scholarly journals NP-Completeness Results for Minimum Planar Spanners

1998 ◽  
Vol Vol. 3 no. 1 ◽  
Author(s):  
Ulrik Brandes ◽  
Dagmar Handke
Keyword(s):  
Related Problem ◽  
Interesting Case ◽  
Edge Weight ◽  
Weighted Graphs ◽  
Np Completeness ◽  
Spanning Subgraph ◽  
Fixed Parameter ◽  
Minimum Number ◽  
International Audience ◽  
Np Hardness

International audience For any fixed parameter t greater or equal to 1, a \emph t-spanner of a graph G is a spanning subgraph in which the distance between every pair of vertices is at most t times their distance in G. A \emph minimum t-spanner is a t-spanner with minimum total edge weight or, in unweighted graphs, minimum number of edges. In this paper, we prove the NP-hardness of finding minimum t-spanners for planar weighted graphs and digraphs if t greater or equal to 3, and for planar unweighted graphs and digraphs if t greater or equal to 5. We thus extend results on that problem to the interesting case where the instances are known to be planar. We also introduce the related problem of finding minimum \emphplanar t-spanners and establish its NP-hardness for similar fixed values of t.

scholarly journals Parameterized Problems Related to Seidel's Switching

2011 ◽  
Vol Vol. 13 no. 2 (Graph and Algorithms) ◽  
Author(s):  
Eva Jelinkova ◽  
Ondrej Suchy ◽  
Petr Hlineny ◽  
Jan Kratochvil
Keyword(s):  
Minimum Degree ◽  
The Other ◽  
Complete Graphs ◽  
Maximum Likelihood Decoding ◽  
Induced Subgraph ◽  
Fixed Parameter Tractable ◽  
Graph Theoretic ◽  
Np Completeness ◽  
Fixed Parameter ◽  
International Audience

Graphs and Algorithms International audience Seidel's switching is a graph operation which makes a given vertex adjacent to precisely those vertices to which it was non-adjacent before, while keeping the rest of the graph unchanged. Two graphs are called switching-equivalent if one can be made isomorphic to the other by a sequence of switches. In this paper, we continue the study of computational complexity aspects of Seidel's switching, concentrating on Fixed Parameter Complexity. Among other results we show that switching to a graph with at most k edges, to a graph of maximum degree at most k, to a k-regular graph, or to a graph with minimum degree at least k are fixed parameter tractable problems, where k is the parameter. On the other hand, switching to a graph that contains a given fixed graph as an induced subgraph is W [1]-complete. We also show the NP-completeness of switching to a graph with a clique of linear size, and of switching to a graph with small number of edges. A consequence of the latter result is the NP-completeness of Maximum Likelihood Decoding of graph theoretic codes based on complete graphs.

scholarly journals Dividing Splittable Goods Evenly and With Limited Fragmentation

Algorithmica ◽  
2019 ◽  
Vol 82 (5) ◽  
pp. 1298-1328
Author(s):  
Peter Damaschke
Keyword(s):  
Related Problem ◽  
Linear Time ◽  
Np Hard ◽  
Weighted Graphs ◽  
Fixed Parameter Tractable ◽  
Averaging Problem ◽  
Fixed Parameter ◽  
Splitting Problem ◽  
Structural Characterizations ◽  
Balanced Solutions

Abstract A splittable good provided in n pieces shall be divided as evenly as possible among m agents, where every agent can take shares from at most F pieces. We call F the fragmentation and mainly restrict attention to the cases $$F=1$$F=1 and $$F=2$$F=2. For $$F=1$$F=1, the max–min and min–max problems are solvable in linear time. The case $$F=2$$F=2 has neat formulations and structural characterizations in terms of weighted graphs. First we focus on perfectly balanced solutions. While the problem is strongly NP-hard in general, it can be solved in linear time if $$m\ge n-1$$m≥n-1, and a solution always exists in this case, in contrast to $$F=1$$F=1. Moreover, the problem is fixed-parameter tractable in the parameter $$2m-n$$2m-n. (Note that this parameter measures the number of agents above the trivial threshold $$m=n/2$$m=n/2.) The structural results suggest another related problem where unsplittable items shall be assigned to subsets so as to balance the average sizes (rather than the total sizes) in these subsets. We give an approximation-preserving reduction from our original splitting problem with fragmentation $$F=2$$F=2 to this averaging problem, and some approximation results in cases when m is close to either n or n / 2.

scholarly journals Krausz dimension and its generalizations in special graph classes

2013 ◽  
Vol Vol. 15 no. 1 (Graph Theory) ◽  
Author(s):  
Olga Glebova ◽  
Yury Metelsky ◽  
Pavel Skums
Keyword(s):  
Open Problem ◽  
Chordal Graphs ◽  
Np Hard ◽  
Induced Subgraphs ◽  
Fixed Parameter Tractable ◽  
Graph Classes ◽  
Fixed Parameter ◽  
Minimum Number ◽  
International Audience ◽  
Split Graphs

