A general algorithmic scheme for combinatorial decompositions with application to modular decompositions of hypergraphs

We consider problems that can be formulated as a task of finding an optimal triangulation of a graph w.r.t. some notion of optimality. We present algorithms parameterized by the size of a minimum edge clique cover (texttt {cc}) to such problems. This parameterization occurs naturally in many problems in this setting, e.g., in the perfect phylogeny problem texttt {cc} is at most the number of taxa, in fractional hypertreewidth texttt {cc} is at most the number of hyperedges, and in treewidth of Bayesian networks texttt {cc} is at most the number of non-root nodes. We show that the number of minimal separators of graphs is at most 2^texttt {cc}, the number of potential maximal cliques is at most 3^texttt {cc}, and these objects can be listed in times O^*(2^texttt {cc}) and O^*(3^texttt {cc}), respectively, even when no edge clique cover is given as input; the O^*(cdot ) notation omits factors polynomial in the input size. These enumeration algorithms imply O^*(3^texttt {cc}) time algorithms for problems such as treewidth, weighted minimum fill-in, and feedback vertex set. For generalized and fractional hypertreewidth we give O^*(4^m) time and O^*(3^m) time algorithms, respectively, where m is the number of hyperedges. When an edge clique cover of size texttt {cc}' is given as a part of the input we give O^*(2^{texttt {cc}'}) time algorithms for treewidth, minimum fill-in, and chordal sandwich. This implies an O^*(2^n) time algorithm for perfect phylogeny, where n is the number of taxa. We also give polynomial space algorithms with time complexities O^*(9^{texttt {cc}'}) and O^*(9^{texttt {cc}+ O(log ^2 texttt {cc})}) for problems in this framework.

Algorithmic aspects of switch cographs

An Algorithm for the Modular Decomposition of Hypergraphs

We consider the fine-grained complexity of sparse graph problems that currently have O(mn) time algorithms, where m is the number of edges and n is the number of vertices in the input graph. This class includes several important path problems on both directed and undirected graphs, including APSP, MWC (Minimum Weight Cycle), Radius, Eccentricities, BC (Betweenness Centrality), etc. We introduce the notion of a sparse reduction which preserves the sparsity of graphs, and we present near linear-time sparse reductions between various pairs of graph problems in the O(mn) class. There are many sub-cubic reductions between graph problems in the O(mn) class, but surprisingly few of these preserve sparsity. In the directed case, our results give a partial order on a large collection of problems in the O(mn) class (along with some equivalences), and many of our reductions are very nontrivial. In the undirected case we give two nontrivial sparse reductions: from MWC to APSP, and from unweighted ANSC (all nodes shortest cycles) to unweighted APSP. We develop a new ‘bit-sampling’ method for these sparse reductions on undirected graphs, which also gives rise to improved or simpler algorithms for cycle finding problems in undirected graphs. We formulate the the notion of MWC hardness, which is based on the assumption that a minimum weight cycle in a directed graph cannot be computed in time polynomially smaller than mn. Our sparse reductions for directed path problems in the O(mn) class establish that several problems in this class, including 2-SiSP (second simple shortest path), s-t Replacement Paths, Radius, Eccentricities and BC are MWC hard. Our sparse reductions give MWC hardness a status for the O(mn) class similar to 3SUM hardness for the quadratic class, since they show sub-mn hardness for a large collection of fundamental and well-studied graph problems that have maintained an O(mn) time bound for over half a century. We also identify Eccentricities and BC as key problems in the O(mn) class which are simultaneously MWC-hard, SETH-hard and k-DSH-hard, where SETH is the Strong Exponential Time Hypothesis, and k-DSH is the hypothesis that a dominating set of size k cannot be computed in time polynomially smaller than nk. Our framework using sparse reductions is very relevant to real-world graphs, which tend to be sparse and for which the O(mn) time algorithms are the ones typically used in practice, and not the O(n3) time algorithms.

We study the bipartite crossing number problem. When the minimum degree and the maximum degree of the graph are close to each other, we derive two polynomial time approximation algorithms for solving this problem, with approximation factors, O(log2 n), and O(log n log log n), from the optimal, respectively, where n is the number of vertices. This problem had been known to be NP-hard, but no approximation algorithm which could generate a provably good solution was known. An important aspect of our work has been relating this problem to the linear arrangement problem. Indeed using this relationship we also present an O(n 1.6) time algorithm for computing the bipartite crossing number of a tree.We also settle down the problem of computing a largest weighted biplanar subgraph of an acyclic graph by providing a linear time algorithm to it. This problem was known to be NP-hard when graph is planar and very sparse, and all weights are 1.KeywordsBipartite GraphLinear Time AlgorithmMedian PropertyLinear ArrangementPolynomial Time Approximation AlgorithmThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The first polynomial time algorithm (\(\mathcal{O}(n^4)\)) for modular decomposition appeared in 1972 [8] and since then there have been incremental improvements, eventually resulting in linear-time algorithms [22,7,23,9]. Although having optimal time complexity these algorithms are quite complicated and difficult to implement. In this paper we present an easily implementable linear-time algorithm for modular decomposition. This algorithm uses the notion of factorizing permutation and a new data-structure, the Ordered Chain Partitions.KeywordsInterval GraphDecomposition TreeRecursive CallStrong ModuleChordal GraphThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Interval graph representation with given interval and intersection lengths

