22,080 research outputs found
Maximum Independent Set when excluding an induced minor: and
Dallard, Milani\v{c}, and \v{S}torgel [arXiv '22] ask if for every class
excluding a fixed planar graph as an induced minor, Maximum Independent Set
can be solved in polynomial time, and show that this is indeed the case when
is any planar complete bipartite graph, or the 5-vertex clique minus one
edge, or minus two disjoint edges. A positive answer would constitute a
far-reaching generalization of the state-of-the-art, when we currently do not
know if a polynomial-time algorithm exists when is the 7-vertex path.
Relaxing tractability to the existence of a quasipolynomial-time algorithm, we
know substantially more. Indeed, quasipolynomial-time algorithms were recently
obtained for the -vertex cycle, [Gartland et al., STOC '21] and the
disjoint union of triangles, [Bonamy et al., SODA '23].
We give, for every integer , a polynomial-time algorithm running in
when is the friendship graph ( disjoint edges
plus a vertex fully adjacent to them), and a quasipolynomial-time algorithm
running in when is (the
disjoint union of triangles and a 4-vertex cycle). The former extends a
classical result on graphs excluding as an induced subgraph [Alekseev,
DAM '07], while the latter extends Bonamy et al.'s result.Comment: 15 pages, 2 figure
Fast approximation of centrality and distances in hyperbolic graphs
We show that the eccentricities (and thus the centrality indices) of all
vertices of a -hyperbolic graph can be computed in linear
time with an additive one-sided error of at most , i.e., after a
linear time preprocessing, for every vertex of one can compute in
time an estimate of its eccentricity such that
for a small constant . We
prove that every -hyperbolic graph has a shortest path tree,
constructible in linear time, such that for every vertex of ,
. These results are based on an
interesting monotonicity property of the eccentricity function of hyperbolic
graphs: the closer a vertex is to the center of , the smaller its
eccentricity is. We also show that the distance matrix of with an additive
one-sided error of at most can be computed in
time, where is a small constant. Recent empirical studies show that
many real-world graphs (including Internet application networks, web networks,
collaboration networks, social networks, biological networks, and others) have
small hyperbolicity. So, we analyze the performance of our algorithms for
approximating centrality and distance matrix on a number of real-world
networks. Our experimental results show that the obtained estimates are even
better than the theoretical bounds.Comment: arXiv admin note: text overlap with arXiv:1506.01799 by other author
Approximation Algorithms for Connected Maximum Cut and Related Problems
An instance of the Connected Maximum Cut problem consists of an undirected
graph G = (V, E) and the goal is to find a subset of vertices S V
that maximizes the number of edges in the cut \delta(S) such that the induced
graph G[S] is connected. We present the first non-trivial \Omega(1/log n)
approximation algorithm for the connected maximum cut problem in general graphs
using novel techniques. We then extend our algorithm to an edge weighted case
and obtain a poly-logarithmic approximation algorithm. Interestingly, in stark
contrast to the classical max-cut problem, we show that the connected maximum
cut problem remains NP-hard even on unweighted, planar graphs. On the positive
side, we obtain a polynomial time approximation scheme for the connected
maximum cut problem on planar graphs and more generally on graphs with bounded
genus.Comment: 17 pages, Conference version to appear in ESA 201
Approximating Connected Maximum Cuts via Local Search
The Connected Max Cut (CMC) problem takes in an undirected graph G(V,E) and finds a subset S ? V such that the induced subgraph G[S] is connected and the number of edges connecting vertices in S to vertices in V?S is maximized. This problem is closely related to the Max Leaf Degree (MLD) problem. The input to the MLD problem is an undirected graph G(V,E) and the goal is to find a subtree of G that maximizes the degree (in G) of its leaves. [Gandhi et al. 2018] observed that an ?-approximation for the MLD problem induces an ?(?)-approximation for the CMC problem.
We present an ?(log log |V|)-approximation algorithm for the MLD problem via local search. This implies an ?(log log |V|)-approximation algorithm for the CMC problem. Thus, improving (exponentially) the best known ?(log |V|) approximation of the Connected Max Cut problem [Hajiaghayi et al. 2015]
Hardness of Exact Distance Queries in Sparse Graphs Through Hub Labeling
A distance labeling scheme is an assignment of bit-labels to the vertices of
an undirected, unweighted graph such that the distance between any pair of
vertices can be decoded solely from their labels. An important class of
distance labeling schemes is that of hub labelings, where a node
stores its distance to the so-called hubs , chosen so that for
any there is belonging to some shortest
path. Notice that for most existing graph classes, the best distance labelling
constructions existing use at some point a hub labeling scheme at least as a
key building block. Our interest lies in hub labelings of sparse graphs, i.e.,
those with , for which we show a lowerbound of
for the average size of the hubsets.
Additionally, we show a hub-labeling construction for sparse graphs of average
size for some , where is the
so-called Ruzsa-Szemer{\'e}di function, linked to structure of induced
matchings in dense graphs. This implies that further improving the lower bound
on hub labeling size to would require a
breakthrough in the study of lower bounds on , which have resisted
substantial improvement in the last 70 years. For general distance labeling of
sparse graphs, we show a lowerbound of , where is the communication complexity of the
Sum-Index problem over . Our results suggest that the best achievable
hub-label size and distance-label size in sparse graphs may be
for some
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