Computing the Treewidth and the Minimum Fill-In with the Modular Decomposition

Using the notion of modular decomposition we extend the class of graphs on which both the treewidth and the minimum fill-in can be solved in polynomial time. We show that if C is a class of graphs that are modularly decomposable into graphs that have a polynomial number of minimal separators, or graphs formed by adding a matching between two cliques, then both the treewidth and the minimum fill-in on C can be solved in polynomial time. For the graphs that are modular decomposable into cycles we give algorithms that use respectively O(n) and O(n3) time for treewidth and minimum fill-in.     

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.     

On packing shortest cycles in graphs

We study the problems to find a maximum packing of shortest edge-disjoint cycles in a graph of given girth g (g-ESCP) and its vertex-disjoint analogue g-VSCP. In the case g=3, Caprara and Rizzi (2001) have shown that g-ESCP can be solved in polynomial time for graphs with maximum degree 4, but is APX-hard for graphs with maximum degree 5, while g-VSCP can be solved in polynomial time for graphs with maximum degree 3, but is APX-hard for graphs with maximum degree 4. For g∈{4,5}, we show that both problems allow polynomial time algorithms for instances with maximum degree 3, but are APX-hard for instances with maximum degree 4. For each g?6, both problems are APX-hard already for graphs with maximum degree 3.     

Optimal Reconstruction of Graphs under the Additive Model

We study the problem of reconstructing unknown graphs under the additive combinatorial search model. The main result concerns the reconstruction of bounded degree graphs, i.e., graphs with the degree of all vertices bounded by a constant d . We show that such graphs can be reconstructed in O(dn) nonadaptive queries, which matches the information-theoretic lower bound. The proof is based on the technique of separating matrices. A central result here is a new upper bound for a general class of separating matrices. As a particular case, we obtain a tight upper bound for the class of d -separating matrices, which settles an open question stated by Lindstr?m in [20]. Finally, we consider several particular classes of graphs. We show how an optimal nonadaptive solution of O(n ² / log n) queries for general graphs can be obtained. We also prove that trees with unbounded vertex degree can be reconstructed in a linear number of queries by a nonadaptive algorithm. Received August 1997; revised January 1999.     

Fast Parallel Reordering and Isomorphism Testing of k -Trees

Abstract. In this paper two problems on the class of k -trees, a subclass of the class of chordal graphs, are considered: the fast reordering problem and the isomorphism problem. An O(log ² n) time parallel algorithm for the fast reordering problem is described that uses O(nk(n-k)/\kern -1ptlog n) processors on a CRCW PRAM proving membership in the class NC for fixed k . An O(nk(k+1)!) time sequential algorithm for the isomorphism problem is obtained representing an improvement over the O(n ² k(k+1)!) algorithm of Sekharan (the second author) [10]. A parallel version of this sequential algorithm is presented that runs in O(log ² n) time using O((nk((k+1)!+n-k))/log n) processors improving on a parallel algorithm of Sekharan for the isomorphism problem [10]. Both the sequential and parallel algorithms use a concept introduced in this paper called the kernel of a k -tree.     

An improved exact algorithm for the domatic number problem

The 3-domatic number problem asks whether a given graph can be partitioned into three dominating sets. We prove that this problem can be solved by a deterministic algorithm in time n^2.695 (up to polynomial factors) and in polynomial space. This result improves the previous bound of n^2.8805, which is due to Björklund and Husfeldt. To prove our result, we combine an algorithm by Fomin et al. with Yamamoto's algorithm for the satisfiability problem. In addition, we show that the 3-domatic number problem can be solved for graphs G with bounded maximum degree Δ(G) by a randomized polynomial-space algorithm, whose running time is better than the previous bound due to Riege and Rothe whenever Δ(G)?5. Our new randomized algorithm employs Schöning's approach to constraint satisfaction problems.     

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.     

