82 research outputs found
Reducing the clique and chromatic number via edge contractions and vertex deletions.
We consider the following problem: can a certain graph parameter of some given graph G be reduced by at least d, for some integer d, via at most k graph operations from some specified set S, for some given integer k? As graph parameters we take the chromatic number and the clique number. We let the set S consist of either an edge contraction or a vertex deletion. As all these problems are NP-complete for general graphs even if d is fixed, we restrict the input graph G to some special graph class. We continue a line of research that considers these problems for subclasses of perfect graphs, but our main results are full classifications, from a computational complexity point of view, for graph classes characterized by forbidding a single induced connected subgraph H
The Firefighter Problem: A Structural Analysis
We consider the complexity of the firefighter problem where b>=1 firefighters
are available at each time step. This problem is proved NP-complete even on
trees of degree at most three and budget one (Finbow et al.,2007) and on trees
of bounded degree b+3 for any fixed budget b>=2 (Bazgan et al.,2012). In this
paper, we provide further insight into the complexity landscape of the problem
by showing that the pathwidth and the maximum degree of the input graph govern
its complexity. More precisely, we first prove that the problem is NP-complete
even on trees of pathwidth at most three for any fixed budget b>=1. We then
show that the problem turns out to be fixed parameter-tractable with respect to
the combined parameter "pathwidth" and "maximum degree" of the input graph
Online Admission Control and Embedding of Service Chains
The virtualization and softwarization of modern computer networks enables the
definition and fast deployment of novel network services called service chains:
sequences of virtualized network functions (e.g., firewalls, caches, traffic
optimizers) through which traffic is routed between source and destination.
This paper attends to the problem of admitting and embedding a maximum number
of service chains, i.e., a maximum number of source-destination pairs which are
routed via a sequence of to-be-allocated, capacitated network functions. We
consider an Online variant of this maximum Service Chain Embedding Problem,
short OSCEP, where requests arrive over time, in a worst-case manner. Our main
contribution is a deterministic O(log L)-competitive online algorithm, under
the assumption that capacities are at least logarithmic in L. We show that this
is asymptotically optimal within the class of deterministic and randomized
online algorithms. We also explore lower bounds for offline approximation
algorithms, and prove that the offline problem is APX-hard for unit capacities
and small L > 2, and even Poly-APX-hard in general, when there is no bound on
L. These approximation lower bounds may be of independent interest, as they
also extend to other problems such as Virtual Circuit Routing. Finally, we
present an exact algorithm based on 0-1 programming, implying that the general
offline SCEP is in NP and by the above hardness results it is NP-complete for
constant L.Comment: early version of SIROCCO 2015 pape
Parameterized Inapproximability of Target Set Selection and Generalizations
In this paper, we consider the Target Set Selection problem: given a graph
and a threshold value for any vertex of the graph, find a minimum
size vertex-subset to "activate" s.t. all the vertices of the graph are
activated at the end of the propagation process. A vertex is activated
during the propagation process if at least of its neighbors are
activated. This problem models several practical issues like faults in
distributed networks or word-to-mouth recommendations in social networks. We
show that for any functions and this problem cannot be approximated
within a factor of in time, unless FPT = W[P],
even for restricted thresholds (namely constant and majority thresholds). We
also study the cardinality constraint maximization and minimization versions of
the problem for which we prove similar hardness results
Extension of Some Edge Graph Problems: Standard and Parameterized Complexity
Le PDF est une version auteur non publiée.We consider extension variants of some edge optimization problems in graphs containing the classical Edge Cover, Matching, and Edge Dominating Set problems. Given a graph G=(V,E) and an edge set U⊆E, it is asked whether there exists an inclusion-wise minimal (resp., maximal) feasible solution E′ which satisfies a given property, for instance, being an edge dominating set (resp., a matching) and containing the forced edge set U (resp., avoiding any edges from the forbidden edge set E∖U). We present hardness results for these problems, for restricted instances such as bipartite or planar graphs. We counter-balance these negative results with parameterized complexity results. We also consider the price of extension, a natural optimization problem variant of extension problems, leading to some approximation results
Most vital segment barriers
We study continuous analogues of "vitality" for discrete network flows/paths,
and consider problems related to placing segment barriers that have highest
impact on a flow/path in a polygonal domain. This extends the graph-theoretic
notion of "most vital arcs" for flows/paths to geometric environments. We give
hardness results and efficient algorithms for various versions of the problem,
(almost) completely separating hard and polynomially-solvable cases
The complexity of the Pk partition problem and related problems in bipartite graphs
International audienceIn this paper, we continue the investigation made in [MT05] about the approximability of Pk partition problems, but focusing here on their complexity. Precisely, we aim at designing the frontier between polynomial and NP-complete versions of the Pk partition problem in bipartite graphs, according to both the constant k and the maximum degree of the input graph. We actually extend the obtained results to more general classes of problems, namely, the minimum k-path partition problem and the maximum Pk packing problem. Moreover, we propose some simple approximation algorithms for those problems
Efficient algorithms for decomposing graphs under degree constraints
AbstractStiebitz [Decomposing graphs under degree constraints, J. Graph Theory 23 (1996) 321–324] proved that if every vertex v in a graph G has degree d(v)⩾a(v)+b(v)+1 (where a and b are arbitrarily given nonnegative integer-valued functions) then G has a nontrivial vertex partition (A,B) such that dA(v)⩾a(v) for every v∈A and dB(v)⩾b(v) for every v∈B. Kaneko [On decomposition of triangle-free graphs under degree constraints, J. Graph Theory 27 (1998) 7–9] and Diwan [Decomposing graphs with girth at least five under degree constraints, J. Graph Theory 33 (2000) 237–239] strengthened this result, proving that it suffices to assume d(v)⩾a+b (a,b⩾1) or just d(v)⩾a+b-1 (a,b⩾2) if G contains no cycles shorter than 4 or 5, respectively.The original proofs contain nonconstructive steps. In this paper we give polynomial-time algorithms that find such partitions. Constructive generalizations for k-partitions are also presented
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