We define the problem as a two-player game between Algorithm and Builder. The game is played in rounds. Each round, Builder presents an interval that is neither contained in nor contains any previously presented inter...
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Given a class A of graphs and a decision problem pi belonging to NP, we say that a full complexity dichotomy of A was obtained if one describes a partition of A into subclasses such that pi is classified as polynomial...
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Given a class A of graphs and a decision problem pi belonging to NP, we say that a full complexity dichotomy of A was obtained if one describes a partition of A into subclasses such that pi is classified as polynomial or NP-complete when restricted to each subclass. The concept of full complexity dichotomy is particularly interesting for the investigation of NP-complete problems: as we partition a class A into NP-complete subclasses and polynomial subclasses, it becomes clearer why the problem is NP-complete in A. The class C of graphs that do not contain a cycle with a unique chord was studied by Trotignon and Vuskovic who proved a structure theorem which led to solving the vertex-colouring problem in polynomial time. In the present survey, we apply the structure theorem to study the complexity of edge-colouring and total-colouring, and show that even for graph classes with strong structure and powerful decompositions, the edge-colouring problem may be difficult. We discuss several surprising complexity dichotomies found in subclasses of C, and the concepts of separating problem proposed by David S. Johnson and the dual concept of separating class. (c) 2019 Elsevier B.V. All rights reserved.
Monitoring the interrelationships between social events and phenomena and forecasting the dynamics of their changes are necessary in the conditions of instability of the modern world. There are many separate methods o...
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We introduce I/O-efficient certifying algorithms for the recognition of bipartite, split, threshold, bipartite chain, and trivially perfect graphs. When the input graph is a member of the respective class, the certify...
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We present the first near-linear work and poly-logarithmic depth algorithm for computing a minimum cut in an undirected graph. Previous parallel algorithms with poly-logarithmic depth required at least quadratic work ...
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We present the first near-linear work and poly-logarithmic depth algorithm for computing a minimum cut in an undirected graph. Previous parallel algorithms with poly-logarithmic depth required at least quadratic work in the number of vertices. In a graph with n vertices and m edges, our randomized algorithm computes the minimum cut with high probability in O(m log(4) n) work and O(log(3) n) depth. This result is obtained by parallelizing a data structure that aggregates weights along paths in a tree, in addition exploiting the connection between minimum cuts and approximate maximum packings of spanning trees. In addition, our algorithm improves upon bounds on the number of cache misses incurred to compute a minimum cut.
The quest for efficient sorting is ongoing, and we will explore a graph-based stable sorting strategy, in particular employing comparison graphs. We use the topological sort to map the comparison graph to a linear dom...
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The quest for efficient sorting is ongoing, and we will explore a graph-based stable sorting strategy, in particular employing comparison graphs. We use the topological sort to map the comparison graph to a linear domain, and we can manipulate our graph such that the resulting topological sort is the sorted array. By taking advantage of the many relations between Hamiltonian paths and topological sorts in comparison graphs, we design a Divide-and-Conquer algorithm that runs in the optimal O(n logn) time. In the process, we construct a new merge process for graphs with relevant invariant properties for our use. Furthermore, this method is more space-efficient than the famous MERGESORT since we modify our fixed graph only. (C) 2020 Elsevier B.V. All rights reserved.
A ranking on a graph G is a function f : V(G) → {1, 2, . . ., k} with the following restriction: if f(u) = f(v) for any u, v ∈ V(G), then on every uv path in G, there exists a vertex w with f(w) > f(u). The optim...
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In the CO-PATH SET problem, the input is a graph G and an integer k, and the goal is to decide whether there is a set of at most k edges whose removal from G results in a graph in which every connected component is a ...
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In the CO-PATH SET problem, the input is a graph G and an integer k, and the goal is to decide whether there is a set of at most k edges whose removal from G results in a graph in which every connected component is a path. In this paper we give a deterministic O*(2k)-time algorithm for CO-PATH SET.(c) 2022 Elsevier B.V. All rights reserved.
We give a polynomial time approximation scheme (PTAS) for the unit demand capacitated vehicle routing problem (CVRP) on trees, for the entire range of the tour capacity. The result extends to the splittable CVRP.
We give a polynomial time approximation scheme (PTAS) for the unit demand capacitated vehicle routing problem (CVRP) on trees, for the entire range of the tour capacity. The result extends to the splittable CVRP.
In this paper we consider the problem of listing the maximal k-degenerate induced subgraphs of a chordal graph, and propose an output-sensitive algorithm using delay O(***(G)) for any n-vertex chordal graph with m edg...
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In this paper we consider the problem of listing the maximal k-degenerate induced subgraphs of a chordal graph, and propose an output-sensitive algorithm using delay O(***(G)) for any n-vertex chordal graph with m edges, where omega(G) <= n is the maximum size of a clique in G. Degeneracy is a well known sparsity measure, and k-degenerate subgraphs are a notion of sparse subgraphs, which generalizes other problems such as independent sets (0-degenerate subgraphs) and forests (1-degenerate subgraphs). Many efficient enumeration algorithms are designed by solving the so-called Extension problem, which asks whether there exists a maximal solution containing a given set of nodes, but no node from a forbidden set. We show that solving this problem is NP-complete for maximal k-degenerate induced subgraphs, motivating the need for additional techniques. (C) 2018 Elsevier B.V. All rights reserved.
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