9 research outputs found
Blazing a Trail via Matrix Multiplications: A Faster Algorithm for Non-Shortest Induced Paths
For vertices and of an -vertex graph , a -trail of is
an induced -path of that is not a shortest -path of . Berger,
Seymour, and Spirkl [Discrete Mathematics 2021] gave the previously only known
polynomial-time algorithm, running in time, to either output a
-trail of or ensure that admits no -trail. We reduce the
complexity to the time required to perform a poly-logarithmic number of
multiplications of Boolean matrices, leading to a largely
improved -time algorithm.Comment: 18 pages, 6 figures, a preliminary version appeared in STACS 202
Finding Large H-Colorable Subgraphs in Hereditary Graph Classes
We study the \textsc{Max Partial -Coloring} problem: given a graph ,
find the largest induced subgraph of that admits a homomorphism into ,
where is a fixed pattern graph without loops. Note that when is a
complete graph on vertices, the problem reduces to finding the largest
induced -colorable subgraph, which for is equivalent (by
complementation) to \textsc{Odd Cycle Transversal}.
We prove that for every fixed pattern graph without loops, \textsc{Max
Partial -Coloring} can be solved:
in -free graphs in polynomial time, whenever is a
threshold graph;
in -free graphs in polynomial time;
in -free graphs in time ;
in -free graphs in time
.
Here, is the number of vertices of the input graph and is
the maximum size of a clique in~. Furthermore, combining the mentioned
algorithms for -free and for -free
graphs with a simple branching procedure, we obtain subexponential-time
algorithms for \textsc{Max Partial -Coloring} in these classes of graphs.
Finally, we show that even a restricted variant of \textsc{Max Partial
-Coloring} is -hard in the considered subclasses of -free
graphs, if we allow loops on
Structure and coloring of (, , diamond)-free graphs
We use and to denote a path and a cycle on t vertices,
respectively. A diamond consists of two triangles that share exactly one edge,
a kite is a graph obtained from a diamond by adding a new vertex adjacent to a
vertex of degree 2 of the diamond, a paraglider is the graph that consists of a
plus a vertex adjacent to three vertices of the , a paw is a graph
obtained from a triangle by adding a pendant edge. A comparable pair
consists of two nonadjacent vertices and such that
or . A universal clique is a clique such that for any two vertices and . A blowup of a
graph H is a graph obtained by substituting a stable set for each vertex, and
correspondingly replacing each edge by a complete bipartite graph. We prove
that 1) there is a unique connected imperfect , kite,
paraglider)-free graph G with \delta(G) \geq \omega(G)+ 1 which has no clique
cutsets, no comparable pairs, and no universal cliques; 2) if G is a connected
imperfect , diamond)-free graph with \delta(G) \geq max{3,
\omega(G)} and without comparable pairs, then G is isomorphic to a graph of a
well defined 12 graph families; and 3) each connected imperfect ,
paw)-free graph is a blowup of . As consequences, we show that \chi(G)
\leq \omega(G)+1 if G is (P7, C5, kite, paraglider)-free, and \chi(G) \leq
max{3, \omega(G)} if G is , H)-free with H being a diamond or a paw.
We also show that \chi(G) \le
Improved Algorithms for Recognizing Perfect Graphs and Finding Shortest Odd and Even Holes
Various classes of induced subgraphs are involved in the deepest results of
graph theory and graph algorithms. A prominent example concerns the {\em
perfection} of that the chromatic number of each induced subgraph of
equals the clique number of . The seminal Strong Perfect Graph Theorem
confirms that the perfection of can be determined by detecting odd holes in
and its complement. Chudnovsky et al. show in 2005 an algorithm
for recognizing perfect graphs, which can be implemented to run in
time for the exponent of square-matrix
multiplication. We show the following improved algorithms.
