scholarly journals Vertex-Partitioning into Fixed Additive Induced-Hereditary Properties is NP-hard

10.37236/1799 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Alastair Farrugia
Keyword(s):  
Planar Graph ◽  
Line Graph ◽  
Perfect Graph ◽  
Np Hard ◽  
Arbitrary Graph ◽  
Np Completeness ◽  
Graph Properties ◽  
Special Cases ◽  
Np Complete ◽  
Hereditary Properties

Can the vertices of an arbitrary graph $G$ be partitioned into $A \cup B$, so that $G[A]$ is a line-graph and $G[B]$ is a forest? Can $G$ be partitioned into a planar graph and a perfect graph? The NP-completeness of these problems are special cases of our result: if ${\cal P}$ and ${\cal Q}$ are additive induced-hereditary graph properties, then $({\cal P}, {\cal Q})$-colouring is NP-hard, with the sole exception of graph $2$-colouring (the case where both ${\cal P}$ and ${\cal Q}$ are the set ${\cal O}$ of finite edgeless graphs). Moreover, $({\cal P}, {\cal Q})$-colouring is NP-complete iff ${\cal P}$- and ${\cal Q}$-recognition are both in NP. This completes the proof of a conjecture of Kratochvíl and Schiermeyer, various authors having already settled many sub-cases.

COMPLEXITY AND APPROXIMABILITY OF DOUBLE DIGEST

2005 ◽  
Vol 03 (02) ◽  
pp. 207-223
Author(s):  
MARK CIELIEBAK ◽  
STEPHAN EIDENBENZ ◽  
GERHARD J. WOEGINGER
Keyword(s):  
Real Life ◽  
Approximation Ratio ◽  
Additional Restriction ◽  
Np Hard ◽  
Model Errors ◽  
Np Completeness ◽  
The Third ◽  
Np Complete ◽  
Feasible Solutions

We revisit the DOUBLE DIGEST problem, which occurs in sequencing of large DNA strings and consists of reconstructing the relative positions of cut sites from two different enzymes. We first show that DOUBLE DIGEST is strongly NP-complete, improving upon previous results that only showed weak NP-completeness. Even the (experimentally more meaningful) variation in which we disallow coincident cut sites turns out to be strongly NP-complete. In the second part, we model errors in data as they occur in real-life experiments: we propose several optimization variations of DOUBLE DIGEST that model partial cleavage errors. We then show that most of these variations are hard to approximate. In the third part, we investigate variations with the additional restriction that coincident cut sites are disallowed, and we show that it is NP-hard to even find feasible solutions in this case, thus making it impossible to guarantee any approximation ratio at all.

scholarly journals Near-Colorings: Non-Colorable Graphs and NP-Completeness

10.37236/3509 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
M. Montassier ◽  
P. Ochem
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Maximum Degree ◽  
Fixed Integer ◽  
Np Completeness ◽  
Vertex Set ◽  
Np Complete

A graph $G$ is $(d_1,...,d_l)$-colorable if the vertex set of $G$ can be partitioned into subsets $V_1,\ldots ,V_l$ such that the graph $G[V_i]$ induced by the vertices of $V_i$ has maximum degree at most $d_i$ for all $1 \leq i \leq l$. In this paper, we focus on complexity aspects of such colorings when $l=2,3$. More precisely, we prove that, for any fixed integers $k,j,g$ with $(k,j) \neq (0,0)$ and $g\geq3$, either every planar graph with girth at least $g$ is $(k,j)$-colorable or it is NP-complete to determine whether a planar graph with girth at least $g$ is $(k,j)$-colorable. Also, for any fixed integer $k$, it is NP-complete to determine whether a planar graph that is either $(0,0,0)$-colorable or non-$(k,k,1)$-colorable is $(0,0,0)$-colorable. Additionally, we exhibit non-$(3,1)$-colorable planar graphs with girth 5 and non-$(2,0)$-colorable planar graphs with girth 7. 

scholarly journals The inapproximability for the $(0,1)$-additive number

2016 ◽  
Vol Vol. 17 no. 3 (Graph Theory) ◽  
Author(s):  
Arash Ahadi ◽  
Ali Dehghan
Keyword(s):  
Planar Graph ◽  
Negative Answer ◽  
Perfect Graphs ◽  
Np Hard ◽  
Additive Number ◽  
Minimum Number ◽  
International Audience ◽  
Np Complete ◽  
Phd Thesis

