Polynomial Time For Graphs Research Articles

Finding the longest isometric cycle in a graph

A cycle in a graph G is isometric if the distance between two vertices in the cycle is equal to their distance in G . Finding the longest isometric cycle of a graph is then a natural variant of the problem of finding a longest cycle. While most variants of the longest cycle problem are NP-complete, we show that quite surprisingly, one can find a longest isometric cycle in a graph in polynomial time.

Recognizing near-bipartite Pfaffian graphs in polynomial time

A matching covered graph is a non-trivial connected graph in which every edge is in some perfect matching. A non-bipartite matching covered graph G is near-bipartite if there are two edges e 1 and e 2 such that G − e 1 − e 2 is bipartite and matching covered. In 2000, Fischer and Little characterized Pfaffian near-bipartite graphs in terms of forbidden subgraphs [I. Fischer, C.H.C. Little, A characterization of Pfaffian near bipartite graphs, J. Combin. Theory Ser. B 82 (2001) 175–222.]. However, their characterization does not imply a polynomial time algorithm to recognize near-bipartite Pfaffian graphs. In this article, we give such an algorithm. We define a more general class of matching covered graphs, which we call weakly near-bipartite graphs. This class includes the near-bipartite graphs. We give a polynomial algorithm for recognizing weakly near-bipartite Pfaffian graphs. We also show that Fischer and Little’s characterization of near-bipartite Pfaffian graphs extends to this wider class.

On the inapproximability of independent domination in [formula omitted]-free perfect graphs

We consider the complexity of approximation for the Independent Dominating Set problem in 2 P 3 -free graphs, i.e., graphs that do not contain two disjoint copies of the chordless path on three vertices as an induced subgraph. We show that, if P ≠ NP , the problem cannot be approximated for 2 P 3 -free graphs in polynomial time within a factor of n 1 − ε for any constant ε > 0 , where n is the number of vertices in the graph. Moreover, we show that the result holds even if the 2 P 3 -free graph is restricted to being weakly chordal (and thereby perfect).

Bandwidth of bipartite permutation graphs in polynomial time

We give the first polynomial-time algorithm that computes the bandwidth of bipartite permutation graphs. Bandwidth is an NP-complete graph layout problem that is notorious for its difficulty even on small graph classes. For example, it remains NP-complete on caterpillars of hair length at most 3, a very restricted subclass of trees. Much attention has been given to designing approximation algorithms for computing the bandwidth, as it is NP-hard to approximate the bandwidth of general graphs with a constant factor guarantee. The problem is considered important even for approximation on restricted classes, with several distinguished results in this direction. Prior to our work, polynomial-time algorithms for exact computation of bandwidth were known only for caterpillars of hair length at most 2, chain graphs, cographs, and most interestingly, interval graphs.

Finding a Simple Polytope from Its Graph in Polynomial Time

We show that one can compute the combinatorial facets of a simple polytope from its graph in polynomial time. Our proof relies on a primal-dual characterization (by Joswig, Kaibel, and Korner in Israel J. Math. 129:109–118, 2002) and a linear program, with an exponential number of constraints, which can be used to construct the solution and can be solved in polynomial time. We show that this allows one to characterize the face lattice of the polytope, via a simple face recognition algorithm. In addition, we use this linear program to construct several interesting polynomial-time computable sets of graphs which may be of independent interest.

A new characterization of [formula omitted]-free graphs

We study P 6 -free graphs, i.e., graphs that do not contain an induced path on six vertices. Our main result is a new characterization of this graph class: a graph G is P 6 -free if and only if each connected induced subgraph of G on more than one vertex contains a dominating induced cycle on six vertices or a dominating (not necessarily induced) complete bipartite subgraph. This characterization is minimal in the sense that there exists an infinite family of P 6 -free graphs for which a smallest connected dominating subgraph is a (not induced) complete bipartite graph. Our characterization of P 6 -free graphs strengthens results of Liu and Zhou, and of Liu, Peng and Zhao. Our proof has the extra advantage of being constructive: we present an algorithm that finds such a dominating subgraph of a connected P 6 -free graph in polynomial time. This enables us to solve the Hypergraph 2- Colorability problem in polynomial time for the class of hypergraphs with P 6 -free incidence graphs.

Optimal vertex elimination in single-expression-use graphs

The source transformation tool for automatic differentiation of Fortran programs ADIFOR uses a preaccumulation technique to speed up tangent-linear codes significantly compared to the standard forward mode. Reverse mode automatic differentiation is applied to all scalar assignments to generate efficient code for the computation of local gradients. It has been well known for some time that reverse mode is not necessarily the optimal choice for the computation of these statement-level gradients as it does not minimize the number of operations required. This article presents an efficient algorithm for the solution of this combinatorial optimization problem. The corresponding software is freely available for downloading on our website. Developers of software for automatic differentiation are invited to integrate the algorithm into their tools. Gradients of scalar multivariate functions can be computed by elimination methods on the linearized computational graph. The combinatorial optimization problem that aims to minimize the number of arithmetic operations performed by the elimination algorithm is known to be NP-complete. In this article we present a polynomial algorithm for solving a relevant subclass of this problem's instances. The proposed method relies on the ability to compute vertex covers in bipartite graphs in polynomial time. A simplified version of this graph algorithm is used in a research prototype of the differentiation-enabled NAGWare Fortran compiler for the preaccumulation of local gradients of scalar assignments in the context of automatic generation of efficient tangent-linear code for numerical programs.

