scholarly journals Smaller Cuts, Higher Lower Bounds

2021 ◽  
Vol 17 (4) ◽  
pp. 1-40
Author(s):  
Amir Abboud ◽  
Keren Censor-Hillel ◽  
Seri Khoury ◽  
Ami Paz
Keyword(s):  
Lower Bounds ◽  
Vertex Cover ◽  
Shortest Paths ◽  
Independent Set ◽  
Maximum Independent Set ◽  
Graph Problems ◽  
All Pairs Shortest Paths ◽  
Dense Graphs ◽  
Hard Problems ◽  
A New Technique

This article proves strong lower bounds for distributed computing in the congest model, by presenting the bit-gadget : a new technique for constructing graphs with small cuts. The contribution of bit-gadgets is twofold. First, developing careful sparse graph constructions with small cuts extends known techniques to show a near-linear lower bound for computing the diameter, a result previously known only for dense graphs. Moreover, the sparseness of the construction plays a crucial role in applying it to approximations of various distance computation problems, drastically improving over what can be obtained when using dense graphs. Second, small cuts are essential for proving super-linear lower bounds, none of which were known prior to this work. In fact, they allow us to show near-quadratic lower bounds for several problems, such as exact minimum vertex cover or maximum independent set, as well as for coloring a graph with its chromatic number. Such strong lower bounds are not limited to NP-hard problems, as given by two simple graph problems in P, which are shown to require a quadratic and near-quadratic number of rounds. All of the above are optimal up to logarithmic factors. In addition, in this context, the complexity of the all-pairs-shortest-paths problem is discussed. Finally, it is shown that graph constructions for congest lower bounds translate to lower bounds for the semi-streaming model, despite being very different in its nature.

scholarly journals Improving Vertex Cover as a Graph Parameter

2015 ◽  
Vol Vol. 17 no.2 (Discrete Algorithms) ◽  
Author(s):  
Robert Ganian
Keyword(s):  
Vertex Cover ◽  
Graph Structure ◽  
Fpt Algorithms ◽  
Graph Problems ◽  
Graph Classes ◽  
Wide Range ◽  
Dense Graphs ◽  
International Audience ◽  
Hard Problems ◽  
Tree Width

International audience Parameterized algorithms are often used to efficiently solve NP-hard problems on graphs. In this context, vertex cover is used as a powerful parameter for dealing with graph problems which are hard to solve even when parameterized by tree-width; however, the drawback of vertex cover is that bounding it severely restricts admissible graph classes. We introduce a generalization of vertex cover called twin-cover and show that FPT algorithms exist for a wide range of difficult problems when parameterized by twin-cover. The advantage of twin-cover over vertex cover is that it imposes a lesser restriction on the graph structure and attains low values even on dense graphs. Apart from introducing the parameter itself, this article provides a number of new FPT algorithms parameterized by twin-cover with a special emphasis on solving problems which are not in FPT even when parameterized by tree-width. It also shows that MS1 model checking can be done in elementary FPT time parameterized by twin-cover and discusses the field of kernelization.

scholarly journals Gumbel-softmax-based Optimization: A Simple General Framework for Optimization Problems on Graphs

2020 ◽  
Author(s):  
Yaoxin Li ◽  
Jing Liu ◽  
Guozheng Lin ◽  
Yueyuan Hou ◽  
Muyun Mou ◽  
...  
Keyword(s):  
Objective Function ◽  
Statistical Physics ◽  
Automatic Differentiation ◽  
Optimization Problems ◽  
Vertex Cover ◽  
Independent Set ◽  
Maximum Independent Set ◽  
Gradient Descent Algorithm ◽  
Hard Problems ◽  
Traditional Approaches

Abstract In computer science, there exist a large number of optimization problems defined on graphs, that is to find a best node state configuration or a network structure such that the designed objective function is optimized under some constraints. However, these problems are notorious for their hardness to solve because most of them are NP-hard or NP-complete. Although traditional general methods such as simulated annealing (SA), genetic algorithms (GA) and so forth have been devised to these hard problems, their accuracy and time consumption are not satisfying in practice. In this work, we proposed a simple, fast, and general algorithm framework based on advanced automatic differentiation technique empowered by deep learning frameworks. By introducing Gumbel-softmax technique, we can optimize the objective function directly by gradient descent algorithm regardless of the discrete nature of variables. We also introduce evolution strategy to parallel version of our algorithm. We test our algorithm on three representative optimization problems on graph including modularity optimization from network science, Sherrington-Kirkpatrick (SK) model from statistical physics, maximum independent set (MIS) and minimum vertex cover (MVC) problem from combinatorial optimization on graph. High-quality solutions can be obtained with much less time consuming compared to traditional approaches.

