8,948 research outputs found
Solving Hamiltonian Cycle by an EPT Algorithm for a Non-sparse Parameter
Many hard graph problems, such as Hamiltonian Cycle, become FPT when
parameterized by treewidth, a parameter that is bounded only on sparse graphs.
When parameterized by the more general parameter clique-width, Hamiltonian
Cycle becomes W[1]-hard, as shown by Fomin et al. [5]. S{\ae}ther and Telle
address this problem in their paper [13] by introducing a new parameter,
split-matching-width, which lies between treewidth and clique-width in terms of
generality. They show that even though graphs of restricted
split-matching-width might be dense, solving problems such as Hamiltonian Cycle
can be done in FPT time.
Recently, it was shown that Hamiltonian Cycle parameterized by treewidth is
in EPT [1, 6], meaning it can be solved in -time. In this
paper, using tools from [6], we show that also parameterized by
split-matching-width Hamiltonian Cycle is EPT. To the best of our knowledge,
this is the first EPT algorithm for any "globally constrained" graph problem
parameterized by a non-trivial and non-sparse structural parameter. To
accomplish this, we also give an algorithm constructing a branch decomposition
approximating the minimum split-matching-width to within a constant factor.
Combined, these results show that the algorithms in [13] for Edge Dominating
Set, Chromatic Number and Max Cut all can be improved. We also show that for
Hamiltonian Cycle and Max Cut the resulting algorithms are asymptotically
optimal under the Exponential Time Hypothesis
An approximation algorithm for the longest cycle problem in solid grid graphs
Although, the Hamiltonicity of solid grid graphs are polynomial-time
decidable, the complexity of the longest cycle problem in these graphs is still
open. In this paper, by presenting a linear-time constant-factor approximation
algorithm, we show that the longest cycle problem in solid grid graphs is in
APX. More precisely, our algorithm finds a cycle of length at least
in 2-connected -node solid grid graphs.
Keywords: Longest cycle, Hamiltonian cycle, Approximation algorithm, Solid
grid graph.Comment: 11 pages, 6 figure
When the Cut Condition is Enough: A Complete Characterization for Multiflow Problems in Series-Parallel Networks
Let be a supply graph and a demand graph defined on the
same set of vertices. An assignment of capacities to the edges of and
demands to the edges of 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 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 within the network with
capacities defined on . We prove a previous conjecture, which states that
when the supply graph is series-parallel, the pair is
cut-sufficient if and only if does not contain an \emph{odd spindle} as
a minor; that is, if it is impossible to contract edges of and delete edges
of and so that becomes the complete bipartite graph , with
odd, and is composed of a cycle connecting the vertices of
degree 2, and an edge connecting the two vertices of degree . 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
Inserting an Edge into a Geometric Embedding
The algorithm of Gutwenger et al. to insert an edge in linear time into a
planar graph with a minimal number of crossings on , is a helpful tool
for designing heuristics that minimize edge crossings in drawings of general
graphs. Unfortunately, some graphs do not have a geometric embedding
such that has the same number of crossings as the embedding .
This motivates the study of the computational complexity of the following
problem: Given a combinatorially embedded graph , compute a geometric
embedding that has the same combinatorial embedding as and that
minimizes the crossings of . We give polynomial-time algorithms for
special cases and prove that the general problem is fixed-parameter tractable
in the number of crossings. Moreover, we show how to approximate the number of
crossings by a factor , where is the maximum vertex degree
of .Comment: Appears in the Proceedings of the 26th International Symposium on
Graph Drawing and Network Visualization (GD 2018
2-Vertex Connectivity in Directed Graphs
We complement our study of 2-connectivity in directed graphs, by considering
the computation of the following 2-vertex-connectivity relations: We say that
two vertices v and w are 2-vertex-connected if there are two internally
vertex-disjoint paths from v to w and two internally vertex-disjoint paths from
w to v. We also say that v and w are vertex-resilient if the removal of any
vertex different from v and w leaves v and w in the same strongly connected
component. We show how to compute the above relations in linear time so that we
can report in constant time if two vertices are 2-vertex-connected or if they
are vertex-resilient. We also show how to compute in linear time a sparse
certificate for these relations, i.e., a subgraph of the input graph that has
O(n) edges and maintains the same 2-vertex-connectivity and vertex-resilience
relations as the input graph, where n is the number of vertices.Comment: arXiv admin note: substantial text overlap with arXiv:1407.304
Maximum Skew-Symmetric Flows and Matchings
The maximum integer skew-symmetric flow problem (MSFP) generalizes both the
maximum flow and maximum matching problems. It was introduced by Tutte in terms
of self-conjugate flows in antisymmetrical digraphs. He showed that for these
objects there are natural analogs of classical theoretical results on usual
network flows, such as the flow decomposition, augmenting path, and max-flow
min-cut theorems. We give unified and shorter proofs for those theoretical
results.
We then extend to MSFP the shortest augmenting path method of Edmonds and
Karp and the blocking flow method of Dinits, obtaining algorithms with similar
time bounds in general case. Moreover, in the cases of unit arc capacities and
unit ``node capacities'' the blocking skew-symmetric flow algorithm has time
bounds similar to those established in Even and Tarjan (1975) and Karzanov
(1973) for Dinits' algorithm. In particular, this implies an algorithm for
finding a maximum matching in a nonbipartite graph in time,
which matches the time bound for the algorithm of Micali and Vazirani. Finally,
extending a clique compression technique of Feder and Motwani to particular
skew-symmetric graphs, we speed up the implied maximum matching algorithm to
run in time, improving the best known bound
for dense nonbipartite graphs.
Also other theoretical and algorithmic results on skew-symmetric flows and
their applications are presented.Comment: 35 pages, 3 figures, to appear in Mathematical Programming, minor
stylistic corrections and shortenings to the original versio
Charting the Complexity Landscape of Waypoint Routing
Modern computer networks support interesting new routing models in which
traffic flows from a source s to a destination t can be flexibly steered
through a sequence of waypoints, such as (hardware) middleboxes or
(virtualized) network functions, to create innovative network services like
service chains or segment routing. While the benefits and technological
challenges of providing such routing models have been articulated and studied
intensively over the last years, much less is known about the underlying
algorithmic traffic routing problems. This paper shows that the waypoint
routing problem features a deep combinatorial structure, and we establish
interesting connections to several classic graph theoretical problems. We find
that the difficulty of the waypoint routing problem depends on the specific
setting, and chart a comprehensive landscape of the computational complexity.
In particular, we derive several NP-hardness results, but we also demonstrate
that exact polynomial-time algorithms exist for a wide range of practically
relevant scenarios
Contracting Graphs to Split Graphs and Threshold Graphs
We study the parameterized complexity of Split Contraction and Threshold
Contraction. In these problems we are given a graph G and an integer k and
asked whether G can be modified into a split graph or a threshold graph,
respectively, by contracting at most k edges. We present an FPT algorithm for
Split Contraction, and prove that Threshold Contraction on split graphs, i.e.,
contracting an input split graph to a threshold graph, is FPT when
parameterized by the number of contractions. To give a complete picture, we
show that these two problems admit no polynomial kernels unless NP\subseteq
coNP/poly.Comment: 14 pages, 4 figure
Claw-free t-perfect graphs can be recognised in polynomial time
A graph is called t-perfect if its stable set polytope is defined by
non-negativity, edge and odd-cycle inequalities. We show that it can be decided
in polynomial time whether a given claw-free graph is t-perfect
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