Problem For Planar Graphs Research Articles

Minimum strictly fundamental cycle bases of planar graphs are hard to find

In this paper, we consider the problem of finding a spanning tree in a graph that minimizes the sum over the lengths of the cycles induced by the chords of the tree. We show the NP-completeness of this problem for planar graphs. The proof will be by reduction of a planar version of the Exact Cover By 3-Sets Problem. Finding such a minimum strictly fundamental cycle basis has various practical applications, e.g. in designing optimal periodic timetables in public transport.

An algorithm and upper bounds for the weighted maximal planar graph problem

In this paper, we investigate the weighted maximal planar graph (WMPG) problem. Given a complete, edge-weighted, simple graph, the WMPG problem involves finding a subgraph with the highest sum of edge weights that is maximal planar, namely, it can be embedded in the plane without any of its edges intersecting, and no additional edge can be added to the subgraph without violating its planarity. We present a new integer linear programming (ILP) model for this problem. We then develop a cutting-plane algorithm to solve the WMPG problem based on the proposed ILP model. This algorithm enables the problem to be solved more efficiently than previously reported algorithms. New upper bounds are also provided, which are useful in evaluating the quality of heuristic solutions or in generating initial solutions for meta-heuristics. Computational results are reported for a set of 417 test instances of size varying from 6 to 100 nodes including 105 instances from the literature and 312 randomly generated instances. The computational results indicate that instances with up to 24 nodes can be solved optimally in reasonable computational time and the new upper bounds for larger instances significantly improve existing upper bounds.

All-Pairs Shortest Path algorithms for planar graph for GPU-accelerated clusters

We present a new approach for solving the All-Pairs Shortest-Path (APSP) problem for planar graphs that exploits the massive on-chip parallelism available in today’s Graphics Processing Units (GPUs). We describe two new algorithms based on our approach. Both algorithms use Floyd–Warshall method, have near optimal complexity in terms of the total number of operations, while their matrix-based structure is regular enough to allow for efficient parallel implementation on the GPUs. By applying a divide-and-conquer approach, we are able to make use of multi-node GPU clusters, resulting in more than an order of magnitude speedup over fastest known Dijkstra-based GPU implementation and a two-fold speedup over a parallel Dijkstra-based CPU implementation.

New analysis and computational study for the planar connected dominating set problem

The connected dominating set (CDS) problem is a well studied NP-hard problem with many important applications. Dorn et al. (Algorithmica 58:790---810 2010) introduce a branch-decomposition based algorithm design technique for NP-hard problems in planar graphs and give an algorithm (DPBF algorithm) which solves the planar CDS problem in $$O(2^{9.822\sqrt{n}}n+n^3)$$O(29.822nn+n3) time and $$O(2^{8.11\sqrt{n}}n+n^3)$$O(28.11nn+n3) time, with a conventional method and fast matrix multiplication in the dynamic programming step of the algorithm, respectively. We show that DPBF algorithm solves the planar CDS problem in $$O(2^{9.8\sqrt{n}}n+n^3)$$O(29.8nn+n3) time with a conventional method and in $$O(2^{8.08\sqrt{n}}n+n^3)$$O(28.08nn+n3) time with a fast matrix multiplication. For a graph $$G$$G, let $${\hbox {bw}}(G)$$bw(G) be the branchwidth of $$G$$G and $$\gamma _c(G)$$źc(G) be the connected dominating number of $$G$$G. We prove $${\hbox {bw}}(G)\le 2\sqrt{10\gamma _c(G)}+32$$bw(G)≤210źc(G)+32. From this result, the planar CDS problem admits an $$O(2^{23.54\sqrt{\gamma _c(G)}}\gamma _c(G)+n^3)$$O(223.54źc(G)źc(G)+n3) time fixed-parameter algorithm. We report computational study results on the practical performance of DPBF algorithm, which show that the size of instances can be solved by the algorithm mainly depends on the branchwidth of the instances, coinciding with the theoretical analysis. For graphs with small or moderate branchwidth, the CDS problem instances with size up to a few thousands edges can be solved in a practical time and memory space.

Min st -Cut Oracle for Planar Graphs with Near-Linear Preprocessing Time

For an undirected n -vertex planar graph G with nonnegative edge weights, we consider the following type of query: given two vertices s and t in G , what is the weight of a min st -cut in G ? We show how to answer such queries in constant time with O ( n log 4 n ) preprocessing time and O ( n log n ) space. We use a Gomory-Hu tree to represent all the pairwise min cuts implicitly. Previously, no subquadratic time algorithm was known for this problem. Since all-pairs min cut and the minimum-cycle basis are dual problems in planar graphs, we also obtain an implicit representation of a minimum-cycle basis in O ( n log 4 n ) time and O ( n log n ) space. Additionally, an explicit representation can be obtained in O ( C ) time and space where C is the size of the basis. These results require that shortest paths are unique. This can be guaranteed either by using randomization without overhead or deterministically with an additional log 2 n factor in the preprocessing times.