Polyhedral Results for 1-Restricted Simple 2-Matchings

A simple 2-matching in a graph is a subgraph whose connected components are nontrivial paths and cycles. A simple 2-matching is called 1-restricted if each connected component has two or more edges. In this paper we consider the problem of finding maximum weight 1-restricted simple 2-matchings (which is a relaxation of the traveling salesman problem). We present an integer programming formulation for this problem, characterize the extreme points of the linear programming relaxation, and characterize the graphs for which the linear programming relaxation has all integral extreme points. We show how to recognize these graphs in polynomial time. We also introduce a new class of blossom-type inequalities that tighten the general linear programming relaxation. A complete description of the convex hull of 1-restricted simple 2-matchings is unknown.

Finding a minimum path cover of a distance-hereditary graph in polynomial time

A path cover of a graph G = ( V , E ) is a set of pairwise vertex-disjoint paths such that the disjoint union of the vertices of these paths equals the vertex set V of G . The path cover problem is, given a graph, to find a path cover having the minimum number of paths. The path cover problem contains the Hamiltonian path problem as a special case since finding a path cover, consisting of a single path, corresponds directly to the Hamiltonian path problem. A graph is a distance-hereditary graph if each pair of vertices is equidistant in every connected induced subgraph containing them. The complexity of the path cover problem on distance-hereditary graphs has remained unknown. In this paper, we propose the first polynomial-time algorithm, which runs in O( | V | 9 ) time, to solve the path cover problem on distance-hereditary graphs.

An O(n3)-Time Recognition Algorithm for hhds-free Graphs

The class of hhds-free graphs properly generalizes the classes of strongly chordal graphs and distance-hereditary graphs, and forms a restriction of the class of perfectly orderable graphs. The problem of recognizing hhds-free graphs in polynomial time was posed by Brandstadt (Problem session, Dagstuhl seminor No. 04221, 2004), and Nikolopoulos and Palios (Proceedings of the 31st International Workshop on Graph Theoretic Concepts in Computer Science, 2005) gave an O(m n 2) algorithm. We present an O(n 3)-time algorithm for the problem. Our algorithm demonstrates a relationship between hhds-free graphs and strongly chordal graphs similar to that which is known to exist between hhd-free graphs and chordal graphs.

On the complexity of the balanced vertex ordering problem

Graphs and Algorithms We consider the problem of finding a balanced ordering of the vertices of a graph. More precisely, we want to minimise the sum, taken over all vertices v, of the difference between the number of neighbours to the left and right of v. This problem, which has applications in graph drawing, was recently introduced by Biedl et al. [Discrete Applied Math. 148:27―48, 2005]. They proved that the problem is solvable in polynomial time for graphs with maximum degree three, but NP-hard for graphs with maximum degree six. One of our main results is to close the gap in these results, by proving NP-hardness for graphs with maximum degree four. Furthermore, we prove that the problem remains NP-hard for planar graphs with maximum degree four and for 5-regular graphs. On the other hand, we introduce a polynomial time algorithm that determines whetherthere is a vertex ordering with total imbalance smaller than a fixed constant, and a polynomial time algorithm that determines whether a given multigraph with even degrees has an 'almost balanced' ordering.

Learning Factor Graphs in Polynomial Time and Sample Complexity

We study the computational and sample complexity of parameter and structure learning in graphical models. Our main result shows that the class of factor graphs with bounded degree can be learned in polynomial time and from a polynomial number of training examples, assuming that the data is generated by a network in this class. This result covers both parameter estimation for a known network structure and structure learning. It implies as a corollary that we can learn factor graphs for both Bayesian networks and Markov networks of bounded degree, in polynomial time and sample complexity. Importantly, unlike standard maximum likelihood estimation algorithms, our method does not require inference in the underlying network, and so applies to networks where inference is intractable. We also show that the error of our learned model degrades gracefully when the generating distribution is not a member of the target class of networks. In addition to our main result, we show that the sample complexity of parameter learning in graphical models has an O(1) dependence on the number of variables in the model when using the KL-divergence normalized by the number of variables as the performance criterion.

