Property Testing in Bounded Degree Graphs

We further develop the study of testing graph properties as initiated by Goldreich, Goldwasser and Ron. Loosely speaking, given an oracle access to a graph, we wish to distinguish the case when the graph has a pre-determined property from the case when it is ``far'' from having this property. Whereas they view graphs as represented by their adjacency matrix and measure the distance between graphs as a fraction of all possible vertex pairs, we view graphs as represented by bounded-length incidence lists and measure the distance between graphs as a fraction of the maximum possible number of edges. Thus, while the previous model is most appropriate for the study of dense graphs, our model is most appropriate for the study of bounded-degree graphs. In particular, we present randomized algorithms for testing whether an unknown bounded-degree graph is connected, k -connected (for k>1 ), cycle-free and Eulerian. Our algorithms work in time polynomial in 1/? , always accept the graph when it has the tested property, and reject with high probability if the graph is ? -far from having the property. For example, the 2-connectivity algorithm rejects (with high probability) any N -vertex d -degree graph for which more than ? dN edges need to be added in order to make the graph 2-edge-connected. In addition we prove lower bounds of Ω(\sqrt N ) on the query complexity of testing algorithms for the bipartite and expander properties.     

Property Testing in Bounded Degree Graphs

Abstract. We further develop the study of testing graph properties as initiated by Goldreich, Goldwasser and Ron. Loosely speaking, given an oracle access to a graph, we wish to distinguish the case when the graph has a pre-determined property from the case when it is ``far' from having this property. Whereas they view graphs as represented by their adjacency matrix and measure the distance between graphs as a fraction of all possible vertex pairs, we view graphs as represented by bounded-length incidence lists and measure the distance between graphs as a fraction of the maximum possible number of edges. Thus, while the previous model is most appropriate for the study of dense graphs, our model is most appropriate for the study of bounded-degree graphs. In particular, we present randomized algorithms for testing whether an unknown bounded-degree graph is connected, k -connected (for k>1 ), cycle-free and Eulerian. Our algorithms work in time polynomial in 1/ɛ , always accept the graph when it has the tested property, and reject with high probability if the graph is ɛ -far from having the property. For example, the 2-connectivity algorithm rejects (with high probability) any N -vertex d -degree graph for which more than ɛ dN edges need to be added in order to make the graph 2-edge-connected. In addition we prove lower bounds of Ω(\sqrt N ) on the query complexity of testing algorithms for the bipartite and expander properties.     

Trimmed Moebius Inversion and Graphs of Bounded Degree

We study ways to expedite Yates’s algorithm for computing the zeta and Moebius transforms of a function defined on the subset lattice. We develop a trimmed variant of Moebius inversion that proceeds point by point, finishing the calculation at a subset before considering its supersets. For an n-element universe U and a family ? of its subsets, trimmed Moebius inversion allows us to compute the number of packings, coverings, and partitions of U with k sets from ? in time within a polynomial factor (in n) of the number of supersets of the members of ?. Relying on an projection theorem of Chung et al. (J. Comb. Theory Ser. A 43:23–37, 1986) to bound the sizes of set families, we apply these ideas to well-studied combinatorial optimisation problems on graphs with maximum degree Δ. In particular, we show how to compute the domatic number in time within a polynomial factor of (2^Δ+1?2) ^n/(Δ+1) and the chromatic number in time within a polynomial factor of (2^Δ+1?Δ?1) ^n/(Δ+1). For any constant Δ, these bounds are O((2?ε) ⁿ ) for ε>0 independent of the number of vertices n.     

Sum Coloring of Bipartite Graphs with Bounded Degree

We consider the Chromatic Sum Problem on bipartite graphs which appears to be much harder than the classical Chromatic Number Problem. We prove that the Chromatic Sum Problem is NP-complete on planar bipartite graphs with $\Delta \leq 5$, but polynomial on bipartite graphs with $\Delta \leq 3$, for which we construct an $O(n^{2})$-time algorithm. Hence, we tighten the borderline of intractability for this problem on bipartite graphs with bounded degree, namely: the case $\Delta =3$ is easy, $% \Delta =5$ is hard. Moreover, we construct a $27/26$-approximation algorithm for this problem thus improving the best known approximation ratio of $10/9$.     

