23,214 research outputs found
Related Orderings of AT-Free Graphs
An ordering of a graph G is a bijection of V(G) to {1, . . . , |V(G)|}. In this thesis, we consider the complexity of two types of ordering problems. The first type of problem we consider aims at minimizing objective functions related to an ordering of the graph. We consider the problems Cutwidth, Imbalance, and Optimal Linear Arrangement. We also consider a problem of another type: S-End-Vertex, where S is one of the following search algorithms: breadth-first search (BFS), lexicographic breadth-first search (LBFS), depth-first search (DFS), and maximal neighbourhood search (MNS). This problem asks if a specified vertex can be the last vertex in an ordering generated by S. We show that, for each type of problem, orderings for one problem may be related to orderings for another problem of that type.
We show that there is always a cutwidth-minimal ordering where equivalence classes of true twins are grouped for any graph, where true twins are vertices with the same closed neighbourhood. This enables a fixed-parameter tractable (FPT) algorithm for Cutwidth on graphs parameterized by the edge clique cover number of the graph and a new parameter, the restricted twin cover number of the graph. The restricted twin cover number of the graph generalizes the vertex cover number of a graph, and is the smallest value k ≥ 0 such that there is a twin cover of the graph T and k−|T| non-trivial components of G−T.
We show that there is also always an imbalance-minimal ordering where equivalence classes of true twins are grouped for any graph. We show a polynomial time algorithm for this problem on superfragile graphs and subsets of proper interval graphs, both subsets of AT-free graphs. An asteroidal triple (AT) is a triple of independent vertices x, y, z such that between every pair of vertices in the triple, there is a path that does not intersect the closed neighbourhood of the third. A graph without an asteroidal triple is said to be AT-free. We also provide closed formulas for Imbalance on some small graph classes.
In the FPT setting, we improve algorithms for Imbalance parameterized by the vertex cover number of the input graph and show that the problem does not have a polynomially sized kernel for the same parameter number unless NP ⊆ coNP/poly.
We show that Optimal Linear Arrangement also has a polynomial algorithm for superfragile graphs and an FPT algorithm with respect to the restricted twin cover number.
Finally, we consider S-End-Vertex, for BFS, LBFS, DFS, and MNS. We perform the first systematic study of the problem on bipartite permutation graphs, a subset of AT-free graphs. We show that for BFS and MNS, the problem has a polynomial time solution. We improve previous results for LBFS, obtaining a linear time algorithm. For DFS, we establish a linear time algorithm. All the results follow from the linear structure of bipartite permutation graphs
Linear-time algorithms for scattering number and Hamilton-connectivity of interval graphs.
We prove that for all inline image an interval graph is inline image-Hamilton-connected if and only if its scattering number is at most k. This complements a previously known fact that an interval graph has a nonnegative scattering number if and only if it contains a Hamilton cycle, as well as a characterization of interval graphs with positive scattering numbers in terms of the minimum size of a path cover. We also give an inline image time algorithm for computing the scattering number of an interval graph with n vertices and m edges, which improves the previously best-known inline image time bound for solving this problem. As a consequence of our two results, the maximum k for which an interval graph is k-Hamilton-connected can be computed in inline image time
Lossy Kernelization
In this paper we propose a new framework for analyzing the performance of
preprocessing algorithms. Our framework builds on the notion of kernelization
from parameterized complexity. However, as opposed to the original notion of
kernelization, our definitions combine well with approximation algorithms and
heuristics. The key new definition is that of a polynomial size
-approximate kernel. Loosely speaking, a polynomial size
-approximate kernel is a polynomial time pre-processing algorithm that
takes as input an instance to a parameterized problem, and outputs
another instance to the same problem, such that . Additionally, for every , a -approximate solution
to the pre-processed instance can be turned in polynomial time into a
-approximate solution to the original instance .
