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 the complexity of signed and minus total domination in graphs

In this paper we present unified methods to solve the minus and signed total domination problems for chordal bipartite graphs and trees in O(n²) and O(n+m) time, respectively. We also prove that the decision problem for the signed total domination problem on doubly chordal graphs is NP-complete. Note that bipartite permutation graphs, biconvex bipartite graphs, and convex bipartite graphs are subclasses of chordal bipartite graphs.     

An adjustable linear time parallel algorithm for maximum weight bipartite matching

We present a parallel algorithm for finding a maximum weight matching in general bipartite graphs with an adjustable time complexity of using O(nmax(2ω,4+ω)) processing elements for ω?1. Parameter ω is not bounded. This is the fastest known strongly polynomial parallel algorithm to solve this problem. This is also the first adjustable parallel algorithm for the maximum weight bipartite matching problem in which the execution time can be reduced by an unbounded factor. We also present a general approach for finding efficient parallel algorithms for the maximum matching problem.     

Adjacency queries in dynamic sparse graphs

We deal with the problem of maintaining a dynamic graph so that queries of the form “is there an edge between u and v?” are processed fast. We consider graphs of bounded arboricity, i.e., graphs with no dense subgraphs, like, for example, planar graphs. Brodal and Fagerberg [G.S. Brodal, R. Fagerberg, Dynamic representations of sparse graphs, in: Proc. 6th Internat. Workshop on Algorithms and Data Structures (WADS'99), in: Lecture Notes in Comput. Sci., vol. 1663, Springer, Berlin, 1999, pp. 342-351] described a very simple linear-size data structure which processes queries in constant worst-case time and performs insertions and deletions in O(1) and O(logn) amortized time, respectively. We show a complementary result that their data structure can be used to get O(logn) worst-case time for query, O(1) amortized time for insertions and O(1) worst-case time for deletions. Moreover, our analysis shows that by combining the data structure of Brodal and Fagerberg with efficient dictionaries one gets O(logloglogn) worst-case time bound for queries and deletions and O(logloglogn) amortized time for insertions, with size of the data structure still linear. This last result holds even for graphs of arboricity bounded by O(logkn), for some constant k.     

Counting edge crossings in a 2-layered drawing

Given a bipartite graph G=(V,W,E) with a bipartition {V,W} of a vertex set and an edge set E, a 2-layered drawing of G in the plane means that the vertices of V and W are respectively drawn as distinct points on two parallel lines and the edges as straight line segments. We consider the problem of counting the number of edge crossings. In this paper, we design two algorithms to this problem based on the dynamic programming and divide-and-conquer approaches. These algorithms run in O(n₁n₂) time and O(m) space and in O(min{n₁n₂,|E|log(min{|V|,|W|})}) time and O(m) space, respectively. Our algorithms outperform the previously fastest Θ(|E|log(min{|V|,|W|})) time algorithm for dense 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.     

Fully dynamic biconnectivity in graphs

We present an algorithm for maintaining the biconnected components of a graph during a sequence of edge insertions and deletions. It requires linear storage and preprocessing time. The amortized running time for insertions and for deletions isO(m ^2/3 ), wherem is the number of edges in the graph. Any query of the form ‘Are the verticesu andv biconnected?’ can be answered in timeO(1). This is the first sublinear algorithm for this problem. We can also output all articulation points separating any two vertices efficiently. If the input is a plane graph, the amortized running time for insertions and deletions drops toO(√n logn) and the query time isO(log² n), wheren is the number of vertices in the graph. The best previously known solution takes timeO(n ^2/3 ) per update or query.     

Algorithms for maximum weight induced paths

A family of graphs is a k-bounded-hole family if every graph in the family has no holes with more than k vertices. The problem of finding in a graph a maximum weight induced path has applications in large communication and neural networks when worst case communication time needs to be evaluated; unfortunately this problem is NP-hard even when restricted to bipartite graphs. We show that this problem has polynomial time algorithms for k-bounded-hole families of graphs, for interval-filament graphs and for graphs decomposable by clique cut-sets or by splits into prime subgraphs for which such algorithms exist.     

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.     

Efficient robust algorithms for the Maximum Weight Stable Set Problem in chair-free graph classes

Modular decomposition of graphs is a powerful tool for designing efficient algorithms for problems on graphs such as Maximum Weight Stable Set (MWS) and Maximum Weight Clique. Using this tool we obtain O(n·m) time algorithms for MWS on chair- and xbull-free graphs which considerably extend an earlier result on bull- and chair-free graphs by De Simone and Sassano (the chair is the graph with vertices a,b,c,d,e and edges ab,bc,cd,be, and the xbull is the graph with vertices a,b,c,d,e,f and edges ab,bc,cd,de,bf,cf). Moreover, our algorithm is robust in the sense that we do not have to check in advance whether the input graphs are indeed chair- and xbull-free.     

Algorithms for transitive closure

Let σ′(n) denote the number of all strongly connected graphs on the n-element set. We prove that σ′(n)?2ⁿ²·(1−n(n−1)/2ⁿ⁻¹). Hence the algorithm computing a transitive closure by a reduction to acyclic graphs has the expected time O(n²), under the assumption of uniform distribution of input graphs. Furthermore, we present a new algorithm constructing the transitive closure of an acyclic graph.     

Dominating set based exact algorithms for 3-coloring

We show that the 3-colorability problem can be solved in O(n^1.296) time on any n-vertex graph with minimum degree at least 15. This algorithm is obtained by constructing a dominating set of the graph greedily, enumerating all possible 3-colorings of the dominating set, and then solving the resulting 2-list coloring instances in polynomial time. We also show that a 3-coloring can be obtained in ²o(n) time for graphs having minimum degree at least ω(n) where ω(n) is any function which goes to ∞. We also show that if the lower bound on minimum degree is replaced by a constant (however large it may be), then neither a ²o(n) time nor a ²o(m) time algorithm is possible (m denotes the number of edges) for 3-colorability unless Exponential Time Hypothesis (ETH) fails. We also describe an algorithm which obtains a 4-coloring of a 3-colorable graph in O(n^1.2535) time.     