限制树宽图上的有界聚类

有界聚类问题源于II3M研究院开发的一个分布式流处理系统,即S系统。问题的输入是一个点赋权和边赋权的无向图,并指定若干个称为终端的顶点。称顶点集合的一个子集为一个子类。子类中所有顶点的权和加上该子类边界上所有边的权和称为该子类的费用。有界聚类问题是要得到所有顶点的一个聚类,要求每个子类的费用不超过给定预算召,每个子类至多包含一个终端,并使得所有子类的总费用最小。对于限制树宽图上的有界聚类问题,给出了拟多项式时间精确算法。利用取整的技巧对该算法进行修正,可在多项式时间之内得到(1+ε)-近似解,其中每个子类的费用不超过(1+ε)B,:是任意小的正数。如果进一步要求每个子类恰好包含一个终端,则所给算法可在多项式时间之内得到(1+ε)-近似解,其中每个子类的费用不超过(2+ε)B。     

Constrained-Path Labellings on Graphs of Bounded Clique-Width

Given a graph G we consider the problem of preprocessing it so that given two vertices x,y and a set X of vertices, we can efficiently report the shortest path (or just its length) between x,y that avoids X. We attach labels to vertices in such a way that this length can be determined from the labels of x,y and the vertices X. For a graph with n vertices of tree-width or clique-width k, we construct labels of size O(k ²log ² n). The constructions extend to directed graphs. The problem is motivated by routing in networks in case of failures or of routing policies which forbid certain paths.     

Bounded Degree Interval Sandwich Problems

The problems of Interval Sandwich (IS) and Intervalizing Colored Graphs (ICG) have received a lot of attention recently, due to their applicability to DNA physical mapping problems with ambiguous data. Most of the results obtained so far on the problems were hardness results. Here we study the problems under assumptions of sparseness, which hold in the biological context. We prove that both problems are polynomial when either (1) the input graph degree and the solution graph clique size are bounded, or (2) the solution graph degree is bounded. In particular, this implies that ICG is polynomial on bounded degree graphs for every fixed number of colors, in contrast with the recent result of Bodlaender and de Fluiter. Received October 2, 1997; revised April 1, 1998.     

Isomorphism for Graphs of Bounded Distance Width

In this paper we study the GRAPH ISOMORPHISM problem on graphs of bounded treewidth, bounded degree, or bounded bandwidth. GRAPH ISOMORPHISM can be solved in polynomial time for graphs of bounded treewidth, pathwidth, or bandwidth, but the exponent depends on the treewidth, pathwidth, or bandwidth. Thus, we look for special cases where ``fixed parameter tractable' polynomial time algorithms can be established. We introduce some new and natural graph parameters: the (rooted) path distance width, which is a restriction of bandwidth, and the (rooted) tree distance width, which is a restriction of treewidth. We give algorithms that solve GRAPH ISOMORPHISM in O(n ² ) time for graphs with bounded rooted path distance width, and in O(n ³ ) time for graphs with bounded rooted tree distance width. Additionally, we show that computing the path distance width of a graph is NP-hard, but both path and tree distance width can be computed in O(n ^k+1 ) time, when they are bounded by a constant k; the rooted path or tree distance width can be computed in O(ne) time. Finally, we study the relationships between the newly introduced parameters and other existing graph parameters. Received February 18, 1997; revised February 23, 1998.     

Dynamic Algorithms for Graphs of Bounded Treewidth

The formalism of monadic second-order (MS) logic has been very successful in unifying a large number of algorithms for graphs of bounded treewidth. We extend the elegant framework of MS logic from static problems to dynamic problems, in which queries about MS properties of a graph of bounded treewidth are interspersed with updates of vertex and edge labels. This allows us to unify and occasionally strengthen a number of scattered previous results obtained in an ad hoc manner and to enable solutions to a wide range of additional problems to be derived automatically. As an auxiliary result of independent interest, we dynamize a data structure of Chazelle for answering queries about products of labels along paths in a tree with edges labeled by elements of a semigroup.     

Resource Allocation in Bounded Degree Trees

We study the bandwidth allocation problem (bap) in bounded degree trees. In this problem we are given a tree and a set of connection requests. Each request consists of a path in the tree, a bandwidth requirement, and a weight. Our goal is to find a maximum weight subset S of requests such that, for every edge e, the total bandwidth of requests in S whose path contains e is at most 1. We also consider the storage allocation problem (sap), in which it is also required that every request in the solution is given the same contiguous portion of the resource in every edge in its path. We present a deterministic approximation algorithm for bap in bounded degree trees with ratio . Our algorithm is based on a novel application of the local ratio technique in which the available bandwidth is divided into narrow strips and requests with very small bandwidths are allocated in these strips. We also present a randomized (2+ε)-approximation algorithm for bap in bounded degree trees. The best previously known ratio for bap in general trees is 5. We present a reduction from sap to bap that works for instances where the tree is a line and the bandwidths are very small. It follows that there exists a deterministic 2.582-approximation algorithm and a randomized (2+ε)-approximation algorithm for sap in the line. The best previously known ratio for this problem is 7. A preliminary version of this paper was presented at the 14th Annual European Symposium on Algorithms (ESA), 2006.     

Linear Time Solvable Optimization Problems on Graphs of Bounded Clique-Width

Hierarchical decompositions of graphs are interesting for algorithmic purposes. There are several types of hierarchical decompositions. Tree decompositions are the best known ones. On graphs of tree-width at most k , i.e., that have tree decompositions of width at most k , where k is fixed, every decision or optimization problem expressible in monadic second-order logic has a linear algorithm. We prove that this is also the case for graphs of clique-width at most k , where this complexity measure is associated with hierarchical decompositions of another type, and where logical formulas are no longer allowed to use edge set quantifications. We develop applications to several classes of graphs that include cographs and are, like cographs, defined by forbidding subgraphs with ``too many' induced paths with four vertices. Received April 13, 1998, and in revised form June 22, 1999, and in final form August 20, 1999.     

