136 research outputs found
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
On Approximability of Steiner Tree in -metrics
In the Continuous Steiner Tree problem (CST), we are given as input a set of
points (called terminals) in a metric space and ask for the minimum-cost tree
connecting them. Additional points (called Steiner points) from the metric
space can be introduced as nodes in the solution. In the Discrete Steiner Tree
problem (DST), we are given in addition to the terminals, a set of facilities,
and any solution tree connecting the terminals can only contain the Steiner
points from this set of facilities. Trevisan [SICOMP'00] showed that CST and
DST are APX-hard when the input lies in the -metric (and Hamming
metric). Chleb\'ik and Chleb\'ikov\'a [TCS'08] showed that DST is NP-hard to
approximate to factor of in the graph metric (and
consequently -metric). Prior to this work, it was unclear if CST
and DST are APX-hard in essentially every other popular metric! In this work,
we prove that DST is APX-hard in every -metric. We also prove that CST
is APX-hard in the -metric. Finally, we relate CST and DST,
showing a general reduction from CST to DST in -metrics. As an
immediate consequence, this yields a -approximation polynomial time
algorithm for CST in -metrics.Comment: Abstract shortened due to arxiv's requirement
LIPIcs, Volume 274, ESA 2023, Complete Volume
LIPIcs, Volume 274, ESA 2023, Complete Volum
LIPIcs, Volume 258, SoCG 2023, Complete Volume
LIPIcs, Volume 258, SoCG 2023, Complete Volum
Decomposing cubic graphs into isomorphic linear forests
A common problem in graph colouring seeks to decompose the edge set of a
given graph into few similar and simple subgraphs, under certain divisibility
conditions. In 1987 Wormald conjectured that the edges of every cubic graph on
vertices can be partitioned into two isomorphic linear forests. We prove
this conjecture for large connected cubic graphs. Our proof uses a wide range
of probabilistic tools in conjunction with intricate structural analysis, and
introduces a variety of local recolouring techniques.Comment: 49 pages, many figure
Sparse graphs with bounded induced cycle packing number have logarithmic treewidth
A graph is -free if it does not contain pairwise vertex-disjoint and
non-adjacent cycles. We show that Maximum Independent Set and 3-Coloring in
-free graphs can be solved in quasi-polynomial time. As a main technical
result, we establish that "sparse" (here, not containing large complete
bipartite graphs as subgraphs) -free graphs have treewidth (even, feedback
vertex set number) at most logarithmic in the number of vertices. This is
proven sharp as there is an infinite family of -free graphs without
-subgraph and whose treewidth is (at least) logarithmic.
Other consequences include that most of the central NP-complete problems
(such as Maximum Independent Set, Minimum Vertex Cover, Minimum Dominating Set,
Minimum Coloring) can be solved in polynomial time in sparse -free graphs,
and that deciding the -freeness of sparse graphs is polynomial time
solvable.Comment: 28 pages, 6 figures. v3: improved complexity result
-Packing Coloring of Cubic Halin Graphs
Given a non-decreasing sequence of
positive integers, an -packing coloring of a graph is a partition of the
vertex set of into subsets such that
for each , the distance between any two distinct vertices
and in is at least . In this paper, we study the problem
of -packing coloring of cubic Halin graphs, and we prove that every cubic
Halin graph is -packing colorable. In addition, we prove that such
graphs are -packing colorable.Comment: 9 page
Twin-width VIII: delineation and win-wins
We introduce the notion of delineation. A graph class is said
delineated if for every hereditary closure of a subclass of
, it holds that has bounded twin-width if and only if
is monadically dependent. An effective strengthening of
delineation for a class implies that tractable FO model checking
on is perfectly understood: On hereditary closures of
subclasses of , FO model checking is fixed-parameter tractable
(FPT) exactly when has bounded twin-width. Ordered graphs
[BGOdMSTT, STOC '22] and permutation graphs [BKTW, JACM '22] are effectively
delineated, while subcubic graphs are not. On the one hand, we prove that
interval graphs, and even, rooted directed path graphs are delineated. On the
other hand, we show that segment graphs, directed path graphs, and visibility
graphs of simple polygons are not delineated. In an effort to draw the
delineation frontier between interval graphs (that are delineated) and
axis-parallel two-lengthed segment graphs (that are not), we investigate the
twin-width of restricted segment intersection classes. It was known that
(triangle-free) pure axis-parallel unit segment graphs have unbounded
twin-width [BGKTW, SODA '21]. We show that -free segment graphs, and
axis-parallel -free unit segment graphs have bounded twin-width, where
is the half-graph or ladder of height . In contrast, axis-parallel
-free two-lengthed segment graphs have unbounded twin-width. Our new
results, combined with the known FPT algorithm for FO model checking on graphs
given with -sequences, lead to win-win arguments. For instance, we derive
FPT algorithms for -Ladder on visibility graphs of 1.5D terrains, and
-Independent Set on visibility graphs of simple polygons.Comment: 51 pages, 19 figure
LIPIcs, Volume 244, ESA 2022, Complete Volume
LIPIcs, Volume 244, ESA 2022, Complete Volum
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