1. The tractability of detecting odd holes was open for decades until the
major breakthrough of Chudnovsky et al. in 2020. Their algorithm is
later implemented by Lai et al. to run in time, leading to the best
formerly known algorithm for recognizing perfect graphs. Our first result is an
algorithm for detecting odd holes, implying an algorithm for
recognizing perfect graphs.
2. Chudnovsky et al. extend in 2021 the algorithms for detecting odd
holes (2020) and recognizing perfect graphs (2005) into the first polynomial
algorithm for obtaining a shortest odd hole, which runs in time. We
reduce the time for finding a shortest odd hole to .
3. Conforti et al. show in 1997 the first polynomial algorithm for detecting
even holes, running in about time. It then takes a line of
intensive efforts in the literature to bring down the complexity to
, , , and finally . On the other hand,
the tractability of finding a shortest even hole has been open for 16 years
until the very recent algorithm of Cheong and Lu in 2022. We
improve the time of finding a shortest even hole to .Comment: 29 pages, 5 figure
Finding Large H-Colorable Subgraphs in Hereditary Graph Classes
âFirst Published in SIAM Journal on Discrete Mathematics in 35, 4, 2021, published by the Society for Industrial and Applied Mathematics (SIAM)â and the copyright notice as stated in the article itself (e.g., âCopyright © by SIAM. Unauthorized reproduction of this article is prohibited.â')We study the Max Partial H-Coloring problem: given a graph G, find the largest induced subgraph of G that admits a homomorphism into H, where H is a fixed pattern graph without loops. Note that when H is a complete graph on k vertices, the problem reduces to finding the largest induced k-colorable subgraph, which for k=2 is equivalent (by complementation) to Odd Cycle Transversal. We prove that for every fixed pattern graph H without loops, Max Partial H-Coloring can be solved in {P5,F}-free graphs in polynomial time, whenever F is a threshold graph; in {P5,bull}-free graphs in polynomial time; in P5-free graphs in time nO(Ï(G)); and in {P6,1âsubdividedclaw}-free graphs in time nO(Ï(G)3). Here, n is the number of vertices of the input graph G and Ï(G) is the maximum size of a clique in G. Furthermore, by combining the mentioned algorithms for P5-free and for {P6,1âsubdividedclaw}-free graphs with a simple branching procedure, we obtain subexponential-time algorithms for Max Partial H-Coloring in these classes of graphs. Finally, we show that even a restricted variant of Max Partial H-Coloring is NP-hard in the considered subclasses of P5-free graphs if we allow loops on H.The first authorâs material is based upon work supported in part by the U.S. ArmyResearch Office under grant W911NF-16-1-0404 and by NSF grant DMS-1763817. The third authorâswork is a part of project TOTAL that has received funding from the European Research Council(ERC) under the European Unionâs Horizon 2020 research and innovation programme (grant 677651).The fourth author was supported by Polish National Science Centre grant 2018/31/D/ST6/00062.The fifth authorâs material is based upon work supported by the National Science Foundation underaward DMS-1802201
Three-in-a-Tree in Near Linear Time
The three-in-a-tree problem is to determine if a simple undirected graph
contains an induced subgraph which is a tree connecting three given vertices.
Based on a beautiful characterization that is proved in more than twenty pages,
Chudnovsky and Seymour [Combinatorica 2010] gave the previously only known
polynomial-time algorithm, running in time, to solve the
three-in-a-tree problem on an -vertex -edge graph. Their three-in-a-tree
algorithm has become a critical subroutine in several state-of-the-art graph
recognition and detection algorithms.
In this paper we solve the three-in-a-tree problem in time,
leading to improved algorithms for recognizing perfect graphs and detecting
thetas, pyramids, beetles, and odd and even holes. Our result is based on a new
and more constructive characterization than that of Chudnovsky and Seymour. Our
new characterization is stronger than the original, and our proof implies a new
simpler proof for the original characterization. The improved characterization
gains the first factor in speed. The remaining improvement is based on
dynamic graph algorithms.Comment: 46 pages, 12 figures, accepted to STOC 202