International audience An <i>additive labeling</i> of a graph $G$ is a function $\ell :V(G) \rightarrow \mathbb{N}$, such that for every two adjacent vertices $v$ and $u$ of $G$, $\Sigma_{w \sim v} \ell (w) \neq \Sigma_{w \sim u} \ell (w)$ ($x \sim y$ means that $x$ is joined to $y$). The additive number of $G$, denoted by $\eta (G)$, is the minimum number $k$ such that $G$ has a additive labeling $\ell : V(G) \rightarrow \mathbb{N}_k$. The additive choosability of a graph $G$, denoted by $\eta_\ell (G)$, is the smallest number $k$ such that $G$ has an additive labeling for any assignment of lists of size $k$ to the vertices of $G$, such that the label of each vertex belongs to its own list. Seamone in his PhD thesis conjectured that for every graph $G$, $\eta(G)= \eta_\ell (G)$. We give a negative answer to this conjecture and we show that for every $k$ there is a graph $G$ such that $\eta_\ell (G) - \eta(G) \geq k$. A $(0,1)$-<i>additive labeling</i> of a graph $G$ is a function $\ell :V(G) \rightarrow \{0,1 \}$, such that for every two adjacent vertices $v$ and $u$ of $G$, $\Sigma_{w \sim v} \ell (w) \neq \Sigma_{w \sim u} \ell (w)$. A graph may lack any $(0,1)$-additive labeling. We show that it is NP-complete to decide whether a $(0,1)$-additive labeling exists for some families of graphs such as perfect graphs and planar triangle-free graphs. For a graph $G$ with some $(0,1)$-additive labelings, the $(0,1)$-additive number of $G$ is defined as $\sigma_1 (G) = \mathrm{min}_{\ell \in \Gamma} \Sigma_{v \in V (G)} \ell (v)$ where $\Gamma$ is the set of $(0,1)$-additive labelings of $G$. We prove that given a planar graph that admits a $(0,1)$-additive labeling, for all $\epsilon > 0$ , approximating the $(0,1)$-additive number within $n^{1-\epsilon}$ is NP-hard.

Determining the thickness of graphs is NP-hard

1983 ◽  
Vol 93 (1) ◽  
pp. 9-23 ◽  
Author(s):  
Anthony Mansfield
Keyword(s):  
Open Problem ◽  
Np Hard ◽  
Arbitrary Graph ◽  
Printed Circuits ◽  
Np Complete ◽  
Intractable Problem

AbstractThe thickness of a graph is a measure of its nonplanarity and has applications in the theory of printed circuits. To determine the thickness of an arbitrary graph is a seemingly intractable problem. This is made precise in this paper where we answer an open problem of Garey and Johnson (2) by proving that it is NP-complete to decide whether a graph has thickness two.

scholarly journals Popular Matching in Roommates Setting Is NP-hard

2021 ◽  
Vol 13 (2) ◽  
pp. 1-20
Author(s):  
Sushmita Gupta ◽  
Pranabendu Misra ◽  
Saket Saurabh ◽  
Meirav Zehavi
Keyword(s):  
Computational Complexity ◽  
Np Hard ◽  
Np Complete ◽  
Open Question ◽  
Popular Matching

An input to the P OPULAR M ATCHING problem, in the roommates setting (as opposed to the marriage setting), consists of a graph G (not necessarily bipartite) where each vertex ranks its neighbors in strict order, known as its preference. In the P OPULAR M ATCHING problem the objective is to test whether there exists a matching M * such that there is no matching M where more vertices prefer their matched status in M (in terms of their preferences) over their matched status in M *. In this article, we settle the computational complexity of the P OPULAR M ATCHING problem in the roommates setting by showing that the problem is NP-complete. Thus, we resolve an open question that has been repeatedly and explicitly asked over the last decade.

The Problem of Scheduling Parallel Computations of Multibody Dynamic Analysis Is NP-Hard

1996 ◽  
Author(s):  
Jin-Fan Liu ◽  
Karim A. Abdel-Malek
Keyword(s):  
Dynamic Analysis ◽  
Spanning Tree ◽  
Minimum Weight ◽  
Computational Cost ◽  
Parallel Computations ◽  
Np Hard ◽  
Multibody Dynamic ◽  
Np Hard Problem ◽  
Np Complete ◽  
Minimum Weight Spanning Tree

Abstract A formulation of a graph problem for scheduling parallel computations of multibody dynamic analysis is presented. The complexity of scheduling parallel computations for a multibody dynamic analysis is studied. The problem of finding a shortest critical branch spanning tree is described and transformed to a minimum radius spanning tree, which is solved by an algorithm of polynomial complexity. The problems of shortest critical branch minimum weight spanning tree (SCBMWST) and the minimum weight shortest critical branch spanning tree (MWSCBST) are also presented. Both problems are shown to be NP-hard by proving that the bounded critical branch bounded weight spanning tree (BCBBWST) problem is NP-complete. It is also shown that the minimum computational cost spanning tree (MCCST) is at least as hard as SCBMWST or MWSCBST problems, hence itself an NP-hard problem. A heuristic approach to solving these problems is developed and implemented, and simulation results are discussed.

