Graphs Of Bounded Genus Research Articles

The energy complexity of diameter and minimum cut computation in bounded-genus networks

This paper investigates the energy complexity of distributed graph problems in multi-hop radio networks, where the energy cost of an algorithm is measured by the maximum number of awake rounds of a vertex. Recent works revealed that some problems, such as broadcast, breadth-first search, and maximal matching, can be solved with energy-efficient algorithms that consume only polylog⁡n energy. However, there exist some problems, such as computing the diameter of the graph, that require Ω(n) energy to solve. To improve energy efficiency for these problems, we focus on a special graph class: bounded-genus graphs. We present algorithms for computing the exact diameter, the exact global minimum cut size, and a (1±ϵ)-approximate s-t minimum cut size with O˜(n) energy for bounded-genus graphs. Our approach is based on a generic framework that divides the vertex set into high-degree and low-degree parts and leverages the structural properties of bounded-genus graphs to control the number of certain connected components in the subgraph induced by the low-degree part.

Edge Separators for Graphs Excluding a Minor

We prove that every $n$-vertex $K_t$-minor-free graph $G$ of maximum degree $\Delta$ has a set $F$ of $O(t^2(\log t)^{1/4}\sqrt{\Delta n})$ edges such that every component of $G - F$ has at most $n/2$ vertices. This is best possible up to the dependency on $t$ and extends earlier results of Diks, Djidjev, Sýkora, and Vrťo (1993) for planar graphs, and of Sýkora and Vrťo (1993) for bounded-genus graphs. Our result is a consequence of the following more general result: The line graph of $G$ is isomorphic to a subgraph of the strong product $H \boxtimes K_{\lfloor p \rfloor}$ for some graph $H$ with treewidth at most $t-2$ and $p = \sqrt{(t-3)\Delta |E(G)|} + \Delta$.

Approximating Maximum Integral Multiflows on Bounded Genus Graphs

We devise the first constant-factor approximation algorithm for finding an integral multi-commodity flow of maximum total value for instances where the supply graph together with the demand edges can be embedded on an orientable surface of bounded genus. This extends recent results for planar instances. Our techniques include an uncrossing algorithm, which is significantly more difficult than in the planar case, a partition of the cycles in the support of an LP solution into free homotopy classes, and a new rounding procedure for freely homotopic non-separating cycles.

Planarity can be verified by an approximate proof labeling scheme in constant-time

Approximate proof labeling schemes were introduced by Censor-Hillel, Paz and Perry [3]. Roughly speaking, a graph property P can be verified by an approximate proof labeling scheme in constant-time if the vertices of a graph having the property can be convinced, in a short period of time not depending on the size of the graph, that they are having the property P or at least they are not far from being having the property P. The main result of this paper is that bounded-degree planar graphs (and also outer-planar graphs, bounded genus graphs, knotlessly embeddable graphs etc.) can be verified by an approximate proof labeling scheme in constant-time.

Adjacency Labelling for Planar Graphs (and Beyond)

We show that there exists an adjacency labelling scheme for planar graphs where each vertex of an n -vertex planar graph G is assigned a (1 + o(1)) log 2 n -bit label and the labels of two vertices u and v are sufficient to determine if uv is an edge of G . This is optimal up to the lower order term and is the first such asymptotically optimal result. An alternative, but equivalent, interpretation of this result is that, for every positive integer n , there exists a graph U n with n 1+o(1) vertices such that every n -vertex planar graph is an induced subgraph of U n . These results generalize to a number of other graph classes, including bounded genus graphs, apex-minor-free graphs, bounded-degree graphs from minor closed families, and k -planar graphs.

