. On the positive side, focusing on r = 4 in oriented graphs, we study the complexity of the problem with respect to two parameterizations: solution size and vertex cover size. For the former, we give a cubic kernel with quadratic number of vertices. This is smaller than the compression size guaranteed by a reduction to the well-known 4-Set Packing. For the latter, we show fixed-parameter tractability using an unapparent integer linear programming formulation of an equivalent problem

Pre-Reduction Graph Products: Hardnesses of Properly Learning DFAs and Approximating EDP on DAGs

The study of graph products is a major research topic and typically concerns the term $f(G*H)$, e.g., to show that $f(G*H)=f(G)f(H)$. In this paper, we study graph products in a non-standard form $f(R[G*H]$ where $R$ is a "reduction", a transformation of any graph into an instance of an intended optimization problem. We resolve some open problems as applications. (1) A tight $n^{1-\epsilon}$-approximation hardness for the minimum consistent deterministic finite automaton (DFA) problem, where $n$ is the sample size. Due to Board and Pitt [Theoretical Computer Science 1992], this implies the hardness of properly learning DFAs assuming $NP\neq RP$ (the weakest possible assumption). (2) A tight $n^{1/2-\epsilon}$ hardness for the edge-disjoint paths (EDP) problem on directed acyclic graphs (DAGs), where $n$ denotes the number of vertices. (3) A tight hardness of packing vertex-disjoint $k$-cycles for large $k$. (4) An alternative (and perhaps simpler) proof for the hardness of properly learning DNF, CNF and intersection of halfspaces [Alekhnovich et al., FOCS 2004 and J. Comput.Syst.Sci. 2008]

Search for the end of a path in the d-dimensional grid and in other graphs

We consider the worst-case query complexity of some variants of certain \cl{PPAD}-complete search problems. Suppose we are given a graph $G$ and a vertex $s \in V(G)$. We denote the directed graph obtained from $G$ by directing all edges in both directions by $G'$. $D$ is a directed subgraph of $G'$ which is unknown to us, except that it consists of vertex-disjoint directed paths and cycles and one of the paths originates in $s$. Our goal is to find an endvertex of a path by using as few queries as possible. A query specifies a vertex $v\in V(G)$, and the answer is the set of the edges of $D$ incident to $v$, together with their directions. We also show lower bounds for the special case when $D$ consists of a single path. Our proofs use the theory of graph separators. Finally, we consider the case when the graph $G$ is a grid graph. In this case, using the connection with separators, we give asymptotically tight bounds as a function of the size of the grid, if the dimension of the grid is considered as fixed. In order to do this, we prove a separator theorem about grid graphs, which is interesting on its own right