Listing all the minimum spanning trees in an undirected graph

Efficient polynomial time algorithms are well known for the minimum spanning tree problem. However, given an undirected graph with integer edge weights, minimum spanning trees may not be unique. In this article, we present an algorithm that lists all the minimum spanning trees included in the graph. The computational complexity of the algorithm is O(N(mn+n ² log n)) in time and O(m) in space, where n, m and N stand for the number of nodes, edges and minimum spanning trees, respectively. Next, we explore some properties of cut-sets, and based on these we construct an improved algorithm, which runs in O(N m log n) time and O(m) space. These algorithms are implemented in C language, and some numerical experiments are conducted for planar as well as complete graphs with random edge weights.     

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.     

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).     

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(n^{1.5 + .5/γ}). Under these conditions, Jerrum and Sinclair showed that a Markov chain of Broder can generate approximately uniform variates in Θ(n^{4.5 + .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(n^{1.5 + .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.     

On the complexity of finding balanced oneway cuts

A bisection of an n-vertex graph is a partition of its vertices into two sets S and T, each of size n/2. The bisection cost is the number of edges connecting the two sets. In directed graphs, the cost is the number of arcs going from S to T. Finding a minimum cost bisection is NP-hard for both undirected and directed graphs. For the undirected case, an approximation of ratio O(log²n) is known. We show that directed minimum bisection is not approximable at all. More specifically, we show that it is NP-hard to tell whether there exists a directed bisection of cost 0, which we call oneway bisection. In addition, we study the complexity of the problem when some slackness in the size of S is allowed, namely, (1/2−ε)n?|S|?(1/2+ε)n. We show that the problem is solvable in polynomial time when , and provide evidence that the problem is not solvable in polynomial time when ε=o(1/(logn)⁴).     

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.     

On Minimum-Area Hulls

We study some minimum-area hull problems that generalize the notion of convex hull to star-shaped and monotone hulls. Specifically, we consider the minimum-area star-shaped hull problem: Given an n -vertex simple polygon P , find a minimum-area, star-shaped polygon P ^* containing P . This problem arises in lattice packings of translates of multiple, nonidentical shapes in material layout problems (e.g., in clothing manufacture), and has been recently posed by Daniels and Milenkovic. We consider two versions of the problem: the restricted version, in which the vertices of P ^* are constrained to be vertices of P , and the unrestricted version, in which the vertices of P ^* can be anywhere in the plane. We prove that the restricted problem falls in the class of ``3sum-hard' (sometimes called ``n ² -hard') problems, which are suspected to admit no solutions in o(n ² ) time. Further, we give an O(n ² ) time algorithm, improving the previous bound of O(n ⁵ ) . We also show that the unrestricted problem can be solved in O(n ² p(n)) time, where p(n) is the time needed to find the roots of two equations in two unknowns, each a polynomial of degree O(n) . We also consider the case in which P ^* is required to be monotone, with respect to an unspecified direction; we refer to this as the minimum-area monotone hull problem. We give a matching lower and upper bound of Θ(n log n) time for computing P ^* in the restricted version, and an upper bound of O(n q(n)) time in the unrestricted version, where q(n) is the time needed to find the roots of two polynomial equations in two unknowns with degrees 2 and O(n) . Received November 1996; revised March 1997.     

A Linear Time Algorithm for the Arc Disjoint Menger Problem in Planar Directed Graphs

Given a graph G=(V,E) and two vertices s,t ∈ V , s\neq t , the Menger problem is to find a maximum number of disjoint paths connecting s and t . Depending on whether the input graph is directed or not, and what kind of disjointness criterion is demanded, this general formulation is specialized to the directed or undirected vertex, and the edge or arc disjoint Menger problem, respectively. For planar graphs the edge disjoint Menger problem has been solved to optimality [W2], while the fastest algorithm for the arc disjoint version is Weihe's general maximum flow algorithm for planar networks [W1], which has running time \bf O (|V| log |V|) . Here we present a linear time, i.e., asymptotically optimal, algorithm for the arc disjoint version in planar directed graphs. Received August 1997; revised January 1999.     