The computational complexity of the backbone coloring problem for planar graphs with connected backbones

In the paper we study the computational complexity of the backbone coloring problem for planar graphs with connected backbones. For every possible value of integer parameters λ≥2 and k≥1 we show that the following problem: Instance:A simple planar graph G, its connected spanning subgraph (backbone) H.Question:Is there a λ-backbone coloring c of G with backbone H such that maxc(V(G))≤k? is either NP-complete or polynomially solvable (by algorithms that run in constant, linear or quadratic time). As a result of these considerations we obtain a complete classification of the computational complexity with respect to the values of λ and k.We also study the problem of computing the backbone chromatic number for two special classes of planar graphs: cacti and thorny graphs. We construct an algorithm that runs in O(n3) time and solves this problem for cacti and another polynomial algorithm that is 1-absolute approximate for thorny graphs.

The degree/diameter problem in maximal planar bipartite graphs

The (Δ,D) (degree/diameter) problem consists of finding the largest possible number of vertices n among all the graphs with maximum degree Δ and diameter D. We consider the (Δ,D) problem for maximal planar bipartite graphs, that are simple planar graphs in which every face is a quadrangle. We obtain that for the (Δ,2) problem, the number of vertices is n=Δ+2; and for the (Δ,3) problem, n=3Δ−1 if Δ is odd and n=3Δ−2 if Δ is even. Then, we study the general case (Δ,D) and obtain that an upper bound on n is approximately 3(2D+1)(Δ−2)⌊D/2⌋ and another one is C(Δ−2)⌊D/2⌋ if Δ≥D and C is a sufficiently large constant. Our upper bound improve for our kind of graphs the one given by Fellows, Hell and Seyffarth for general planar graphs. We also give a lower bound on n for maximal planar bipartite graphs, which is approximately (Δ−2)k if D=2k, and 3(Δ−3)k if D=2k+1, for Δ and D sufficiently large in both cases.

Node-Weighted Steiner Tree and Group Steiner Tree in Planar Graphs

We improve the approximation ratios for two optimization problems in planar graphs. For node-weighted Steiner tree, a classical network-optimization problem, the best achievable approximation ratio in general graphs is Θ (log n ), and nothing better was previously known for planar graphs. We give a constant-factor approximation for planar graphs. Our algorithm generalizes to allow as input any nontrivial minor-closed graph family, and also generalizes to address other optimization problems such as Steiner forest, prize-collecting Steiner tree, and network-formation games. The second problem we address is group Steiner tree: given a graph with edge weights and a collection of groups (subsets of nodes), find a minimum-weight connected subgraph that includes at least one node from each group. The best approximation ratio known in general graphs is O (log 3 n ), or O (log 2 n ) when the host graph is a tree. We obtain an O (log n polyloglog n ) approximation algorithm for the special case where the graph is planar embedded and each group is the set of nodes on a face. We obtain the same approximation ratio for the minimum-weight tour that must visit each group.

A strengthened analysis of a local algorithm for the minimum dominating set problem in planar graphs

In recent years growing interest in local distributed algorithms has widely been observed. This results from their high resistance to errors and damage, as well as from their good performance, which is independent of the size of the network. A local deterministic distributed algorithm finding an approximation of a Minimum Dominating Set in planar graphs has been presented by Lenzen et al., and they proved that the algorithm returns a 130-approximation of the Minimum Dominating Set. In this article we will show that the algorithm is two times more effective than was previously assumed, and we prove that the algorithm by Lenzen et al. outputs a 52-approximation to a Minimum Dominating Set. Therefore the gap between the lower bound and the approximation ratio of the best yet local deterministic distributed algorithm is reduced by half.

Computational Study on a PTAS for Planar Dominating Set Problem

The dominating set problem is a core NP-hard problem in combinatorial optimization and graph theory, and has many important applications. Baker [JACM 41,1994] introduces a k-outer planar graph decomposition-based framework for designing polynomial time approximation scheme (PTAS) for a class of NP-hard problems in planar graphs. It is mentioned that the framework can be applied to obtain an O(2ckn) time, c is a constant, (1+1/k)-approximation algorithm for the planar dominating set problem. We show that the approximation ratio achieved by the mentioned application of the framework is not bounded by any constant for the planar dominating set problem. We modify the application of the framework to give a PTAS for the planar dominating set problem. With k-outer planar graph decompositions, the modified PTAS has an approximation ratio (1 + 2/k). Using 2k-outer planar graph decompositions, the modified PTAS achieves the approximation ratio (1+1/k) in O(22ckn) time. We report a computational study on the modified PTAS. Our results show that the modified PTAS is practical.