Collective Tree Spanners in Graphs with Bounded Parameters

In this paper we study collective additive tree spanners for special families of graphs including planar graphs, graphs with bounded genus, graphs with bounded tree-width, graphs with bounded clique-width, and graphs with bounded chordality. We say that a graph G=(V,E) admits a system of μ collective additive tree r -spanners if there is a system $\mathcal{T}(G)In this paper we study collective additive tree spanners for special families of graphs including planar graphs, graphs with bounded genus, graphs with bounded tree-width, graphs with bounded clique-width, and graphs with bounded chordality. We say that a graph G=(V,E) admits a system of μ collective additive tree r -spanners if there is a system T(G)\mathcal{T}(G) of at most μ spanning trees of G such that for any two vertices x,y of G a spanning tree T ? T(G)T\in\mathcal{T}(G) exists such that d _T (x,y)≤d _G (x,y)+r. We describe a general method for constructing a “small” system of collective additive tree r-spanners with small values of r for “well” decomposable graphs, and as a byproduct show (among other results) that any weighted planar graph admits a system of O(?n)O(\sqrt{n}) collective additive tree 0-spanners, any weighted graph with tree-width at most k−1 admits a system of klog ₂ n collective additive tree 0-spanners, any weighted graph with clique-width at most k admits a system of klog _3/2 n collective additive tree (2w)(2\mathsf{w}) -spanners, and any weighted graph with size of largest induced cycle at most c admits a system of log ₂ n collective additive tree (2?c/2?w)(2\lfloor c/2\rfloor\mathsf{w}) -spanners and a system of 4log ₂ n collective additive tree (2(?c/3?+1)w)(2(\lfloor c/3\rfloor +1)\mathsf {w}) -spanners (here, w\mathsf{w} is the maximum edge weight in G). The latter result is refined for weighted weakly chordal graphs: any such graph admits a system of 4log ₂ n collective additive tree (2w)(2\mathsf{w}) -spanners. Furthermore, based on this collection of trees, we derive a compact and efficient routing scheme for those families of graphs.     

I/O-Efficient Algorithms for Graphs of Bounded Treewidth

We present an algorithm that takes I/Os (sort(N)=Θ((N/(DB))log  _M/B (N/B)) is the number of I/Os it takes to sort N data items) to compute a tree decomposition of width at most k, for any graph G of treewidth at most k and size N, where k is a constant. Given such a tree decomposition, we use a dynamic programming framework to solve a wide variety of problems on G in I/Os, including the single-source shortest path problem and a number of problems that are NP-hard on general graphs. The tree decomposition can also be used to obtain an optimal separator decomposition of G. We use such a decomposition to perform depth-first search in G in  I/Os. As important tools that are used in the tree decomposition algorithm, we introduce flippable DAGs and present an algorithm that computes a perfect elimination ordering of a k-tree in I/Os. The second contribution of our paper, which is of independent interest, is a general and simple framework for obtaining I/O-efficient algorithms for a number of graph problems that can be solved using greedy algorithms in internal memory. We apply this framework in order to obtain an improved algorithm for finding a maximal matching and the first deterministic I/O-efficient algorithm for finding a maximal independent set of an arbitrary graph. Both algorithms take I/Os. The maximal matching algorithm is used in the tree decomposition algorithm. An abstract of this paper was presented at the 12th Annual ACM-SIAM Symposium on Discrete Algorithms, Proceedings, pp. 89–90, 2001. Research of A. Maheshwari supported by NSERC. Part of this work was done while the second author was a Ph.D. student at the School of Computer Science of Carleton University.     

Efficient Orthogonal Drawings of High Degree Graphs

Most of the work that appears in the two-dimensional orthogonal graph drawing literature deals with graphs whose maximum degree is four. In this paper we present an algorithm for orthogonal drawings of simple graphs with degree higher than four. Vertices are represented by rectangular boxes of perimeter less than twice the degree of the vertex. Our algorithm is based on creating groups / pairs of vertices of the graph. The orthogonal drawings produced by our algorithm have area at most (m-1) ( m / 2 +2) . Two important properties of our algorithm are that the drawings exhibit a small total number of bends (less than m ), and that there is at most one bend per edge. Received January 15, 1997; revised February 1, 1998.     

Constructing Plane Spanners of Bounded Degree and Low Weight

Given a set S of n points in the plane, we give an O(n log n)-time algorithm that constructs a plane t-spanner for S, with t ≈ 10, such that the degree of each point of S is bounded from above by 27, and the total edge length is proportional to the weight of a minimum spanning tree of S. Previously, no algorithms were known for constructing plane t-spanners of bounded degree.