441 research outputs found
Approximating the Regular Graphic TSP in near linear time
We present a randomized approximation algorithm for computing traveling
salesperson tours in undirected regular graphs. Given an -vertex,
-regular graph, the algorithm computes a tour of length at most
, with high probability, in time. This improves upon a recent result by Vishnoi (\cite{Vishnoi12}, FOCS
2012) for the same problem, in terms of both approximation factor, and running
time. The key ingredient of our algorithm is a technique that uses
edge-coloring algorithms to sample a cycle cover with cycles with
high probability, in near linear time.
Additionally, we also give a deterministic
factor approximation algorithm
running in time .Comment: 12 page
Bounded degree graphs and hypergraphs with no full rainbow matchings
Given a multi-hypergraph that is edge-colored into color classes , a full rainbow matching is a matching of that contains
exactly one edge from each color class . One way to guarantee the
existence of a full rainbow matching is to have the size of each color class
be sufficiently large compared to the maximum degree of . In this
paper, we apply a simple iterative method to construct edge-colored
multi-hypergraphs with a given maximum degree, large color classes, and no full
rainbow matchings. First, for every and , we construct
edge-colored -uniform multi-hypergraphs with maximum degree such
that each color class has size and there is no full
rainbow matching, which demonstrates that a theorem of Aharoni, Berger, and
Meshulam (2005) is best possible. Second, we construct properly edge-colored
multigraphs with no full rainbow matchings which disprove conjectures of
Delcourt and Postle (2022). Finally, we apply results on full rainbow matchings
to list edge-colorings and prove that a color degree generalization of Galvin's
theorem (1995) does not hold
List covering of regular multigraphs
A graph covering projection, also known as a locally bijective homomorphism,
is a mapping between vertices and edges of two graphs which preserves
incidencies and is a local bijection. This notion stems from topological graph
theory, but has also found applications in combinatorics and theoretical
computer science.
It has been known that for every fixed simple regular graph of valency
greater than 2, deciding if an input graph covers is NP-complete. In recent
years, topological graph theory has developed into heavily relying on multiple
edges, loops, and semi-edges, but only partial results on the complexity of
covering multigraphs with semi-edges are known so far. In this paper we
consider the list version of the problem, called \textsc{List--Cover}, where
the vertices and edges of the input graph come with lists of admissible
targets. Our main result reads that the \textsc{List--Cover} problem is
NP-complete for every regular multigraph of valency greater than 2 which
contains at least one semi-simple vertex (i.e., a vertex which is incident with
no loops, with no multiple edges and with at most one semi-edge). Using this
result we almost show the NP-co/polytime dichotomy for the computational
complexity of \textsc{ List--Cover} of cubic multigraphs, leaving just five
open cases.Comment: Accepted to IWOCA 202
Low-Memory Algorithms for Online and W-Streaming Edge Coloring
For edge coloring, the online and the W-streaming models seem somewhat
orthogonal: the former needs edges to be assigned colors immediately after
insertion, typically without any space restrictions, while the latter limits
memory to sublinear in the input size but allows an edge's color to be
announced any time after its insertion. We aim for the best of both worlds by
designing small-space online algorithms for edge-coloring. We study the problem
under both (adversarial) edge arrivals and vertex arrivals. Our results
significantly improve upon the memory used by prior online algorithms while
achieving an -competitive ratio. In particular, for -node graphs with
maximum vertex-degree under edge arrivals, we obtain an online
-coloring in space. This is also the
first W-streaming edge-coloring algorithm for -coloring in sublinear
memory. All prior works either used linear memory or colors.
We also achieve a smooth color-space tradeoff: for any , we get an
-coloring in space,
improving upon the state of the art that used space for
the same number of colors. The improvements stem from extensive use of random
permutations that enable us to avoid previously used colors. Most of our
algorithms can be derandomized and extended to multigraphs, where edge coloring
is known to be considerably harder than for simple graphs.Comment: 32 pages, 1 figur
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