We present improved algorithms for short cycle decomposition of a graph. Short cycle decompositions were introduced in the recent work of Chu et al, and were used to make progress on several questions in graph sparsification. For all constants $\delta \in (0,1]$, we give an $O(mn^\delta)$ time algorithm that, given a graph $G,$ partitions its edges into cycles of length $O(\log n)^\frac{1}{\delta}$, with $O(n)$ extra edges not in any cycle. This gives the first subquadratic, in fact almost linear time, algorithm achieving polylogarithmic cycle lengths. We also give an $m \cdot \exp(O(\sqrt{\log n}))$ time algorithm that partitions the edges of a graph into cycles of length $\exp(O(\sqrt{\log n} \log\log n))$, with $O(n)$ extra edges not in any cycle. This improves on the short cycle decomposition algorithms given in Chu et al in terms of all parameters, and is significantly simpler. As a result, we obtain faster algorithms and improved guarantees for several problems in graph sparsification -- construction of resistance sparsifiers, graphical spectral sketches, degree preserving sparsifiers, and approximating the effective resistances of all edges.

This paper considers the problem of locating a path-shaped facility of a specific size on a weighted tree network. The criterion for optimality used in this paper is the minimax criterion in which the weighted distance to the farthest vertex from the facility is minimised. The minimax criterion gives acceptable results from the point of view of the fast response for the clients who are located far away from the facility. This location problem usually has applications in computer science, information science, and operations research. An O(n) time algorithm is proposed for finding the optimal location of a path-shaped facility of a bounded length on a weighted tree network, where n is the number of vertices in the tree.

A fully dynamic algorithm for modular decomposition and recognition of cographs

Complexity of independency and cliquy trees

We present an O(nrG) time algorithm for computing and maintaining a minimum length shortest watchman tour that sees a simple polygon under monotone visibility in direction \(\theta \), while \(\theta \) varies in \([0,180^{\circ })\), obtaining the directions for the tour to be the shortest one over all tours, where n is the number of vertices, r is the number of reflex vertices, and \(G\le r\) is the maximum number of gates of the polygon used at any time in the algorithm.

We study the complexity of Power Edge Set (PES), a problem dedicated to the monitoring of an electric network. In such context we propose some new complexity results. We show that PES remains \(\mathcal {NP}\)-hard in planar graphs with degree at most five. This result is extended to bipartite planar graphs with degree at most six. We also show that PES is hard to approximate within a factor lower than Open image in new window in the bipartite case (resp. \(17/15-\epsilon \)), unless \(\mathcal {P}=\mathcal {NP}\), (resp. under \(\mathcal {UGC}\)). We also show that, assuming \(\mathcal {ETH}\), there is no \(2^{o(\sqrt{n})}\)-time algorithm and no \(2^{o(k)}n^{O(1)}\)-time parameterized algorithm, where n is the number of vertices and k the number of PMUs placed. These results improve the current best known bounds.

This paper addresses the minmax regret 1-sink location problem on dynamic flow path networks with parametric weights. We are given a dynamic flow network consisting of an undirected path with positive edge lengths, positive edge capacities, and nonnegative vertex weights. A path can be considered as a road, an edge length as the distance along the road and a vertex weight as the number of people at the site. An edge capacity limits the number of people that can enter the edge per unit time. We consider the problem of locating a sink in the network, to which all the people evacuate from the vertices as quickly as possible. In our model, each weight is represented by a linear function in a common parameter t, and the decision maker who determines the location of a sink does not know the value of t. We formulate the sink location problem under such uncertainty as the minmax regret problem. Given t and a sink location x, the cost of x under t is the sum of arrival times at x for all the people determined by t. The regret for x under t is the gap between the cost of x under t and the optimal cost under t. The task of the problem is formulated as the one to find a sink location that minimizes the maximum regret over all t. For the problem, we propose an \(O(n^4 2^{\alpha (n)} \alpha (n) \log n)\) time algorithm where n is the number of vertices in the network and \(\alpha (\cdot )\) is the inverse Ackermann function. Also for the special case in which every edge has the same capacity, we show that the complexity can be reduced to \(O(n^3 2^{\alpha (n)} \alpha (n) \log n)\).

We present an algorithm for maintaining maximal matching in a graph under addition and deletion of edges. Our algorithm is randomized and it takes expected amortized $O(\log n)$ time for each edge update, where $n$ is the number of vertices in the graph. While there exists a trivial $O(n)$ time algorithm for each edge update, the previous best known result for this problem is due to Ivkovicź and Lloyd [Lecture Notes in Comput. Sci. 790, Springer-Verlag, London, 1994, pp. 99--111]. For a graph with $n$ vertices and $m$ edges, they gave an $O( {(n+ m)}^{0.7072})$ update time algorithm which is sublinear only for a sparse graph. For the related problem of maximum matching, Onak and Rubinfeld [Proceedings of STOC'10, Cambridge, MA, 2010, pp. 457--464] designed a randomized algorithm that achieves expected amortized $O(\log^2 n)$ time for each update for maintaining a $c$-approximate maximum matching for some unspecified large constant $c$. In contrast, we can maintain a factor 2 approximate maximum matching in expected amortized $O(\log n )$ time per update as a direct corollary of the maximal matching scheme. This in turn also implies a 2-approximate vertex cover maintenance scheme that takes expected amortized $O(\log n )$ time per update.