Exact Sampling from Perfect Matchings of Dense Regular Bipartite Graphs

We present the first algorithm for generating random variates exactly uniformly from the set of perfect matchings of a bipartite graph with a polynomial expected running time over a nontrivial set of graphs. Previous Markov chain results obtain approximately uniform variates for arbitrary graphs in polynomial time, but their general running time is Θ(n10 (ln n)2). Other algorithms (such as Kasteleyn's O(n3) algorithm for planar graphs) concentrated on restricted versions of the problem. Our algorithm employs acceptance/rejection together with a new upper limit on the permanent of a form similar to Bregman's theorem. For graphs with 2n nodes, where the degree of every node is γn for a constant γ, the expected running time is O(n1.5 + .5/γ). Under these conditions, Jerrum and Sinclair showed that a Markov chain of Broder can generate approximately uniform variates in Θ(n4.5 + .5/γ ln n) time, making our algorithm significantly faster on this class of graphs. The problem of counting the number of perfect matchings in these types of graphs is # P complete. In addition, we give a 1 + σ approximation algorithm for finding the permanent of 0–1 matrices with identical row and column sums that runs in O(n1.5 + .5/γ (1/σ2) log (1/δ))$, where the probability that the output is within 1 + \sigma$ of the permanent is at least 1 – δ.

Finding augmenting chains in extensions of claw-free graphs

Finding augmenting chains is in the heart of the maximum matching problem, which is equivalent to the maximum stable set problem in the class of line graphs. Due to the celebrated result of Edmonds, augmenting chains can be found in line graphs in polynomial time. Minty and Sbihi generalized this result to claw-free graphs. In this paper we extend it to larger classes. As a particular consequence, a new polynomially solvable case for the maximum stable set problem has been detected.

Statistical Significance of Tree Similarity Scores

New methodologies for performing pairwise tree-matching on carbohydrate sugar chain data were introduced in [3], in which well-known sequence alignment algorithms [12] were extended and an already known polynomial-time graph algorithm for finding the maximum common subtree (MCST) of two trees [7] was used to implement what is called the KEGG Carbohydrate Matcher, or KCaM. These new methodologies are currently available on the web via KEGG Glycan [9, 15]. We make note of some appealing work related to not only KCaM but biological tree-structure matching in general.

The edge-disjoint paths problem is NP-complete for series–parallel graphs

Many combinatorial problems are NP-complete for general graphs. However, when restricted to series–parallel graphs or partial k-trees, many of these problems can be solved in polynomial time, mostly in linear time. On the other hand, very few problems are known to be NP-complete for series–parallel graphs or partial k-trees. These include the subgraph isomorphism problem and the bandwidth problem. However, these problems are NP-complete even for trees. In this paper, we show that the edge-disjoint paths problem is NP-complete for series–parallel graphs and for partial 2-trees although the problem is trivial for trees and can be solved for outerplanar graphs in polynomial time.

Recognizing Circulant Graphs in Polynomial Time: An Application of Association Schemes

In this paper we present a time-polynomial recognition algorithm for certain classes of circulant graphs. Our approach uses coherent configurations and Schur rings generated by circulant graphs for elucidating their symmetry properties and eventually finding a cyclic automorphism.

Note - Uncrossing a Family of Set-Pairs

Frank and Jordan [1] proved an important min-max result on covering a crossing family of set-pairs. As an application, among others they can solve the unweighted node-connectivity augmentation problem for directed graphs in polynomial time. In this paper, we show how to solve the dual packing problem in polynomial time. To decompose a fractional dual optimum as a convex combination of integer vertices, besides the ellipsoid method, we use a polynomial-time algorithm for uncrossing a family of set-pairs. Our main result is this uncrossing algorithm.

Degree-preserving trees

We consider the degree-preserving spanning tree (DPST) problem: Given a connected graph G, find a spanning tree T of G such that as many vertices of T as possible have the same degree in T as in G. This problem is a graph-theoretical translation of a problem arising in the system-theoretical context of identifiability in networks, a concept which has applications in, for example, water distribution networks and electrical networks. We show that the DPST problem is NP-complete, even when restricted to split graphs or bipartite planar graphs, but that it can be solved in polynomial time for graphs with a bounded asteroidal number and for graphs with a bounded treewidth. For the class of interval graphs, we give a linear time algorithm. For the class of cocomparability graphs, we give an O(n4) algorithm. Furthermore, we present linear time approximation algorithms for planar graphs of a worst-case performance ratio of 1 - e for every e > 0.

Degree‐preserving trees

We consider the degree-preserving spanning tree (DPST) problem: Given a connected graph G, find a spanning tree T of G such that as many vertices of T as possible have the same degree in T as in G. This problem is a graph-theoretical translation of a problem arising in the system-theoretical context of identifiability in networks, a concept which has applications in, for example, water distribution networks and electrical networks. We show that the DPST problem is NP-complete, even when restricted to split graphs or bipartite planar graphs, but that it can be solved in polynomial time for graphs with a bounded asteroidal number and for graphs with a bounded treewidth. For the class of interval graphs, we give a linear time algorithm. For the class of cocomparability graphs, we give an O(n4) algorithm. Furthermore, we present linear time approximation algorithms for planar graphs of a worst-case performance ratio of 1 − ϵ for every ϵ > 0. © 2000 John Wiley & Sons, Inc.