Graph Theory International audience A Krausz (k,m)-partition of a graph G is a decomposition of G into cliques, such that any vertex belongs to at most k cliques and any two cliques have at most m vertices in common. The m-Krausz dimension kdimm(G) of the graph G is the minimum number k such that G has a Krausz (k,m)-partition. In particular, 1-Krausz dimension or simply Krausz dimension kdim(G) is a well-known graph-theoretical parameter. In this paper we prove that the problem "kdim(G)≤3" is polynomially solvable for chordal graphs, thus partially solving the open problem of P. Hlineny and J. Kratochvil. We solve another open problem of P. Hlineny and J. Kratochvil by proving that the problem of finding Krausz dimension is NP-hard for split graphs and complements of bipartite graphs. We show that the problem of finding m-Krausz dimension is NP-hard for every m≥1, but the problem "kdimm(G)≤k" is is fixed-parameter tractable when parameterized by k and m for (∞,1)-polar graphs. Moreover, the class of (∞,1)-polar graphs with kdimm(G)≤k is characterized by a finite list of forbidden induced subgraphs for every k,m≥1.

scholarly journals Sparse graphs and an augmentation problem

2021 ◽  
Author(s):  
Csaba Király ◽  
András Mihálykó
Keyword(s):  
Polynomial Algorithm ◽  
Rigidity Theory ◽  
Sparse Graphs ◽  
Spanning Subgraph ◽  
Minimum Number ◽  
Np Hardness ◽  
Tight Graphs

AbstractFor two integers $$k>0$$ k > 0 and $$\ell $$ ℓ , a graph $$G=(V,E)$$ G = ( V , E ) is called $$(k,\ell )$$ ( k , ℓ ) -tight if $$|E|=k|V|-\ell $$ | E | = k | V | - ℓ and $$i_G(X)\le k|X|-\ell $$ i G ( X ) ≤ k | X | - ℓ for each $$X\subseteq V$$ X ⊆ V for which $$i_G(X)\ge 1$$ i G ( X ) ≥ 1 , where $$i_G(X)$$ i G ( X ) denotes the number of induced edges by X. G is called $$(k,\ell )$$ ( k , ℓ ) -redundant if $$G-e$$ G - e has a spanning $$(k,\ell )$$ ( k , ℓ ) -tight subgraph for all $$e\in E$$ e ∈ E . We consider the following augmentation problem. Given a graph $$G=(V,E)$$ G = ( V , E ) that has a $$(k,\ell )$$ ( k , ℓ ) -tight spanning subgraph, find a graph $$H=(V,F)$$ H = ( V , F ) with the minimum number of edges, such that $$G\cup H$$ G ∪ H is $$(k,\ell )$$ ( k , ℓ ) -redundant. We give a polynomial algorithm and a min-max theorem for this augmentation problem when the input is $$(k,\ell )$$ ( k , ℓ ) -tight. For general inputs, we give a polynomial algorithm when $$k\ge \ell $$ k ≥ ℓ and show the NP-hardness of the problem when $$k<\ell $$ k < ℓ . Since $$(k,\ell )$$ ( k , ℓ ) -tight graphs play an important role in rigidity theory, these algorithms can be used to make several types of rigid frameworks redundantly rigid by adding a smallest set of new bars.

scholarly journals Strong chromatic index of products of graphs

2007 ◽  
Vol Vol. 9 no. 1 (Graph and Algorithms) ◽  
Author(s):  
Olivier Togni
Keyword(s):  
Cartesian Product ◽  
Complete Graphs ◽  
Chromatic Index ◽  
Induced Matching ◽  
Colour Class ◽  
Strong Chromatic Index ◽  
Minimum Number ◽  
International Audience

Graphs and Algorithms International audience The strong chromatic index of a graph is the minimum number of colours needed to colour the edges in such a way that each colour class is an induced matching. In this paper, we present bounds for strong chromatic index of three different products of graphs in term of the strong chromatic index of each factor. For the cartesian product of paths, cycles or complete graphs, we derive sharper results. In particular, strong chromatic indices of d-dimensional grids and of some toroidal grids are given along with approximate results on the strong chromatic index of generalized hypercubes.

scholarly journals A note on compact and compact circular edge-colorings of graphs

2008 ◽  
Vol Vol. 10 no. 3 (Graph and Algorithms) ◽  
Author(s):  
Dariusz Dereniowski ◽  
Adam Nadolski
Keyword(s):  
Approximation Algorithm ◽  
Decision Problem ◽  
Edge Coloring ◽  
Exact Algorithm ◽  
Weighted Graphs ◽  
Edge Colorings ◽  
Odd Cycle ◽  
International Audience ◽  
Np Complete ◽  
Odd Cycles

