Constant factor approximation algorithms for the densest k-subgraph problem on proper interval graphs and bipartite permutation graphs

The densest k-subgraph problem asks for a k-vertex subgraph with the maximum number of edges. This problem is NP-hard on bipartite graphs, chordal graphs, and planar graphs. A 3-approximation algorithm is known for chordal graphs. We present -approximation algorithms for proper interval graphs and bipartite permutation graphs. The latter result relies on a new characterisation of bipartite permutation graphs which may be of independent interest.     

Finding a dominating set on bipartite graphs

Finding a dominating set of minimum cardinality is an NP-hard graph problem, even when the graph is bipartite. In this paper we are interested in solving the problem on graphs having a large independent set. Given a graph G with an independent set of size z, we show that the problem can be solved in time O^∗(²n−z), where n is the number of vertices of G. As a consequence, our algorithm is able to solve the dominating set problem on bipartite graphs in time O^∗(²n/2). Another implication is an algorithm for general graphs whose running time is O(n^1.7088).     

On maximum induced matchings in bipartite graphs

The problem of finding a maximum induced matching is known to be NP-hard in general bipartite graphs. We strengthen this result by reducing the problem to some special classes of bipartite graphs such as bipartite graphs with maximum degree 3 or C₄-free bipartite graphs. On the other hand, we describe a new polynomially solvable case for the problem in bipartite graphs which deals with a generalization of bi-complement reducible graphs.     

The labeled perfect matching in bipartite graphs

In this paper, we deal with both the complexity and the approximability of the labeled perfect matching problem in bipartite graphs. Given a simple graph G=(V,E) with |V|=2n vertices such that E contains a perfect matching (of size n), together with a color (or label) function , the labeled perfect matching problem consists in finding a perfect matching on G that uses a minimum or a maximum number of colors.     

Some optimization problems on weak-bisplit graphs

A new decomposition scheme for bipartite graphs namely canonical decomposition was introduced by Fouquet et al. [Internat. J. Found. Comput. Sci. 10 (1999) 513-533]. The so-called weak-bisplit graphs are totally decomposable following this decomposition. We present here some optimization problems for general bipartite graphs which have efficient solutions when dealing with weak-bisplit graphs.     

Push-relabel based algorithms for the maximum transversal problem

We investigate the push-relabel algorithm for solving the problem of finding a maximum cardinality matching in a bipartite graph in the context of the maximum transversal problem. We describe in detail an optimized yet easy-to-implement version of the algorithm and fine-tune its parameters. We also introduce new performance-enhancing techniques. On a wide range of real-world instances, we compare the push-relabel algorithm with state-of-the-art algorithms based on augmenting paths and pseudoflows. We conclude that a carefully tuned push-relabel algorithm is competitive with all known augmenting path-based algorithms, and superior to the pseudoflow-based ones.     

Variations of the maximum leaf spanning tree problem for bipartite graphs

The maximum leaf spanning tree problem is known to be NP-complete. In [M.S. Rahman, M. Kaykobad, Complexities of some interesting problems on spanning trees, Inform. Process. Lett. 94 (2005) 93-97], a variation on this problem was posed. This variation restricts the problem to bipartite graphs and asks, for a fixed integer K, whether or not the graph contains a spanning tree with at least K leaves in one of the partite sets. We show not only that this problem is NP-complete, but that it remains NP-complete for planar bipartite graphs of maximum degree 4. We also consider a generalization of a related decision problem, which is known to be polynomial-time solvable. We show the problem is still polynomial-time solvable when generalized to weighted graphs.     

Parallel algorithms for the hamiltonian cycle and hamiltonian path problems in semicomplete bipartite digraphs

We give anO(log⁴ n)-timeO(n ²)-processor CRCW PRAM algorithm to find a hamiltonian cycle in a strong semicomplete bipartite digraph,B, provided that a factor ofB (i.e., a collection of vertex disjoint cycles covering the vertex set ofB) is computed in a preprocessing step. The factor is found (if it exists) using a bipartite matching algorithm, hence placing the whole algorithm in the class Random-NC. We show that any parallel algorithm which can check the existence of a hamiltonian cycle in a strong semicomplete bipartite digraph in timeO(r(n)) usingp(n) processors can be used to check the existence of a perfect matching in a bipartite graph in timeO(r(n)+n ² /p(n)) usingp(n) processors. Hence, our problem belongs to the class NC if and only if perfect matching in bipartite graphs belongs to NC. We also consider the problem of finding a hamiltonian path in a semicomplete bipartite digraph.     

Edge coloring of bipartite graphs with constraints

A classical result from graph theory states that the edges of an l-regular bipartite graph can be colored using exactly l colors so that edges that share an endpoint are assigned different colors. In this paper, we study two constrained versions of the bipartite edge coloring problem.

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A linear algorithm for the all-bidirectional-edges problem on planar graphs

The all-bidirectional-edges problem is to find an edge-labeling of an undirected networkG=(V, E), with a source and a sink, such that an edgee=uv inE is labeled u, v or u, u (or both) depending on the existence of a (simple) path from the source to the sink traversinge, that visits the verticesu andv in the orderu, v orv, u respectively. The best-known algorithm for this problem requiresO(¦V¦·¦E¦) time [5]. We show that the problem is solvable optimally on a planar graph.     

