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A new point of NP-hardness for 2-to-1 Label Cover [article]

Per Austrin and Ryan O'Donnell and John Wright
2012 arXiv   pre-print
We show that given a satisfiable instance of the 2-to-1 Label Cover problem, it is NP-hard to find a (23/24 + )-satisfying assignment.  ...  Of particular interest is the d = 2 case, i.e., 2-to-1 Label Cover. Label Cover serves as the starting point for most NP-hardness of approximation results.  ...  Our results In this paper, we focus on proving NP-hardness for the 2-to-1 Label Cover problem.  ... 
arXiv:1204.5666v1 fatcat:a5b4kdi6ojc7blmjtrbuqmpdei

A New Point of NP-Hardness for 2-to-1 Label Cover [chapter]

Per Austrin, Ryan O'Donnell, John Wright
2012 Lecture Notes in Computer Science  
For every integer d ≥ 2 and > 0, there is a label set size q such that it is NP-hard to (1, )-decide the d-to-1 Label Cover problem.  ...  We show that given a satisfiable instance of the 2-to-1 Label Cover problem, it is NP-hard to find a ( 23 24 + )-satisfying assignment.  ...  Of particular interest is the d = 2 case, i.e., 2-to-1 Label Cover. Label Cover serves as the starting point for most NP-hardness of approximation results.  ... 
doi:10.1007/978-3-642-32512-0_1 fatcat:6qhlb4q4uzdqffgf5dnoj3qkwm

Near-Optimal Sample Compression for Nearest Neighbors

Lee-Ad Gottlieb, Aryeh Kontorovich, Pinhas Nisnevitch
2018 IEEE Transactions on Information Theory  
23: C(q, i − 1) ← ∅, N (q, i − 1) ← {p} Initialize lists of p: 24: NP-hard to find a better compressed set Reduction:• From Label Cover problem,• Reduction holds for Euclidean sets too Label cover:•  ...  Input: Graph, set of valid labels• Output: Valid labelling • Minimization version: Can use multiple colors, minimize number of labels • Dinur and Safra (2004) showed NP-hard to approximate within a factor  ... 
doi:10.1109/tit.2018.2822267 fatcat:itmer3jfajfojna2mod246r26q

Improved Hardness of Approximating Chromatic Number [article]

Sangxia Huang
2013 arXiv   pre-print
We prove that for sufficiently large K, it is NP-hard to color K-colorable graphs with less than 2^K^1/3 colors. This improves the previous result of K versus K^O(log K) in Khot [14].  ...  Acknowledgements The author is grateful to Siu On Chan for pointing out Theorem E.1 in his paper, leading to a gap of K vs. 2 K 1/3 , improving the original gap of K vs. 2 K 1/5 .  ...  Theorem 1. For all sufficiently large K, it is NP-hard to color a K-colorable graph with 2 K 1/3 colors.  ... 
arXiv:1301.5216v3 fatcat:akcrrznhqjhynm3puyq7feoocu

Improved Hardness of Approximating Chromatic Number [chapter]

Sangxia Huang
2013 Lecture Notes in Computer Science  
We prove that for sufficiently large K, it is NP-hard to color K-colorable graphs with less than 2 K 1/3 colors. This improves the previous result of K versus K O(log K) in Khot [14] .  ...  Acknowledgements The author is grateful to Siu On Chan for pointing out Theorem E.1 in his paper, leading to a gap of K vs. 2 K 1/3 , improving the original gap of K vs. 2 K 1/5 .  ...  Theorem 1. For all sufficiently large K, it is NP-hard to color a K-colorable graph with 2 K 1/3 colors.  ... 
doi:10.1007/978-3-642-40328-6_17 fatcat:nqgtmn7hrbbfbisaxzzhpsozjy

On NP-Hardness of the Paired de Bruijn Sound Cycle Problem [chapter]

Evgeny Kapun, Fedor Tsarev
2013 Lecture Notes in Computer Science  
In this paper we show that the problem of checking if there is a sound cycle in a paired de Bruijn graph is NP-hard in general case.  ...  The paired de Bruijn graph is an extension of de Bruijn graph incorporating mate pair information for genome assembly proposed by Mevdedev et al.  ...  The problem of checking whether a paired de Bruijn graph contains a covering sound cycle is NP-hard for any fixed |Σ| ≥ 2.  ... 
doi:10.1007/978-3-642-40453-5_6 fatcat:lj45sd7qfvflhlgcrvrgozh4yi

On NP-Hardness of the Paired de Bruijn Sound Cycle Problem [article]

Evgeny Kapun, Fedor Tsarev
2013 arXiv   pre-print
In this paper we show that the problem of checking if there is a sound cycle in a paired de Bruijn graph is NP-hard in general case.  ...  The paired de Bruijn graph is an extension of de Bruijn graph incorporating mate pair information for genome assembly proposed by Mevdedev et al.  ...  The problem of checking whether a paired de Bruijn graph contains a covering sound cycle is NP-hard for any fixed |Σ| ≥ 2.  ... 
arXiv:1307.7806v1 fatcat:4tnyye5bvvh6nhzops2egvpqn4

NP-Hardness of Coloring 2-Colorable Hypergraph with Poly-Logarithmically Many Colors