scholarly journals Gumbel-softmax-based optimization: a simple general framework for optimization problems on graphs

2021 ◽  
Vol 8 (1) ◽  
Author(s):  
Yaoxin Li ◽  
Jing Liu ◽  
Guozheng Lin ◽  
Yueyuan Hou ◽  
Muyun Mou ◽  
...  
Keyword(s):  
Objective Function ◽  
Statistical Physics ◽  
Automatic Differentiation ◽  
Optimization Problems ◽  
Vertex Cover ◽  
Independent Set ◽  
Computational Social Science ◽  
Maximization Problem ◽  
Maximum Independent Set ◽  
Gradient Descent Algorithm

AbstractIn computer science, there exist a large number of optimization problems defined on graphs, that is to find a best node state configuration or a network structure, such that the designed objective function is optimized under some constraints. However, these problems are notorious for their hardness to solve, because most of them are NP-hard or NP-complete. Although traditional general methods such as simulated annealing (SA), genetic algorithms (GA), and so forth have been devised to these hard problems, their accuracy and time consumption are not satisfying in practice. In this work, we proposed a simple, fast, and general algorithm framework based on advanced automatic differentiation technique empowered by deep learning frameworks. By introducing Gumbel-softmax technique, we can optimize the objective function directly by gradient descent algorithm regardless of the discrete nature of variables. We also introduce evolution strategy to parallel version of our algorithm. We test our algorithm on four representative optimization problems on graph including modularity optimization from network science, Sherrington–Kirkpatrick (SK) model from statistical physics, maximum independent set (MIS) and minimum vertex cover (MVC) problem from combinatorial optimization on graph, and Influence Maximization problem from computational social science. High-quality solutions can be obtained with much less time-consuming compared to the traditional approaches.

scholarly journals A Generalization of Generalized Paley Graphs and New Lower Bounds for $R(3,q)$

10.37236/474 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Kang Wu ◽  
Wenlong Su ◽  
Haipeng Luo ◽  
Xiaodong Xu
Keyword(s):  
Finite Fields ◽  
Lower Bounds ◽  
Independent Set ◽  
Maximum Independent Set ◽  
Ramsey Numbers ◽  
Cyclic Graphs ◽  
Paley Graphs

Generalized Paley graphs are cyclic graphs constructed from quadratic or higher residues of finite fields. Using this type of cyclic graphs to study the lower bounds for classical Ramsey numbers, has high computing efficiency in both looking for parameter sets and computing clique numbers. We have found a new generalization of generalized Paley graphs, i.e. automorphism cyclic graphs, also having the same advantages. In this paper we study the properties of the parameter sets of automorphism cyclic graphs, and develop an algorithm to compute the order of the maximum independent set, based on which we get new lower bounds for $8$ classical Ramsey numbers: $R(3,22) \geq 131$, $R(3,23) \geq 137$, $R(3,25) \geq 154$, $R(3,28) \geq 173$, $R(3,29) \geq 184$, $R(3,30) \geq 190$, $R(3,31) \geq 199$, $R(3,32) \geq 214$. Furthermore, we also get $R(5,23) \geq 521$ based on $R(3,22) \geq 131$. These nine results above improve their corresponding best known lower bounds.

scholarly journals Gumbel-softmax-based Optimization: A Simple General Framework for Optimization Problems on Graphs

2020 ◽  
Author(s):  
Yaoxin Li ◽  
Jing Liu ◽  
Guozheng Lin ◽  
Yueyuan Hou ◽  
Muyun Mou ◽  
...  
Keyword(s):  
Objective Function ◽  
Statistical Physics ◽  
Automatic Differentiation ◽  
Optimization Problems ◽  
Vertex Cover ◽  
Independent Set ◽  
Computational Social Science ◽  
Maximization Problem ◽  
Maximum Independent Set ◽  
Gradient Descent Algorithm

Abstract In computer science, there exist a large number of optimization problems defined on graphs, that is to find a best node state configuration or a network structure such that the designed objective function is optimized under some constraints. However, these problems are notorious for their hardness to solve because most of them are NP-hard or NP-complete. Although traditional general methods such as simulated annealing (SA), genetic algorithms (GA) and so forth have been devised to these hard problems, their accuracy and time consumption are not satisfying in practice. In this work, we proposed a simple, fast, and general algorithm framework based on advanced automatic differentiation technique empowered by deep learning frameworks. By introducing Gumbel-softmax technique, we can optimize the objective function directly by gradient descent algorithm regardless of the discrete nature of variables. We also introduce evolution strategy to parallel version of our algorithm. We test our algorithm on four representative optimization problems on graph including modularity optimization from network science, Sherrington-Kirkpatrick (SK) model from statistical physics, maximum independent set (MIS) and minimum vertex cover (MVC) problem from combinatorial optimization on graph, and Influence Maximization problem from computational social science. High-quality solutions can be obtained with much less time consuming compared to traditional approaches.