Planar graph vertex partition for linear problem kernels

A simple partition of the vertex set of a graph is introduced to analyze kernels for planar graph problems in which vertices and edges not in a solution have small distance to the solution. This method directly leads to improved kernel sizes for several problems, without needing new reduction rules. Moreover, new kernelization algorithms are developed for Connected Vertex Cover, Edge Dominating Set, and Maximum Triangle Packing problems, further improving the kernel sizes for these problems.

On the power of unambiguity in log-space

We report progress on the NL versus UL problem. We show that counting the number of s-t paths in graphs where the number of s-v paths for any v is bounded by a polynomial can be done in FUL: the unambiguous log-space function class. Several new upper bounds follow from this including \({{{ReachFewL} \subseteq {UL}}}\) and \({{{LFew} \subseteq {UL}^{FewL}}}\) We investigate the complexity of min-uniqueness—a central notion in studying the NL versus UL problem. In this regard we revisit the class OptL[log n] and introduce UOptL[log n], an unambiguous version of OptL[log n]. We investigate the relation between UOptL[log n] and other existing complexity classes. We consider the unambiguous hierarchies over UL and UOptL[log n]. We show that the hierarchy over UOptL[log n] collapses. This implies that \({{{ULH} \subseteq {L}^{{promiseUL}}}}\) thus collapsing the UL hierarchy. We show that the reachability problem over graphs embedded on 3 pages is complete for NL. This contrasts with the reachability problem over graphs embedded on 2 pages, which is log-space equivalent to the reachability problem in planar graphs and hence is in UL.

On Finding Min-Min Disjoint Paths

The Min-Min problem of finding a disjoint-path pair with the length of the shorter path minimized is known to be NP-hard and admits no K-approximation for any K>1 in the general case (Xu et al. in IEEE/ACM Trans. Netw. 14:147–158, 2006). In this paper, we first show that Bhatia et al.’s NP-hardness proof (Bhatia et al. in J. Comb. Optim. 12:83–96, 2006), a claim of correction to Xu et al.’s proof (Xu et al. in IEEE/ACM Trans. Netw. 14:147–158, 2006), for the edge-disjoint Min-Min problem in the general undirected graphs is incorrect by giving a counter example that is an unsatisfiable 3SAT instance but classified as a satisfiable 3SAT instance in the proof of Bhatia et al. (J. Comb. Optim. 12:83–96, 2006). We then gave a correct proof of NP-hardness of this problem in undirected graphs. Finally we give a polynomial-time algorithm for the vertex disjoint Min-Min problem in planar graphs by showing that the vertex disjoint Min-Min problem is polynomially solvable in st-planar graph G=(V,E) whose corresponding auxiliary graph G(V,E∪{e(st)}) can be embedded into a plane, and a planar graph can be decomposed into several st-planar graphs whose Min-Min paths collectively contain a Min-Min disjoint-path pair between s and t in the original graph G. To the best of our knowledge, these are the first polynomial algorithms for the Min-Min problems in planar graphs.

NP-completeness and APX-completeness of restrained domination in graphs

Let G=(V,E) be a simple graph. A vertex set S⊆V is a restrained dominating set if every vertex not in S is adjacent to a vertex in S and to a vertex in V−S. In this paper, we investigate the NP-completeness of the restrained domination problem in planar graphs and split graphs. Meanwhile, it is proved that the restrained domination problem is APX-complete for bounded-degree graphs.

Greenʼs theorem and isolation in planar graphs

We show a simple application of Greenʼs theorem from multivariable calculus to the isolation problem in planar graphs. In particular, we give a log-space construction of a skew-symmetric, polynomially-bounded edge weight function for directed planar graphs, such that the weight of any simple cycle in the graph is non-zero with respect to this weight function. As a direct consequence of the above weight function, we are able to isolate a directed path between two fixed vertices, in a directed planar graph. We also show that given a bipartite planar graph, we can obtain an edge weight function (using the above function) in log-space, which isolates a perfect matching in the given graph. Earlier this was known to be true only for grid graphs – which is a proper subclass of planar graphs.We also look at the problem of obtaining a straight line embedding of a planar graph in log-space. Although we do not quite achieve this goal, we give a piecewise straight line embedding of the given planar graph in log-space.

Maximum size of a planar graph with given degree and even diameter

We offer the exact solution of the degree–diameter problem for planar graphs in the case of even diameter 2 d and large degree Δ ≥ 6 ( 12 d + 1 ) . New graph examples are constructed improving the lower bounds for Δ ≥ 5 .