Low-Congestion shortcuts without embedding

Distributed optimization algorithms are frequently faced with solving sub-problems on disjoint connected parts of a network. Unfortunately, the diameter of these parts can be significantly larger than the diameter of the underlying network, leading to slow running times. Recent work by [Ghaffari and Hauepler; SODA'16] showed that this phenomenon can be seen as the broad underlying reason for the pervasive $\Omega(\sqrt{n} + D)$ lower bounds that apply to most optimization problems in the CONGEST model. On the positive side, this work also introduced low-congestion shortcuts as an elegant solution to circumvent this problem in certain topologies of interest. Particularly, they showed that there exist good shortcuts for any planar network and more generally any bounded genus network. This directly leads to fast $O(D \log^{O(1)} n)$ distributed algorithms for MST and Min-Cut approximation, given that one can efficiently construct these shortcuts in a distributed manner. Unfortunately, the shortcut construction of [Ghaffari and Hauepler; SODA'16] relies heavily on having access to a genus embedding of the network. Computing such an embedding distributedly, however, is a hard problem - even for planar networks. No distributed embedding algorithm for bounded genus graphs is in sight. In this work, we side-step this problem by defining a restricted and more structured form of shortcuts and giving a novel construction algorithm which efficiently finds a shortcut which is, up to a logarithmic factor, as good as the best shortcut that exists for a given network. This new construction algorithm directly leads to an $O(D \log^{O(1)} n)$-round algorithm for solving optimization problems like MST for any topology for which good restricted shortcuts exist - without the need to compute any embedding. This includes the first efficient algorithm for bounded genus graphs.

Approximation algorithms for connected maximum cut and related problems

An instance of the Connected Maximum Cut problem consists of an undirected graph G=(V,E) and the goal is to find a subset of vertices S⊆V that maximizes the number of edges in the cut δ(S) such that the induced graph G[S] is connected. We present the first non-trivial Ω(1log⁡n) approximation algorithm for the Connected Maximum Cut problem in general graphs using novel techniques. We then extend our algorithm to edge weighted case and obtain a poly-logarithmic approximation algorithm. Interestingly, in contrast to the classical Max-Cut problem that can be solved in polynomial time on planar graphs, we show that the Connected Maximum Cut problem remains NP-hard on unweighted, planar graphs. On the positive side, we obtain a polynomial time approximation scheme for the Connected Maximum Cut problem on planar graphs and more generally on bounded genus graphs.

Distributed Dominating Set Approximations beyond Planar Graphs

The Minimum Dominating Set (MDS) problem is a fundamental and challenging problem in distributed computing. While it is well known that minimum dominating sets cannot be well approximated locally on general graphs, in recent years there has been much progress on computing good local approximations on sparse graphs and in particular on planar graphs. In this article, we study distributed and deterministic MDS approximation algorithms for graph classes beyond planar graphs. In particular, we show that existing approximation bounds for planar graphs can be lifted to bounded genus graphs and more general graphs, which we call locally embeddable graphs, and present (1) a local constant-time, constant-factor MDS approximation algorithm on locally embeddable graphs, and (2) a local O (log * n )-time (1+ϵ)-approximation scheme for any ϵ > 0 on graphs of bounded genus. Our main technical contribution is a new analysis of a slightly modified variant of an existing algorithm by Lenzen et al. [21]. Interestingly, unlike existing proofs for planar graphs, our analysis does not rely on direct topological arguments but on combinatorial density arguments only.

Network Sparsification for Steiner Problems on Planar and Bounded-Genus Graphs

We propose polynomial-time algorithms that sparsify planar and bounded-genus graphs while preserving optimal or near-optimal solutions to Steiner problems. Our main contribution is a polynomial-time algorithm that, given an unweighted undirected graph G embedded on a surface of genus g and a designated face f bounded by a simple cycle of length k , uncovers a set F ⊆ E ( G ) of size polynomial in g and k that contains an optimal Steiner tree for any set of terminals that is a subset of the vertices of f . We apply this general theorem to prove that: — Given an unweighted graph G embedded on a surface of genus g and a terminal set S ⊆ V ( G ), one can in polynomial time find a set F ⊆ E ( G ) that contains an optimal Steiner tree T for S and that has size polynomial in g and | E ( T )|. — An analogous result holds for an optimal Steiner forest for a set S of terminal pairs. — Given an unweighted planar graph G and a terminal set S ⊆ V ( G ), one can in polynomial time find a set F ⊆ E ( G ) that contains an optimal (edge) multiway cut C separating S (i.e., a cutset that intersects any path with endpoints in different terminals from S ) and that has size polynomial in | C |. In the language of parameterized complexity, these results imply the first polynomial kernels for S teiner T ree and S teiner F orest on planar and bounded-genus graphs (parameterized by the size of the tree and forest, respectively) and for (E dge ) M ultiway C ut on planar graphs (parameterized by the size of the cutset). Additionally, we obtain a weighted variant of our main contribution: a polynomial-time algorithm that, given an undirected plane graph G with positive edge weights, a designated face f bounded by a simple cycle of weight w ( f ), and an accuracy parameter ε > 0, uncovers a set F ⊆ E ( G ) of total weight at most poly(ε -1 ) w ( f ) that, for any set of terminal pairs that lie on f , contains a Steiner forest within additive error ε w ( f ) from the optimal Steiner forest.