COMPLEXITY ASPECTS OF VISIBILITY GRAPHS

1995 ◽  
Vol 05 (03) ◽  
pp. 289-312 ◽  
Author(s):  
YAW-LING LIN ◽  
STEVEN S. SKIENA
Keyword(s):  
Optimization Problems ◽  
Dominating Set ◽  
Vertex Cover ◽  
Independent Set ◽  
Visibility Graph ◽  
Outerplanar Graph ◽  
Maximum Independent Set ◽  
Graph Recognition ◽  
Visibility Graphs ◽  
Necessary And Sufficient

In this paper, we consider two distinct problems related to complexity aspects of the visibility graphs of simple polygons. Recognizing visibility graphs is a long-standing open problem. It is not even known whether visibility graph recognition is in NP. That visibility graph recognition is in NP would be established if we could demonstrate that any n vertex visibility graph is realized by a polygon which can be drawn on an exponentially-sized grid. This motivates a study of the area requirements for realizing visibility graphs. In this paper, we prove: • Θ(n3) area is necessary and sufficient to realize the complete visibility graph Kn. • There exist visibility graphs which require exponential area to realize. • Any maximal outerplanar graph of diameter d can be realized in O(d2 · 2d) area, which can be as small as O(n log2 n) for a balanced mop. Linear maximal outer-planar graphs can be realized in O(n8) area. The second part of this paper considers the complexity of specific optimization problems on visibility graphs. Given a polygon P, we show that finding a maximum independent set, minimum vertex cover, or maximum dominating set in the visibility graph of P are all NP-complete. Further we show that for polygons P1 and P2, the problem of testing if they have isomorphic visibility graphs is isomorphism-complete. These problems remain hard when given the visibility graphs as input.

scholarly journals Layered Graphs: A Class that Admits Polynomial Time Solutions for Some Hard Problems

2018 ◽  
Author(s):  
Bhadrachalam Chitturi ◽  
Srijith Balachander ◽  
Sandeep Satheesh ◽  
Krithic Puthiyoppil
Keyword(s):  
Polynomial Time ◽  
Dominating Set ◽  
Vertex Cover ◽  
Independent Set ◽  
Chordal Graphs ◽  
Minimum Cardinality ◽  
Graph Class ◽  
Hard Problems ◽  
Connected Vertex Cover ◽  
Circular Arc Graphs

The independent set, IS, on a graph G = ( V , E ) is V * ⊆ V such that no two vertices in V * have an edge between them. The MIS problem on G seeks to identify an IS with maximum cardinality, i.e. MIS. V * ⊆ V is a vertex cover, i.e. VC of G = ( V , E ) if every e ∈ E is incident upon at least one vertex in V * . V * ⊆ V is dominating set, DS, of G = ( V , E ) if ∀ v ∈ V either v ∈ V * or ∃ u ∈ V * and ( u , v ) ∈ E . The MVC problem on G seeks to identify a vertex cover with minimum cardinality, i.e. MVC. Likewise, MCV seeks a connected vertex cover, i.e. VC which forms one component in G, with minimum cardinality, i.e. MCV. A connected DS, CDS, is a DS that forms a connected component in G. The problems MDS and MCD seek to identify a DS and a connected DS i.e. CDS respectively with minimum cardinalities. MIS, MVC, MDS, MCV and MCD on a general graph are known to be NP-complete. Polynomial time algorithms are known for bipartite graphs, chordal graphs, cycle graphs, comparability graphs, claw-free graphs, interval graphs and circular arc graphs for some of these problems. We introduce a novel graph class, layered graph, where each layer refers to a subgraph containing at most some k vertices. Inter layer edges are restricted to the vertices in adjacent layers. We show that if k = Θ ( log ∣ V ∣ ) then MIS, MVC and MDS can be computed in polynomial time and if k = O ( ( log ∣ V ∣ ) α ) , where α < 1 , then MCV and MCD can be computed in polynomial time. If k = Θ ( ( log ∣ V ∣ ) 1 + ϵ ) , for ϵ > 0 , then MIS, MVC and MDS require quasi-polynomial time. If k = Θ ( log ∣ V ∣ ) then MCV, MCD require quasi-polynomial time. Layered graphs do have constraints such as bipartiteness, planarity and acyclicity.

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