Computing rank-width exactly

We prove that the rank-width of an n-vertex graph can be computed exactly in time O(n²n³log²nloglogn). To improve over a trivial O(n³logn)-time algorithm, we develop a general framework for decompositions on which an optimal decomposition can be computed efficiently. This framework may be used for other width parameters, including the branch-width of matroids and the carving-width of graphs.     

Locally connected spanning trees in cographs, complements of bipartite graphs and doubly chordal graphs

A spanning tree T of a graph G=(V,E) is called a locally connected spanning tree if the set of all neighbors of v in T induces a connected subgraph of G for all v∈V. The problem of recognizing whether a graph admits a locally connected spanning tree is known to be NP-complete even when the input graphs are restricted to chordal graphs. In this paper, we propose linear time algorithms for finding locally connected spanning trees in cographs, complements of bipartite graphs and doubly chordal graphs, respectively.     

On a simple randomized algorithm for finding a 2-factor in sparse graphs

We analyze the performance of a simple randomized algorithm for finding 2-factors in directed Hamiltonian graphs of out-degree at most two and in undirected Hamiltonian graphs of degree at most three. For the directed case, the algorithm finds a 2-factor in O(n²) expected time. The analysis of our algorithm is based on random walks on the line and interestingly resembles the analysis of a randomized algorithm for the 2-SAT problem given by Papadimitriou [On selecting a satisfying truth assignment, in: Proc. 32nd Annual IEEE Symp. on the Foundations of Computer Science (FOCS), 1991, p. 163]. For the undirected case, the algorithm finds a 2-factor in O(n³) expected time. We also analyze random versions of these graphs and show that cycles of length Ω(n/logn) can be found with high probability in polynomial time. This partially answers an open question of Broder et al. [Finding hidden Hamilton cycles, Random Structures Algorithms 5 (1994) 395] on finding hidden Hamiltonian cycles in sparse random graphs and improves on a result of Karger et al. [On approximating the longest path in a graph, Algorithmica 18 (1997) 82].     

On simultaneous straight-line grid embedding of a planar graph and its dual

Simultaneous representations of planar graphs and their duals normally require that the dual vertices to be placed inside their corresponding primal faces, and the edges of the dual graph to cross only their corresponding primal edges. Erten and Kobourov [C. Erten, S.G. Kobourov, Simultaneous embedding of a planar graph and its dual on the grid, Theory Computer Systems 38 (2005) 313-327] provided a linear time algorithm on simultaneous straight-line grid embedding of a 3-connected planar graph and its dual such that all the vertices are placed on grid points and each edge is drawn as one straight-line segment except for one which is drawn using two segments. Their drawing size is (2n−2)×(2n−2), where n is the total number of vertices in the graph and its dual. They raised an open question on whether there is a large class of planar graphs that allows this simultaneous straight-line grid embedding on a smaller grid. We answer this open question by giving a linear time simultaneous straight-line grid embedding algorithm for a 3-connected planar graph and its dual on a grid of size (n−1)×n.     

Exact exponential-time algorithms for finding bicliques

Due to a large number of applications, bicliques of graphs have been widely considered in the literature. This paper focuses on non-induced bicliques. Given a graph G=(V,E) on n vertices, a pair (X,Y), with X,Y⊆V, X∩Y=∅, is a non-induced biclique if {x,y}∈E for any x∈X and y∈Y. The NP-complete problem of finding a non-induced (k₁,k₂)-biclique asks to decide whether G contains a non-induced biclique (X,Y) such that |X|=k₁ and |Y|=k₂. In this paper, we design a polynomial-space O(n^1.6914)-time algorithm for this problem. It is based on an algorithm for bipartite graphs that runs in time O(n^1.30052). In deriving this algorithm, we also exhibit a relation to the spare allocation problem known from memory chip fabrication. As a byproduct, we show that the constraint bipartite vertex cover problem can be solved in time O(n^1.30052).     

Trading uninitialized space for time

The design of efficient graph algorithms usually precludes the test of edge existence, because an efficient support of that operation already requires time for the initialization of an adjacency-matrix representation. We describe an alternative representation of static directed graphs taking Θ(n+m) initialization time and using Θ(n²) space, which supports the efficient implementation of all usual operations on static graphs. The sparse graph representation allows the design of efficient graph algorithms using both iteration over all vertices adjacent with a given vertex and edge-existence operations, although at the expense of additional (uninitialized) space which may, nevertheless, be used for other purposes. To the best of our knowledge, the representation leads to the first graph algorithms with the disconcerting property that the time complexity is better than the space complexity.     

New length bounds for cycle bases

Based on a recent work by Abraham, Bartal and Neiman (2007), we construct a strictly fundamental cycle basis of length O(n²) for any unweighted graph, whence proving the conjecture of Deo et al. (1982).For weighted graphs, we construct cycle bases of length O(W⋅lognloglogn), where W denotes the sum of the weights of the edges. This improves the upper bound that follows from the result of Elkin et al. (2005) by a logarithmic factor and, for comparison from below, some natural classes of large girth graphs are known to exhibit minimum cycle bases of length Ω(W⋅logn).We achieve this bound for weighted graphs by not restricting ourselves to strictly fundamental cycle bases—as it is inherent to the approach of Elkin et al.—but rather also considering weakly fundamental cycle bases in our construction. This way we profit from some nice properties of Hierarchically Well-Separated Trees that were introduced by Bartal (1998).