Distributed [formula omitted] constant approximation of MDS in bounded genus graphs

We give a distributed algorithm which finds a constant approximation of a minimum dominating set in graphs of constant genus. The algorithm works in the CONGESTBC model (distributed, synchronous, with short messages and broadcast type of transmissions) and runs in a constant number of distributed rounds.

A transfer principle and applications to eigenvalue estimates for graphs

In this paper, we prove a variant of the Burger–Brooks transfer principle which, combined with recent eigenvalue bounds for surfaces, allows to obtain upper bounds on the eigenvalues of graphs as a function of their genus. More precisely, we show the existence of a universal constants C such that the k th eigenvalue \lambda_k^{\mathrm {nr}} of the normalized Laplacian of a graph G of (geometric) genus g on n vertices satisfies \lambda_k^{\mathrm {nr}}(G) \leq C \frac{d_{\mathrm {max}}(g+k)}{n}, where d_{\mathrm {max}} denotes the maximum valence of vertices of the graph. Our result is tight up to a change in the value of the constant C , and improves recent results of Kelner, Lee, Price and Teng on bounded genus graphs. To show that the transfer theorem might be of independent interest, we relate eigenvalues of the Laplacian on a metric graph to the eigenvalues of its simple graph models, and discuss an application to the mesh partitioning problem, extending results of Miller–Teng–Thurston–Vavasis and Spielman–Teng to arbitrary meshes.

On Algorithms Employing Treewidth for $L$-bounded Cut Problems

Given a graph $G=(V,E)$ with two distinguished vertices $s,t\in V$ and an integer parameter $L>0$, an $L$-bounded cut is a subset $F$ of edges (vertices) such that the every path between $s$ and $t$ in $G\setminus F$ has length more than $L$. The task is to find an $L$-bounded cut of minimum cardinality. Though the problem is very simple to state and has been studied since the beginning of the 70's, it is not much understood yet. The problem is known to be $\cal{NP}$-hard to approximate within a small constant factor even for $L\geq 4$ (for $L\geq 5$ for the vertex-deletion version). On the other hand, the best known approximation algorithm for general graphs has approximation ratio only $\mathcal{O}({n^{2/3}})$ in the edge case, and $\mathcal{O}({\sqrt{n}})$ in the vertex case, where $n$ denotes the number of vertices. We show that for planar graphs, it is possible to solve both the edge- and the vertex-deletion version of the problem optimally in $\mathcal{O}((L+2)^{3L}n)$ time. That is, the problem is fixed-parameter tractable (FPT) with respect to $L$ on planar graphs. Furthermore, we show that the problem remains FPT even for bounded genus graphs, a super class of planar graphs. Our second contribution deals with approximations of the vertex-deletion version of the problem. We describe an algorithm that for a given graph $G$, its tree decomposition of width $\tau$ and vertices $s$ and $t$ computes a $\tau$-approximation of the minimum $L$-bounded $s-t$ vertex cut; if the decomposition is not given, then the approximation ratio is $\mathcal{O}(\tau \sqrt{\log \tau})$. For graphs with treewidth bounded by $\mathcal{O}(n^{1/2-\epsilon})$ for any $\epsilon>0$, but not by a constant, this is the best approximation in terms of $n$ that we are aware of.

