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تسجيل مستخدم جديدGraph Spanners: A Tutorial Review
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This tutorial review provides a guiding reference to researchers who want to have an overview of the large body of literature about graph spanners. It reviews the current literature covering various research streams about graph spanners, such as different formulations, sparsity and lightness results, computational complexity, dynamic algorithms, and applications. As an additional contribution, we offer a list of open problems on graph spanners.
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A emph{spanner} of a graph $G$ is a subgraph $H$ that approximately preserves shortest path distances in $G$. Spanners are commonly applied to compress computation on metric spaces corresponding to weighted input graphs. Classic spanner constructions
can seamlessly handle edge weights, so long as error is measured emph{multiplicatively}. In this work, we investigate whether one can similarly extend constructions of spanners with purely emph{additive} error to weighted graphs. These extensions are not immediate, due to a key lemma about the size of shortest path neighborhoods that fails for weighted graphs. Despite this, we recover a suitable amortized version, which lets us prove direct extensions of classic $+2$ and $+4$ unweighted spanners (both all-pairs and pairwise) to $+2W$ and $+4W$ weighted spanners, where $W$ is the maximum edge weight. Specifically, we show that a weighted graph $G$ contains all-pairs (pairwise) $+2W$ and $+4W$ weighted spanners of size $O(n^{3/2})$ and $widetilde{O}(n^{7/5})$ ($O(np^{1/3})$ and $O(np^{2/7})$) respectively. For a technical reason, the $+6$ unweighted spanner becomes a $+8W$ weighted spanner; closing this error gap is an interesting remaining open problem. That is, we show that $G$ contains all-pairs (pairwise) $+8W$ weighted spanners of size $O(n^{4/3})$ ($O(np^{1/4})$).
Many hard algorithmic problems dealing with graphs, circuits, formulas and constraints admit polynomial-time upper bounds if the underlying graph has small treewidth. The same problems often encourage reducing the maximal degree of vertices to simpli
fy theoretical arguments or address practical concerns. Such degree reduction can be performed through a sequence of splittings of vertices, resulting in an _expansion_ of the original graph. We observe that the treewidth of a graph may increase dramatically if the splittings are not performed carefully. In this context we address the following natural question: is it possible to reduce the maximum degree to a constant without substantially increasing the treewidth? Our work answers the above question affirmatively. We prove that any simple undirected graph G=(V, E) admits an expansion G=(V, E) with the maximum degree <= 3 and treewidth(G) <= treewidth(G)+1. Furthermore, such an expansion will have no more than 2|E|+|V| vertices and 3|E| edges; it can be computed efficiently from a tree-decomposition of G. We also construct a family of examples for which the increase by 1 in treewidth cannot be avoided.
A cactus graph is a graph in which any two cycles are edge-disjoint. We present a constructive proof of the fact that any plane graph $G$ contains a cactus subgraph $C$ where $C$ contains at least a $frac{1}{6}$ fraction of the triangular faces of $G
$. We also show that this ratio cannot be improved by showing a tight lower bound. Together with an algorithm for linear matroid parity, our bound implies two approximation algorithms for computing dense planar structures inside any graph: (i) A $frac{1}{6}$ approximation algorithm for, given any graph $G$, finding a planar subgraph with a maximum number of triangular faces; this improves upon the previous $frac{1}{11}$-approximation; (ii) An alternate (and arguably more illustrative) proof of the $frac{4}{9}$ approximation algorithm for finding a planar subgraph with a maximum number of edges. Our bound is obtained by analyzing a natural local search strategy and heavily exploiting the exchange arguments. Therefore, this suggests the power of local search in handling problems of this kind.
Inductive $k$-independent graphs generalize chordal graphs and have recently been advocated in the context of interference-avoiding wireless communication scheduling. The NP-hard problem of finding maximum-weight induced $c$-colorable subgraphs, whic
h is a generalization of finding maximum independent sets, naturally occurs when selecting $c$ sets of pairwise non-conflicting jobs (modeled as graph vertices). We investigate the parameterized complexity of this problem on inductive $k$-independent graphs. We show that the Independent Set problem is W[1]-hard even on 2-simplicial 3-minoes---a subclass of inductive 2-independent graphs. In contrast, we prove that the more general Maximum $c$-Colorable Subgraph problem is fixed-parameter tractable on edge-wise unions of cluster and chordal graphs, which are 2-simplicial. In both cases, the parameter is the solution size. Aside from this, we survey other graph classes between inductive 1-inductive and inductive 2-inductive graphs with applications in scheduling.
There has been significant recent progress on algorithms for approximating graph spanners, i.e., algorithms which approximate the best spanner for a given input graph. Essentially all of these algorithms use the same basic LP relaxation, so a variety
of papers have studied the limitations of this approach and proved integrality gaps for this LP in a variety of settings. We extend these results by showing that even the strongest lift-and-project methods cannot help significantly, by proving polynomial integrality gaps even for $n^{Omega(epsilon)}$ levels of the Lasserre hierarchy, for both the directed and undirected spanner problems. We also extend these integrality gaps to related problems, notably Directed Steiner Network and Shallow-Light Steiner Network.
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