NP vs QMA_\log(2)

2010 ◽  
Vol 10 (1&2) ◽  
pp. 141-151
Author(s):  
S. Beigi
Keyword(s):  
Quantum Algorithms ◽  
Np Hard ◽  
Complete Problems ◽  
Hard Problems ◽  
Np Complete ◽  
Np Hard Problems

Although it is believed unlikely that $\NP$-hard problems admit efficient quantum algorithms, it has been shown that a quantum verifier can solve NP-complete problems given a "short" quantum proof; more precisely, NP\subseteq QMA_{\log}(2) where QMA_{\log}(2) denotes the class of quantum Merlin-Arthur games in which there are two unentangled provers who send two logarithmic size quantum witnesses to the verifier. The inclusion NP\subseteq QMA_{\log}(2) has been proved by Blier and Tapp by stating a quantum Merlin-Arthur protocol for 3-coloring with perfect completeness and gap 1/24n^6. Moreover, Aaronson et al. have shown the above inclusion with a constant gap by considering $\widetilde{O}(\sqrt{n})$ witnesses of logarithmic size. However, we still do not know if QMA_{\log}(2) with a constant gap contains NP. In this paper, we show that 3-SAT admits a QMA_{\log}(2) protocol with the gap 1/n^{3+\epsilon}} for every constant \epsilon>0.

scholarly journals Rooted $K_4$-Minors

10.37236/3476 ◽  
2013 ◽  
Vol 20 (2) ◽  
Author(s):  
Ruy Fabila-Monroy ◽  
David R. Wood
Keyword(s):  
Planar Graph ◽  
Pairwise Disjoint ◽  
Single Face ◽  
Special Cases ◽  
Connected Subgraphs ◽  
Connected Planar Graph

Let $a,b,c,d$ be four vertices in a graph $G$. A $K_4$ minor rooted at $a,b,c,d$ consists of four pairwise-disjoint pairwise-adjacent connected subgraphs of $G$, respectively containing $a,b,c,d$. We characterise precisely when $G$ contains a $K_4$-minor rooted at $a,b,c,d$ by describing six classes of obstructions, which are the edge-maximal graphs containing no $K_4$-minor rooted at $a,b,c,d$. The following two special cases illustrate the full characterisation: (1) A 4-connected non-planar graph contains a $K_4$-minor rooted at $a,b,c,d$ for every choice of $a,b,c,d$. (2) A 3-connected planar graph contains a $K_4$-minor rooted at $a,b,c,d$ if and only if $a,b,c,d$ are not on a single face.

scholarly journals The Complexity Landscape of Resource-Constrained Scheduling

2020 ◽  
Author(s):  
Robert Ganian ◽  
Thekla Hamm ◽  
Guillaume Mescoff
Keyword(s):  
Artificial Intelligence ◽  
Polynomial Time ◽  
Project Scheduling ◽  
Scheduling Problem ◽  
Comprehensive Understanding ◽  
Resource Constrained ◽  
Special Cases ◽  
Project Scheduling Problem ◽  
Np Complete ◽  
New Algorithms

The Resource-Constrained Project Scheduling Problem (RCPSP) and its extension via activity modes (MRCPSP) are well-established scheduling frameworks that have found numerous applications in a broad range of settings related to artificial intelligence. Unsurprisingly, the problem of finding a suitable schedule in these frameworks is known to be NP-complete; however, aside from a few results for special cases, we have lacked an in-depth and comprehensive understanding of the complexity of the problems from the viewpoint of natural restrictions of the considered instances. In the first part of our paper, we develop new algorithms and give hardness-proofs in order to obtain a detailed complexity map of (M)RCPSP that settles the complexity of all 1024 considered variants of the problem defined in terms of explicit restrictions of natural parameters of instances. In the second part, we turn to implicit structural restrictions defined in terms of the complexity of interactions between individual activities. In particular, we show that if the treewidth of a graph which captures such interactions is bounded by a constant, then we can solve MRCPSP in polynomial time.

scholarly journals FOLDING EQUILATERAL PLANE GRAPHS

2013 ◽  
Vol 23 (02) ◽  
pp. 75-92 ◽  
Author(s):  
ZACHARY ABEL ◽  
ERIK D. DEMAINE ◽  
MARTIN L. DEMAINE ◽  
SARAH EISENSTAT ◽  
JAYSON LYNCH ◽  
...  
Keyword(s):  
Plane Graph ◽  
Analogous Problem ◽  
Central Vertex ◽  
Polyhedral Complex ◽  
Plane Graphs ◽  
Single Vertex ◽  
Np Completeness ◽  
Np Complete ◽  
Polyhedral Manifold

We consider two types of folding applied to equilateral plane graph linkages. First, under continuous folding motions, we show how to reconfigure any linear equilateral tree (lying on a line) into a canonical configuration. By contrast, it is known that such reconfiguration is not always possible for linear (nonequilateral) trees and for (nonlinear) equilateral trees. Second, under instantaneous folding motions, we show that an equilateral plane graph has a noncrossing linear folded state if and only if it is bipartite. Furthermore, we show that the equilateral constraint is necessary for this result, by proving that it is strongly NP-complete to decide whether a (nonequilateral) plane graph has a linear folded state. Equivalently, we show strong NP-completeness of deciding whether an abstract metric polyhedral complex with one central vertex has a noncrossing flat folded state. By contrast, the analogous problem for a polyhedral manifold with one central vertex (single-vertex origami) is only weakly NP-complete.

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