We consider nodeweighted survivable network design (SNDP) in planar graphs and minorclosed families of graphs. The input consists of a nodeweighted undirected graph G = ( V , E ) and integer connectivity requirements r ( uv ) for each unordered pair of nodes uv . The goal is to find a minimum weighted subgraph H of G such that H contains r ( uv ) disjoint paths between u and v for each node pair uv . Three versions of the problem are edgeconnectivity SNDP (ECSNDP), elementconnectivity SNDP (ElemSNDP), and vertexconnectivity SNDP (VCSNDP), depending on whether the paths are required to be edge, element, or vertex disjoint, respectively. Our main result is an O ( k )approximation algorithm for ECSNDP and ElemSNDP when the input graph is planar or more generally if it belongs to a proper minorclosed family of graphs; here, k = max uv r ( uv ) is the maximum connectivity requirement. This improves upon the O ( k log n )approximation known for nodeweighted ECSNDP and ElemSNDP in general graphs [31]. We also obtain an O (1) approximation for nodeweighted VCSNDP when the connectivity requirements are in {0, 1, 2}; for higher connectivity our resultmore »
Codegree Conditions for Tiling Complete kPartite kGraphs and Loose Cycles
Abstract Given two k graphs ( k uniform hypergraphs) F and H , a perfect F tiling (or F factor) in H is a set of vertexdisjoint copies of F that together cover the vertex set of H . For all complete k partite k graphs K , Mycroft proved a minimum codegree condition that guarantees a K factor in an n vertex k graph, which is tight up to an error term o ( n ). In this paper we improve the error term in Mycroft’s result to a sublinear term that relates to the Turán number of K when the differences of the sizes of the vertex classes of K are coprime. Furthermore, we find a construction which shows that our improved codegree condition is asymptotically tight in infinitely many cases, thus disproving a conjecture of Mycroft. Finally, we determine exact minimum codegree conditions for tiling K (k) (1, … , 1, 2) and tiling loose cycles, thus generalizing the results of Czygrinow, DeBiasio and Nagle, and of Czygrinow, respectively.
 Award ID(s):
 1700622
 Publication Date:
 NSFPAR ID:
 10161153
 Journal Name:
 Combinatorics, Probability and Computing
 Volume:
 28
 Issue:
 06
 Page Range or eLocationID:
 840 to 870
 ISSN:
 09635483
 Sponsoring Org:
 National Science Foundation
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