Our main technical contribution are -approximate kernels of
polynomial size for three problems, namely Connected Vertex Cover, Disjoint
Cycle Packing and Disjoint Factors. These problems are known not to admit any
polynomial size kernels unless . Our approximate
kernels simultaneously beat both the lower bounds on the (normal) kernel size,
and the hardness of approximation lower bounds for all three problems. On the
negative side we prove that Longest Path parameterized by the length of the
path and Set Cover parameterized by the universe size do not admit even an
-approximate kernel of polynomial size, for any , unless
. In order to prove this lower bound we need to combine
in a non-trivial way the techniques used for showing kernelization lower bounds
with the methods for showing hardness of approximationComment: 58 pages. Version 2 contain new results: PSAKS for Cycle Packing and
approximate kernel lower bounds for Set Cover and Hitting Set parameterized
by universe siz
Welfare Maximization with Deferred Acceptance Auctions in Reallocation Problems
We design approximate weakly group strategy-proof mechanisms for resource
reallocation problems using Milgrom and Segal's deferred acceptance auction
framework: the radio spectrum and network bandwidth reallocation problems in
the procurement auction setting and the cost minimization problem with set
cover constraints in the selling auction setting. Our deferred acceptance
auctions are derived from simple greedy algorithms for the underlying
optimization problems and guarantee approximately optimal social welfare (cost)
of the agents retaining their rights (contracts). In the reallocation problems,
we design procurement auctions to purchase agents' broadcast/access rights to
free up some of the resources such that the unpurchased rights can still be
exercised with respect to the remaining resources. In the cost minimization
problem, we design a selling auction to sell early termination rights to agents
with existing contracts such that some minimal constraints are still satisfied
with remaining contracts. In these problems, while the "allocated" agents
transact, exchanging rights and payments, the objective and feasibility
constraints are on the "rejected" agents.Comment: A short version to appear in the 23rd European Symposium on
Algorithms (ESA 2015
A More Fine-Grained Complexity Analysis of Finding the Most Vital Edges for Undirected Shortest Paths
We study the NP-hard Shortest Path Most Vital Edges problem arising in the
context of analyzing network robustness. For an undirected graph with positive
integer edge lengths and two designated vertices and , the goal is to
delete as few edges as possible in order to increase the length of the (new)
shortest -path as much as possible. This scenario has been studied from the
viewpoint of parameterized complexity and approximation algorithms. We
contribute to this line of research by providing refined computational
tractability as well as hardness results. We achieve this by a systematic
investigation of various problem-specific parameters and their influence on the
computational complexity. Charting the border between tractability and
intractability, we also identify numerous challenges for future research
Optimizing Budget Allocation in Graphs
In the classical facility location problem we consider a graph with fixed
weights on the edges of . The goal is then to find an optimal positioning
for a set of facilities on the graph with respect to some objective function.
We introduce a new framework for facility location problems, where the weights
on the graph edges are not fixed, but rather should be assigned. The goal is to
find a valid assignment for which the resulting weighted graph optimizes the
facility location objective function. We present algorithms for finding the
optimal {\em budget allocation} for the center point problem and for the median
point problem on trees. Our algorithms run in linear time, both for the case
where a candidate vertex is given as part of the input, and for the case where
finding a vertex that optimizes the solution is part of the problem. We also
present a hardness result for the general graph case of the center point
problem, followed by an approximation algorithm on graphs - with
general metric spaces
Egalitarian Graph Orientations
Given an undirected graph, one can assign directions to each of the edges of
the graph, thus orienting the graph. To be as egalitarian as possible, one may
wish to find an orientation such that no vertex is unfairly hit with too many
arcs directed into it. We discuss how this objective arises in problems
resulting from telecommunications. We give optimal, polynomial-time algorithms
for: finding an orientation that minimizes the lexicographic order of the
indegrees and finding a strongly-connected orientation that minimizes the
maximum indegree. We show that minimizing the lexicographic order of the
indegrees is NP-hard when the resulting orientation is required to be acyclic
Hydras: Directed Hypergraphs and Horn Formulas
We introduce a new graph parameter, the hydra number, arising from the
minimization problem for Horn formulas in propositional logic. The hydra number
of a graph is the minimal number of hyperarcs of the form
required in a directed hypergraph , such that for
every pair , the set of vertices reachable in from is
the entire vertex set if , and it is otherwise.
Here reachability is defined by forward chaining, a standard marking algorithm.
Various bounds are given for the hydra number. We show that the hydra number
of a graph can be upper bounded by the number of edges plus the path cover
number of the line graph of a spanning subgraph, which is a sharp bound in
several cases. On the other hand, we construct single-headed graphs for which
that bound is off by a constant factor. Furthermore, we characterize trees with
low hydra number, and give a lower bound for the hydra number of trees based on
the number of vertices that are leaves in the tree obtained from by
deleting its leaves. This bound is sharp for some families of trees. We give
bounds for the hydra number of complete binary trees and also discuss a related
minimization problem.Comment: 17 pages, 4 figure
Minimizing Branching Vertices in Distance-preserving Subgraphs
It is -hard to determine the minimum number of branching
vertices needed in a single-source distance-preserving subgraph of an
undirected graph. We show that this problem can be solved in polynomial time if
the input graph is an interval graph.
In earlier work, it was shown that every interval graph with terminal
vertices admits an all-pairs distance-preserving subgraph with
branching vertices. We consider graphs that can be expressed as the strong
product of two interval graphs, and present a polynomial time algorithm that
takes such a graph with terminals as input, and outputs an all-pairs
distance-preserving subgraph of it with branching vertices. This bound
is tight.Comment: 21 pages, 6 figure
A Survey of Shortest-Path Algorithms
A shortest-path algorithm finds a path containing the minimal cost between
two vertices in a graph. A plethora of shortest-path algorithms is studied in
the literature that span across multiple disciplines. This paper presents a
survey of shortest-path algorithms based on a taxonomy that is introduced in
the paper. One dimension of this taxonomy is the various flavors of the
shortest-path problem. There is no one general algorithm that is capable of
solving all variants of the shortest-path problem due to the space and time
complexities associated with each algorithm. Other important dimensions of the
taxonomy include whether the shortest-path algorithm operates over a static or
a dynamic graph, whether the shortest-path algorithm produces exact or
approximate answers, and whether the objective of the shortest-path algorithm
is to achieve time-dependence or is to only be goal directed. This survey
studies and classifies shortest-path algorithms according to the proposed
taxonomy. The survey also presents the challenges and proposed solutions
associated with each category in the taxonomy
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