A linear time algorithm for the induced disjoint paths problem in planar graphs

In this paper, we consider a problem which we call the induced disjoint paths problem (IDPP) for planar graphs. We are given a planar graph G and a collection of vertex pairs { ( s 1 , t 1 ) , … , ( s k , t k ) } . The objective is to find k paths P 1 , … , P k such that P i is a path from s i to t i and P i and P j have neither common vertices nor adjacent vertices for any distinct i , j . This problem setting is a generalization of the disjoint paths problem, since if we subdivide each edge, then desired disjoint paths would be induced disjoint paths. The problem is motivated by not only the disjoint paths problem but also the recognition of an induced subgraph. The latter has been developed in recent years, and this is actually connected to the Strong Perfect Graph Theorem (Chudnovsky, et al., 2006) [1], and the recognition of the perfect graphs (Chudnovsky, et al., 2005) [2]. Our main result in this paper is to give a linear time algorithm for the IDPP for planar graphs. This generalizes the result by Reed, Robertson, Schrijver and Seymour (1993) [14]. This also gives a polynomial time algorithm to find an induced circuit through given k vertices in planar graphs if one exists when k is fixed. The case k = 2 was previously proved by McDiarmid, Reed, Schrijver and Shepherd (1994) [10].

Graph Coloring Applied to Secure Computation in Non-Abelian Groups

We study the natural problem of secure n-party computation (in the computationally unbounded attack model) of circuits over an arbitrary finite non-Abelian group (G,?), which we call G-circuits. Besides its intrinsic interest, this problem is also motivating by a completeness result of Barrington, stating that such protocols can be applied for general secure computation of arbitrary functions. For flexibility, we are interested in protocols which only require black-box access to the group G (i.e. the only computations performed by players in the protocol are a group operation, a group inverse, or sampling a uniformly random group element). Our investigations focus on the passive adversarial model, where up to t of the n participating parties are corrupted. Our results are as follows. We initiate a novel approach for the construction of black-box protocols for G-circuits based on k-of-k threshold secret-sharing schemes, which are efficiently implementable over any black-box (non-Abelian) group G. We reduce the problem of constructing such protocols to a combinatorial coloring problem in planar graphs. We then give three constructions for such colorings. Our first approach leads to a protocol with optimal resilience t<n/2, but it requires exponential communication complexity $O({\binom{2 t+1}{t}}^{2} \cdot N_{g})$ group elements and round complexity $O(\binom{2 t + 1}{t} \cdot N_{g})$ , for a G-circuit of size N g . Nonetheless, using this coloring recursively, we obtain another protocol to t-privately compute G-circuits with communication complexity $\mathcal{P}\mathit{oly}(n)\cdot N_{g}$ for any t?O(n 1?∈ ) where ∈ is any positive constant. For our third protocol, there is a probability ? (which can be made arbitrarily small) for the coloring to be flawed in term of security, in contrast to the first two techniques, where the colorings are always secure (we call this protocol probabilistic, and those earlier protocols deterministic). This third protocol achieves optimal resilience t<n/2. It has communication complexity O(n 5.056(n+log?? ?1)2?N g ) and the number of rounds is O(n 2.528?(n+log?? ?1)?N g ).

Maximum independent sets in 3- and 4-regular Hamiltonian graphs

Smooth 4-regular Hamiltonian graphs are generalizations of cycle-plus-triangles graphs. While the latter have been shown to be 3-choosable, 3-colorability of the former is NP-complete. In this paper we first show that the independent set problem for 3-regular Hamiltonian planar graphs is NP-complete, and using this result we show that this problem is also NP-complete for smooth 4-regular Hamiltonian graphs. We also show that this problem remains NP-complete if we restrict the problem to the existence of large independent sets (i.e., independent sets whose size is at least one third of the order of the graphs).

The hardness of routing two pairs on one face

We prove the NP-completeness of the integer multiflow problem in planar graphs, with the following restrictions: there are only two classes of parallel demand edges, both lying on the infinite face of the routing graph. This was one of the open challenges concerning disjoint paths, explicitly asked by Muller (Math Program 105 (2–3):275–288, 2006). It also strengthens Schwarzler’s recent proof of one of the open problems of Schrijver’s book (Combinatorial Optimization: Polyhedra and Efficiency. Springer, Berlin 2003), about the complexity of the edge-disjoint paths problem with terminals on the outer boundary of a planar graph. We also give a directed acyclic reduction. This proves that the arc-disjoint paths problem is NP-complete in directed acyclic graphs, even with only two classes of demand arcs.