Graphs and Algorithms International audience We study two variants of edge-coloring of edge-weighted graphs, namely compact edge-coloring and circular compact edge-coloring. First, we discuss relations between these two coloring models. We prove that every outerplanar bipartite graph admits a compact edge-coloring and that the decision problem of the existence of compact circular edge-coloring is NP-complete in general. Then we provide a polynomial time 1:5-approximation algorithm and pseudo-polynomial exact algorithm for compact circular coloring of odd cycles and prove that it is NP-hard to optimally color these graphs. Finally, we prove that if a path P2 is joined by an edge to an odd cycle then the problem of the existence of a compact circular coloring becomes NP-complete.

Multiplicative Parameterization Above a Guarantee

2021 ◽  
Vol 13 (3) ◽  
pp. 1-16
Author(s):  
Fedor V. Fomin ◽  
Petr A. Golovach ◽  
Daniel Lokshtanov ◽  
Fahad Panolan ◽  
Saket Saurabh ◽  
...  
Keyword(s):  
Minimum Size ◽  
Input Graph ◽  
Fixed Parameter Tractable ◽  
Long Cycle ◽  
Fixed Parameter ◽  
Parameterized Problem ◽  
Fixed Constant ◽  
Fixed Parameter Algorithm ◽  
Np Hardness ◽  
Additive Form

Parameterization above a guarantee is a successful paradigm in Parameterized Complexity. To the best of our knowledge, all fixed-parameter tractable problems in this paradigm share an additive form defined as follows. Given an instance ( I,k ) of some (parameterized) problem π with a guarantee g(I) , decide whether I admits a solution of size at least (or at most) k + g(I) . Here, g(I) is usually a lower bound on the minimum size of a solution. Since its introduction in 1999 for M AX SAT and M AX C UT (with g(I) being half the number of clauses and half the number of edges, respectively, in the input), analysis of parameterization above a guarantee has become a very active and fruitful topic of research. We highlight a multiplicative form of parameterization above (or, rather, times) a guarantee: Given an instance ( I,k ) of some (parameterized) problem π with a guarantee g(I) , decide whether I admits a solution of size at least (or at most) k · g(I) . In particular, we study the Long Cycle problem with a multiplicative parameterization above the girth g(I) of the input graph, which is the most natural guarantee for this problem, and provide a fixed-parameter algorithm. Apart from being of independent interest, this exemplifies how parameterization above a multiplicative guarantee can arise naturally. We also show that, for any fixed constant ε > 0, multiplicative parameterization above g(I) 1+ε of Long Cycle yields para-NP-hardness, thus our parameterization is tight in this sense. We complement our main result with the design (or refutation of the existence) of fixed-parameter algorithms as well as kernelization algorithms for additional problems parameterized multiplicatively above girth.

scholarly journals ON SPANNERS OF GEOMETRIC GRAPHS

2009 ◽  
Vol 20 (01) ◽  
pp. 135-149 ◽  
Author(s):  
JOACHIM GUDMUNDSSON ◽  
MICHIEL SMID
Keyword(s):  
Real Number ◽  
Upper Bound ◽  
Weighted Graph ◽  
Geometric Graph ◽  
Geometric Graphs ◽  
Np Hard ◽  
Minimum Number ◽  
Np Hardness

Given a connected geometric graph G, we consider the problem of constructing a t-spanner of G having the minimum number of edges. We prove that for every real number t with [Formula: see text], there exists a connected geometric graph G with n vertices, such that every t-spanner of G contains Ω(n1+1/t) edges. This bound almost matches the known upper bound, which states that every connected weighted graph with n vertices contains a t-spanner with O(n1+2/(t-1)) edges. We also prove that the problem of deciding whether a given geometric graph contains a t-spanner with at most K edges is NP-hard. Previously, this NP-hardness result was only known for non-geometric graphs.

scholarly journals On subset sum problem in branch groups

2020 ◽  
Vol Volume 12, issue 1 ◽  
Author(s):  
Andrey Nikolaev ◽  
Alexander Ushakov
Keyword(s):  
Final Version ◽  
Subset Sum Problem ◽  
Grigorchuk Group ◽  
Np Completeness ◽  
Subset Sum ◽  
Np Complete ◽  
Np Hardness

We consider a group-theoretic analogue of the classic subset sum problem. In this brief note, we show that the subset sum problem is NP-complete in the first Grigorchuk group. More generally, we show NP-hardness of that problem in weakly regular branch groups, which implies NP-completeness if the group is, in addition, contracting. Comment: v3: final version for journal of Groups, Complexity, Cryptology. arXiv admin note: text overlap with arXiv:1703.07406

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