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Register a new userBalanced Districting on Grid Graphs with Provable Compactness and Contiguity
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Added by
Cyrus Hettle
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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Given a graph $G = (V,E)$ with vertex weights $w(v)$ and a desired number of parts $k$, the goal in graph partitioning problems is to partition the vertex set V into parts $V_1,ldots,V_k$. Metrics for compactness, contiguity, and balance of the parts $V_i$ are frequent objectives, with much existing literature focusing on compactness and balance. Revisiting an old method known as striping, we give the first polynomial-time algorithms with guaranteed contiguity and provable bicriteria approximations for compactness and balance for planar grid graphs. We consider several types of graph partitioning, including when vertex weights vary smoothly or are stochastic, reflecting concerns in various real-world instances. We show significant improvements in experiments for balancing workloads for the fire department and reducing over-policing using 911 call data from South Fulton, GA.
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In this work, we continue the study of vertex colorings of graphs, in which adjacent vertices are allowed to be of the same color as long as each monochromatic connected component is of relatively small cardinality. We focus on colorings with two and three available colors and present improved bounds on the size of the monochromatic connected components for two meaningful subclasses of planar graphs, namely maximal outerplanar graphs and complete planar 3-trees.
In this paper, we prove that, given a clique-width $k$-expression of an $n$-vertex graph, textsc{Hamiltonian Cycle} can be solved in time $n^{mathcal{O}(k)}$. This improves the naive algorithm that runs in time $n^{mathcal{O}(k^2)}$ by Espelage et al. (WG 2001), and it also matches with the lower bound result by Fomin et al. that, unless the Exponential Time Hypothesis fails, there is no algorithm running in time $n^{o(k)}$ (SIAM. J. Computing 2014). We present a technique of representative sets using two-edge colored multigraphs on $k$ vertices. The essential idea is that, for a two-edge colored multigraph, the existence of an Eulerian trail that uses edges with different colors alternately can be determined by two information: the number of colored edges incident with each vertex, and the connectedness of the multigraph. With this idea, we avoid the bottleneck of the naive algorithm, which stores all the possible multigraphs on $k$ vertices with at most $n$ edges.
We study aperiodic balanced sequences over finite alphabets. A sequence v of this type is fully characterised by a Sturmian sequence u and two constant gap sequences y and y. We show that the language of v is eventually dendric and we focus on return words to its factors. We deduce a method computing critical exponent and asymptotic critical exponent of balanced sequences provided the associated Sturmian sequence u has a quadratic slope. The method is based on looking for the shortest return words to bispecial factors in v. We illustrate our method on several examples, in particular we confirm a conjecture of Rampersad, Shallit and Vandomme that two specific sequences have the least critical exponent among all balanced sequences over 9- resp. 10-letter alphabets.
We develop an algorithmic framework for graph colouring that reduces the problem to verifying a local probabilistic property of the independent sets. With this we give, for any fixed $kge 3$ and $varepsilon>0$, a randomised polynomial-time algorithm for colouring graphs of maximum degree $Delta$ in which each vertex is contained in at most $t$ copies of a cycle of length $k$, where $1/2le tle Delta^frac{2varepsilon}{1+2varepsilon}/(logDelta)^2$, with $lfloor(1+varepsilon)Delta/log(Delta/sqrt t)rfloor$ colours. This generalises and improves upon several notable results including those of Kim (1995) and Alon, Krivelevich and Sudakov (1999), and more recent ones of Molloy (2019) and Achlioptas, Iliopoulos and Sinclair (2019). This bound on the chromatic number is tight up to an asymptotic factor $2$ and it coincides with a famous algorithmic barrier to colouring random graphs.
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