On the Sum Coloring Problem on Interval Graphs

In this paper we study the following NP-complete problem: given an interval graph G = (V,E) , find a node p -coloring such that the cost is minimal, where denotes a partition of V whose subsets are ordered by nonincreasing cardinality. We present an O(m χ (G) + n log n) time ε -approximate algorithm (ε < 2) to solve the problem, where n , m , and χ #(G) are the number of nodes of the interval graph, its number of cliques, and its chromatic number, respectively. The algorithm is shown to solve the problem exactly on some classes of interval graphs, namely, the proper and the containment interval graphs, and the intersection graphs of sets of ``short' intervals. The problem of determining the minimum number of colors needed to achieve the minimum over all p -colorings of G is also addressed. Received February 1, 1996; revised August 22, 1997.     

Parallel Algorithms for Maximum Matching in Complements of Interval Graphs and Related Problems

Given a set of n intervals representing an interval graph, the problem of finding a maximum matching between pairs of disjoint (nonintersecting) intervals has been considered in the sequential model. In this paper we present parallel algorithms for computing maximum cardinality matchings among pairs of disjoint intervals in interval graphs in the EREW PRAM and hypercube models. For the general case of the problem, our algorithms compute a maximum matching in O( log ³ n) time using O(n/ log ² n) processors on the EREW PRAM and using n processors on the hypercubes. For the case of proper interval graphs, our algorithm runs in O( log n ) time using O(n) processors if the input intervals are not given already sorted and using O(n/ log n ) processors otherwise, on the EREW PRAM. On n -processor hypercubes, our algorithm for the proper interval case takes O( log n log log n ) time for unsorted input and O( log n ) time for sorted input. Our parallel results also lead to optimal sequential algorithms for computing maximum matchings among disjoint intervals. In addition, we present an improved parallel algorithm for maximum matching between overlapping intervals in proper interval graphs. Received November 20, 1995; revised September 3, 1998.     

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.     

Optimal Sublogarithmic Time Parallel Algorithms on Rooted Forests

The problem of finding a sublogarithmic time optimal parallel algorithm for 3 -colouring rooted forests has been open for long. We settle this problem by obtaining an O(( log log n) log ^* ( log ^* n)) time optimal parallel algorithm on a TOLERANT Concurrent Read Concurrent Write (CRCW) Parallel Random Access Machine (PRAM). Furthermore, we show that if f(n) is the running time of the best known algorithm for 3 -colouring a rooted forest on a COMMON or TOLERANT CRCW PRAM, a fractional independent set of the rooted forest can be found in O(f(n)) time with the same number of processors, on the same model. Using these results, it is shown that decomposable top-down algebraic computation and, hence, depth computation (ranking), 2 -colouring and prefix summation on rooted forests can be done in O( log n) optimal time on a TOLERANT CRCW PRAM. These algorithms have been obtained by proving a result of independent interest, one concerning the self-simulation property of TOLERANT: an N -processor TOLERANT CRCW PRAM that uses an address space of size O(N) only, can be simulated on an n -processor TOLERANT PRAM in O(N/n) time, with no asymptotic increase in space or cost, when n=O(N/ log log N) . Received May 20, 1997; revised June 15, 1998.     

Approximation Algorithms for Connected Dominating Sets

The dominating set problem in graphs asks for a minimum size subset of vertices with the following property: each vertex is required to be either in the dominating set, or adjacent to some vertex in the dominating set. We focus on the related question of finding a connected dominating set of minimum size, where the graph induced by vertices in the dominating set is required to be connected as well. This problem arises in network testing, as well as in wireless communication. Two polynomial time algorithms that achieve approximation factors of 2H(Δ)+2 and H(Δ)+2 are presented, where Δ is the maximum degree and H is the harmonic function. This question also arises in relation to the traveling tourist problem, where one is looking for the shortest tour such that each vertex is either visited or has at least one of its neighbors visited. We also consider a generalization of the problem to the weighted case, and give an algorithm with an approximation factor of (c _n +1) \ln n where c _n ln k is the approximation factor for the node weighted Steiner tree problem (currently c _n = 1.6103 ). We also consider the more general problem of finding a connected dominating set of a specified subset of vertices and provide a polynomial time algorithm with a (c+1) H(Δ) +c-1 approximation factor, where c is the Steiner approximation ratio for graphs (currently c = 1.644 ). Received June 22, 1996; revised February 28, 1997.