Smaller Extended Formulations for the Spanning Tree Polytope of Bounded-Genus Graphs

We give an $O(g^{1/2} n^{3/2} + g^{3/2} n^{1/2})$-size extended formulation for the spanning tree polytope of an $n$-vertex graph embedded on a surface of genus $g$, improving on the known $O(n^2 + g n)$-size extended formulations following from Wong and Martin.

Multicuts in Planar and Bounded-Genus Graphs with Bounded Number of Terminals

Given an undirected, edge-weighted graph G together with pairs of vertices, called pairs of terminals, the minimum multicut problem asks for a minimum-weight set of edges such that, after deleting these edges, the two terminals of each pair belong to different connected components of the graph. Relying on topological techniques, we provide a polynomial-time algorithm for this problem in the case where G is embedded on a fixed surface of genus g (e.g., when G is planar) and has a fixed number t of terminals. The running time is a polynomial of degree O(sqrt{g^2+gt}) in the input size. In the planar case, our result corrects an error in an extended abstract by Bentz [Int. Workshop on Parameterized and Exact Computation, 109-119, 2012]. The minimum multicut problem is also a generalization of the multiway cut problem, a.k.a. multiterminal cut problem; even for this special case, no dedicated algorithm was known for graphs embedded on surfaces.

Formula omitted]-set problem in graphs

A subset D⊆V of a graph G=(V,E) is a (1,j)-set (Chellali et al., 2013) if every vertex v∈V∖D is adjacent to at least 1 but not more than j vertices in D. The cardinality of a minimum (1,j)-set of G, denoted as γ(1,j)(G), is called the (1,j)-domination number of G. In this paper, using probabilistic methods, we obtain an upper bound on γ(1,j)(G) for j≥O(logΔ), where Δ is the maximum degree of the graph. The proof of this upper bound yields a randomized linear time algorithm. We show that the associated decision problem is NP-complete for choral graphs but, answering a question of Chellali et al., provide a linear-time algorithm for trees for a fixed j. Apart from this, we design a polynomial time algorithm for finding γ(1,j)(G) for a fixed j in a split graph, and show that (1,j)-set problem is fixed parameter tractable in bounded genus graphs and bounded treewidth graphs.

On the generalised colouring numbers of graphs that exclude a fixed minor

The colouring number col(G) of a graph G is the minimum integer k such that there exists a linear ordering of the vertices of G in which each vertex v has back-degree at most k, i.e. v has at most k neighbours u with u<v. The colouring number is a structural measure that measures the edge density of subgraphs of G. For r≥1, the numbers colr(G) and wcolr(G) generalise the colouring number, where col1(G) and wcol1(G) are equivalent to col(G). For increasing values of r these measures converge to the well-known structural measures tree-width and tree-depth. For an n-vertex graph, coln(G) is equal to the tree-width of G and wcoln(G) is equal to the tree-depth of G.We show that if G excludes Kt as a minor, then colr(G)≤(t2)⋅(2r+1) and wcolr(G)≤(t2)r⋅(2r+1).It is easily observed that if G is planar, then colr(G)≤5r+3. The technically most demanding part of the paper is to show that for those graphs, wcolr(G)≤5r5. These results generalise to bounded genus graphs, i.e. if G is of genus g, then colr(G)≤(2g+3)(2r+1) and wcolr(G)≤2g(2r+1)+5r5.

Linear-Time Compression of Bounded-Genus Graphs into Information-Theoretically Optimal Number of Bits

