223 research outputs found
On the Existence of Hamiltonian Paths for History Based Pivot Rules on Acyclic Unique Sink Orientations of Hypercubes
An acyclic USO on a hypercube is formed by directing its edges in such as way
that the digraph is acyclic and each face of the hypercube has a unique sink
and a unique source. A path to the global sink of an acyclic USO can be modeled
as pivoting in a unit hypercube of the same dimension with an abstract
objective function, and vice versa. In such a way, Zadeh's 'least entered rule'
and other history based pivot rules can be applied to the problem of finding
the global sink of an acyclic USO. In this paper we present some theoretical
and empirical results on the existence of acyclic USOs for which the various
history based pivot rules can be made to follow a Hamiltonian path. In
particular, we develop an algorithm that can enumerate all such paths up to
dimension 6 using efficient pruning techniques. We show that Zadeh's original
rule admits Hamiltonian paths up to dimension 9 at least, and prove that most
of the other rules do not for all dimensions greater than 5
Colouring Complete Multipartite and Kneser-type Digraphs
The dichromatic number of a digraph is the smallest such that can
be partitioned into acyclic subdigraphs, and the dichromatic number of an
undirected graph is the maximum dichromatic number over all its orientations.
Extending a well-known result of Lov\'{a}sz, we show that the dichromatic
number of the Kneser graph is and that the
dichromatic number of the Borsuk graph is if is large
enough. We then study the list version of the dichromatic number. We show that,
for any and , the list
dichromatic number of is . This extends a recent
result of Bulankina and Kupavskii on the list chromatic number of ,
where the same behaviour was observed. We also show that for any ,
and , the list dichromatic number of the complete
-partite graph with vertices in each part is , extending
a classical result of Alon. Finally, we give a directed analogue of Sabidussi's
theorem on the chromatic number of graph products.Comment: 15 page
On the oriented chromatic number of dense graphs
Let be a graph with vertices, edges, average degree , and maximum degree . The \emph{oriented chromatic number} of is the maximum, taken over all orientations of , of the minimum number of colours in a proper vertex colouring such that between every pair of colour classes all edges have the same orientation. We investigate the oriented chromatic number of graphs, such as the hypercube, for which . We prove that every such graph has oriented chromatic number at least . In the case that , this lower bound is improved to . Through a simple connection with harmonious colourings, we prove a general upper bound of \Oh{\Delta\sqrt{n}} on the oriented chromatic number. Moreover this bound is best possible for certain graphs. These lower and upper bounds are particularly close when is ()-regular for some constant , in which case the oriented chromatic number is between and
Almost optimal asynchronous rendezvous in infinite multidimensional grids
Two anonymous mobile agents (robots) moving in an asynchronous manner have to meet in an infinite grid of dimension δ> 0, starting from two arbitrary positions at distance at most d. Since the problem is clearly infeasible in such general setting, we assume that the grid is embedded in a δ-dimensional Euclidean space and that each agent knows the Cartesian coordinates of its own initial position (but not the one of the other agent). We design an algorithm permitting the agents to meet after traversing a trajectory of length O(d δ polylog d). This bound for the case of 2d-grids subsumes the main result of [12]. The algorithm is almost optimal, since the Ω(d δ) lower bound is straightforward. Further, we apply our rendezvous method to the following network design problem. The ports of the δ-dimensional grid have to be set such that two anonymous agents starting at distance at most d from each other will always meet, moving in an asynchronous manner, after traversing a O(d δ polylog d) length trajectory. We can also apply our method to a version of the geometric rendezvous problem. Two anonymous agents move asynchronously in the δ-dimensional Euclidean space. The agents have the radii of visibility of r1 and r2, respectively. Each agent knows only its own initial position and its own radius of visibility. The agents meet when one agent is visible to the other one. We propose an algorithm designing the trajectory of each agent, so that they always meet after traveling a total distance of O( ( d)), where r = min(r1, r2) and for r ≥ 1. r)δpolylog ( d r
Complexity and Algorithms for ISOMETRIC PATH COVER on Chordal Graphs and Beyond
A path is isometric if it is a shortest path between its endpoints. In this article, we consider the graph covering problem Isometric Path Cover, where we want to cover all the vertices of the graph using a minimum-size set of isometric paths. Although this problem has been considered from a structural point of view (in particular, regarding applications to pursuit-evasion games), it is little studied from the algorithmic perspective. We consider Isometric Path Cover on chordal graphs, and show that the problem is NP-hard for this class. On the positive side, for chordal graphs, we design a 4-approximation algorithm and an FPT algorithm for the parameter solution size. The approximation algorithm is based on a reduction to the classic path covering problem on a suitable directed acyclic graph obtained from a breadth first search traversal of the graph. The approximation ratio of our algorithm is 3 for interval graphs and 2 for proper interval graphs. Moreover, we extend the analysis of our approximation algorithm to k-chordal graphs (graphs whose induced cycles have length at most k) by showing that it has an approximation ratio of k+7 for such graphs, and to graphs of treelength at most ?, where the approximation ratio is at most 6?+2
A survey of graph burning
Graph burning is a deterministic, discrete-time process that models how
influence or contagion spreads in a graph. Associated to each graph is its
burning number, which is a parameter that quantifies how quickly the influence
spreads. We survey results on graph burning, focusing on bounds, conjectures,
and algorithms related to the burning number. We will discuss state-of-the-art
results on the burning number conjecture, burning numbers of graph classes, and
algorithmic complexity. We include a list of conjectures, variants, and open
problems on graph burning
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