On the longest path algorithm for reconstructing trees from distance matrices

Culberson and Rudnicki [J.C. Culberson, P. Rudnicki, A fast algorithm for constructing trees from distance matrices, Inform. Process. Lett. 30 (4) (1989) 215-220] gave an algorithm that reconstructs a degree d restricted tree from its distance matrix. According to their analysis, it runs in time O(dnlogdn) for topological trees. However, this turns out to be false; we show that the algorithm takes time in the topological case, giving tight examples.     

Edge crossings in drawings of bipartite graphs

Systems engineers have recently shown interest in algorithms for drawing directed graphs so that they are easy to understand and remember. Each of the commonly used methods has a step which aims to adjust the drawing to decrease the number of arc crossings. We show that the most popular strategy involves an NP-complete problem regarding the minimization of the number of arcs in crossings in a bipartite graph. The performance of the commonly employed barycenter heuristic for this problem is analyzed. An alternative method, the median heuristic, is proposed and analyzed. The new method is shown to compare favorably with the old in terms of performance guarantees. As a bonus, we show that the median heuristic performs well with regard to the total length of the arcs in the drawing.     

The computational complexity of the backbone coloring problem for bounded-degree graphs with connected backbones

Given a graph G, a spanning subgraph H of G   and an integer λ≥2$\lambda \ge 2$, a λ-backbone coloring of G with backbone H is a proper vertex coloring of G   using colors 1,2,…$1,2,\dots$, in which the color difference between vertices adjacent in H is greater than or equal to λ. The backbone coloring problem is that of finding such a coloring whose maximum color does not exceed a given limit k  . In this paper, we study the backbone coloring problem for bounded-degree graphs with connected backbones and we give a complete computational complexity classification of this problem. We present a polynomial algorithm for optimal backbone coloring for subcubic graphs with arbitrary backbones. We also prove that the backbone coloring problem for graphs with arbitrary backbones and with fixed maximum degree (at least 4) is NP-complete. Furthermore, we show that for the special case of graphs with fixed maximum degree at least 5 and λ≥4$\lambda \ge 4$ the problem remains NP-complete even for spanning tree backbones.     

Constrained minimum vertex cover in bipartite graphs: complexity and parameterized algorithms

Motivated by the research in reconfigurable memory array structures, this paper studies the complexity and algorithms for the constrained minimum vertex cover problem on bipartite graphs (min-cvcb) defined as follows: given a bipartite graph G=(V,E) with vertex bipartition V=U∪L and two integers k_u and k_l, decide whether there is a minimum vertex cover in G with at most k_u vertices in U and at most k_l vertices in L. It is proved in this paper that the min-cvcb problem is NP-complete. This answers a question posed by Hasan and Liu. A parameterized algorithm is developed for the problem, in which classical results in matching theory and recently developed techniques in parameterized computation theory are nicely combined and extended. The algorithm runs in time O(1.26^k_u+k_l+(k_u+k_l)|G|) and significantly improves previous algorithms for the problem.     

On approximating the longest path in a graph

We consider the problem of approximating the longest path in undirected graphs. In an attempt to pin down the best achievable performance ratio of an approximation algorithm for this problem, we present both positive and negative results. First, a simple greedy algorithm is shown to find long paths in dense graphs. We then consider the problem of finding paths in graphs that are guaranteed to have extremely long paths. We devise an algorithm that finds paths of a logarithmic length in Hamiltonian graphs. This algorithm works for a much larger class of graphs (weakly Hamiltonian), where the result is the best possible. Since the hard case appears to be that of sparse graphs, we also consider sparse random graphs. Here we show that a relatively long path can be obtained, thereby partially answering an open problem of Broderet al. To explain the difficulty of obtaining better approximations, we also prove hardness results. We show that, for any ε<1, the problem of finding a path of lengthn-n ^ε in ann-vertex Hamiltonian graph isNP-hard. We then show that no polynomial-time algorithm can find a constant factor approximation to the longest-path problem unlessP=NP. We conjecture that the result can be strengthened to say that, for some constant δ>0, finding an approximation of ration ^δ is alsoNP-hard. As evidence toward this conjecture, we show that if any polynomial-time algorithm can approximate the longest path to a ratio of , for any ε>0, thenNP has a quasi-polynomial deterministic time simulation. The hardness results apply even to the special case where the input consists of bounded degree graphs. D. Karger was supported by an NSF Graduate Fellowship, NSF Grant CCR-9010517, and grants from the Mitsubishi Corporation and OTL. R. Motwani was supported by an Alfred P. Sloan Research Fellowship, an IBM Faculty Development Award, grants from Mitsubishi and OTL, NSF Grant CCR-9010517, and NSF Young Investigator Award CCR-9357849, with matching funds from IBM, the Schlumberger Foundation, the Shell Foundation, and the Xerox Corporation, G. D. S. Ramkumar was supported by a grant from the Toshiba Corporation. Communicated by M. X. Goemans.     

A parallel algorithm for generating bicompatible elimination orderings of proper interval graphs

In this paper, we first show how a certain ordering of vertices, called bicompatible elimination ordering (BCO), of a proper interval graph can be used to solve efficiently several problems in proper interval graphs. We, then, propose an NC parallel algorithm (i.e., polylogarithmic-time employing a polynomial number of processors) in SIMD CRCW PRAM (Single Instruction Stream Multiple Data Stream Concurrent Read Concurrent Write Parallel Random Access Machine) model of parallel computation to compute a BCO of a proper interval graph. To the best of our knowledge, this is the first NC parallel algorithm to compute a BCO of a proper interval graph.