A $\textit{compression scheme}$ $A$ for a class $\mathbb{G}$ of graphs consists of an encoding algorithm $\textit{Encode}_A$ that computes a binary string $\textit{Code}_A(G)$ for any given graph $G$ in $\mathbb{G}$ and a decoding algorithm $\textit{Decode}_A$ that recovers $G$ from $\textit{Code}_A(G)$. A compression scheme $A$ for $\mathbb{G}$ is $\textit{optimal}$ if both $\textit{Encode}_A$ and $\textit{Decode}_A$ run in linear time and the number of bits of $\textit{Code}_A(G)$ for any $n$-node graph $G$ in $\mathbb{G}$ is information-theoretically optimal to within lower-order terms. Trees and plane triangulations were the only known nontrivial graph classes that admit optimal compression schemes. Based upon Goodrich's separator decomposition for planar graphs and Djidjev and Venkatesan's planarizers for bounded-genus graphs, we give an optimal compression scheme for any hereditary (i.e., closed under taking subgraphs) class $\mathbb{G}$ under the premise that any $n$-node graph of $\mathbb{G}$ to be encoded comes with a genus-$o(\frac{n}{\log^2 n})$ embedding. By Mohar's linear-time algorithm that embeds a bounded-genus graph on a genus-$O(1)$ surface, our result implies that any hereditary class of genus-$O(1)$ graphs admits an optimal compression scheme. For instance, our result yields the first-known optimal compression schemes for planar graphs, plane graphs, graphs embedded on genus-$1$ surfaces, graphs with genus $2$ or less, $3$-colorable directed plane graphs, $4$-outerplanar graphs, and forests with degree at most $5$. For non-hereditary graph classes, we also give a methodology for obtaining optimal compression schemes. From this methodology, we give the first known optimal compression schemes for triangulations of genus-$O(1)$ surfaces and floorplans.

A simple reduction from maximum weight matching to maximum cardinality matching

Let mcm(m,n) and mwm(m,n,N) be the complexities of computing a maximum cardinality matching and a maximum weight matching, and let mcmbi, mwmbi be their counterparts for bipartite graphs, where m, n, and N are the edge count, vertex count, and maximum integer edge weight. Kao, Lam, Sung, and Ting (2001) [1] gave a general reduction showing mwmbi(m,n,N)=O(N⋅mcmbi(m,n)) and Huang and Kavitha (2012) [2] recently proved the analogous result for general graphs, that mwm(m,n,N)=O(N⋅mcm(m,n)).We show that Gabowʼs mwmbi and mwm algorithms from 1983 [3] and 1985 [4] can be modified to replicate the results of Kao et al. and Huang and Kavitha, but with dramatically simpler proofs. We also show that our reduction leads to new bounds on the complexity of mwm on sparse graph classes, e.g., (bipartite) planar graphs, bounded genus graphs, and H-minor-free graphs.

Polynomial-Time Approximation Schemes for Subset-Connectivity Problems in Bounded-Genus Graphs

We present the first polynomial-time approximation schemes (PTASes) for the following subset-connectivity problems in edge-weighted graphs of bounded-genus: Steiner tree, low-connectivity survivable-network design, and subset TSP. The schemes run in \(\mathcal{O}(n \log n)\) time for graphs embedded on both orientable and nonorientable surfaces. This work generalizes the PTAS framework from planar graphs to bounded-genus graphs: any problem that is shown to be approximable by the planar PTAS framework of Borradaile et al. (ACM Trans. Algorithms 5(3), 2009) will also be approximable in bounded-genus graphs by our extension.

Approximation Schemes for Steiner Forest on Planar Graphs and Graphs of Bounded Treewidth

We give the first polynomial-time approximation scheme (PTAS) for the Steiner forest problem on planar graphs and, more generally, on graphs of bounded genus. As a first step, we show how to build a Steiner forest spanner for such graphs. The crux of the process is a clustering procedure called prize-collecting clustering that breaks down the input instance into separate subinstances which are easier to handle; moreover, the terminals in different subinstances are far from each other. Each subinstance has a relatively inexpensive Steiner tree connecting all its terminals, and the subinstances can be solved (almost) separately. Another building block is a PTAS for Steiner forest on graphs of bounded treewidth. Surprisingly, Steiner forest is NP-hard even on graphs of treewidth 3. Therefore, our PTAS for bounded-treewidth graphs needs a nontrivial combination of approximation arguments and dynamic programming on the tree decomposition. We further show that Steiner forest can be solved in polynomial time for series-parallel graphs (graphs of treewidth at most two) by a novel combination of dynamic programming and minimum cut computations, completing our thorough complexity study of Steiner forest in the range of bounded-treewidth graphs, planar graphs, and bounded-genus graphs.