Amey Bhangale, Michael Wagner
2018 International Colloquium on Automata, Languages and Programming  
We give very short and simple proofs of the following statements: Given a 2-colorable 4-uniform hypergraph on n vertices, 1. It is NP-hard to color it with log δ n colors for some δ > 0. 2.  ...  It is quasi-NP-hard to color it with O log 1−o(1) n colors.  ...  Can we go beyond poly(log n) coloring NP-hardness factor for coloring a c-colorable k-uniform hypergraph for some constants c ≥ 2 and k ≥ 3?  ... 
doi:10.4230/lipics.icalp.2018.15 dblp:conf/icalp/Bhangale18 fatcat:mgbf47aqozgf5dbcg6a33wvnba

Improved 3LIN Hardness via Linear Label Cover

Prahladh Harsha, Subhash Khot, Euiwoong Lee, Devanathan Thiruvenkatachari, Michael Wagner
2019 International Workshop on Approximation Algorithms for Combinatorial Optimization  
The previous best hardness using a polynomial time reduction achieves ε = (log log n) −c , which is obtained by the Label Cover hardness of Moshkovitz and Raz [  ...  , finds an assignment that satisfies atleast ( 1 2 + ε)-fraction of clauses unless NP ⊆ BPP.  ...  For any constant c > 0, for δ = 1 /(log n) c , Gap Label Cover(1, δ) is NP-hard when the Label Cover instance satisfies |Σ A |, |Σ B | ≤ |A| + |B|.  ... 
doi:10.4230/lipics.approx-random.2019.9 dblp:conf/approx/HarshaKLT19 fatcat:vjy5cuwaqbambgcza7ev3gjlu4

The Projection Games Conjecture and the NP-Hardness of ln n-Approximating Set-Cover

Dana Moshkovitz
2015 Theory of Computing  
Our work implies that it is NP-hard to approximate SET-COVER on instances of size N to within (1 − α) ln N for arbitrarily small α > 0.  ...  We establish a tight NP-hardness result for approximating the SET-COVER problem based on a strong PCP theorem.  ...  Acknowledgments The motivation to prove the SET-COVER result came from discussions with Ran Raz.  ... 
doi:10.4086/toc.2015.v011a007 dblp:journals/toc/Moshkovitz15 fatcat:sg65cikhdre7vdecuqrsmmvjqi

On the hardness of approximating label-cover

Irit Dinur, Shmuel Safra
2004 Information Processing Letters  
The LABEL-COVER problem, defined by S. Arora, L. Babai, J. Stern  ...  We prove that LABEL-COVER is NP-hard to approximate to within 2 (log n)1 1−δ where δ = (log log n) −c for any c < 1/2.  ...  Our reduction will imply that LABEL-COVER is NP-hard to approximate to within factor g c (n) for any c < 1/2. Our starting point is the PCP theorem from [4] . Theorem 1 (PCP Theorem [4] ).  ... 
doi:10.1016/j.ipl.2003.11.007 fatcat:wecu2hhnczbddkgpleetf54rm4

Optimal Hub Labeling is NP-complete [article]

Mathias Weller
2014 arXiv   pre-print
We give a reduction from the well-known Vertex Cover problem on graphs to prove that finding an optimal hub labeling is indeed NP-hard.  ...  The problem of assigning as few such hubs as possible was conjectured to be NP-hard, but no proof was known to date.  ...  With Lemma 1, Lemma 2 and Lemma 3 imply the main theorem which, since Vertex Cover is NP-hard on planar graphs, holds also for apex graphs. Theorem 1.  ... 
arXiv:1407.8373v1 fatcat:7mxmoytfmbce5a4ymb2q75gga4

On the Inapproximability of Disjoint Paths and Minimum Steiner Forest with Bandwidth Constraints

Bin Ma, Lusheng Wang
2000 Journal of computer and system sciences (Print)  
within ratio 2 log 1&= n unless NP DTIME [2 polylog n ], the integer multicommodity flow problem in directed graphs cannot be approximated within ratio 2 log 1&= n unless NP DTIME[2 polylog n ], the max  ...  We show-that the max directed vertex-disjoint paths problem cannot be approximated within ratio 2 log 1&= n unless NP DTIME[2 polylog n ], the max directed edge-disjoint paths problem cannot be approximated  ...  For any =>0, max label cover cannot be approximated within ratio 2 log 1&= n unless NP DTIME[2 polylog n ]. The main theorem in this subsection is that DVDP is hard to approximate.  ... 
doi:10.1006/jcss.1999.1661 fatcat:ctryptac2vg53akhauyrjqgb24

New NP-hardness results for 3-Coloring and 2-to-1 Label Cover [article]

Per Austrin, Ryan O'Donnell, Li-Yang Tan, John Wright
2012 arXiv   pre-print
In a related result, we show that given a satisfiable instance of the 2-to-1 Label Cover problem, it is NP-hard to find a (23/24 + )-satisfying assignment.  ...  We show that given a 3-colorable graph, it is NP-hard to find a 3-coloring with (16/17 + ) of the edges bichromatic.  ...  Of particular interest is the d = 2 case, i.e., 2-to-1 Label Cover. Label Cover serves as the starting point for most NP-hardness of approximation results.  ... 
arXiv:1210.5648v2 fatcat:qjxagczemfdppbflroqazi6geq

Minimum Constraint Removal Problem for Line Segments is NP-hard [article]

Bahram Sadeghi Bigham
2021 arXiv   pre-print
In this paper, using a reduction from Subset Sum problem, in three steps, we show that the problem is NP-hard for both weighted and unweighted line segments.  ...  It has been proved that MCR problem is NP-hard when constraints have arbitrary shapes or even they are in shape of convex polygons.  ...  Also, one can present approximation or heuristic algorithms to solve new types of NP-hard cases of the problem.  ... 
arXiv:2107.03140v1 fatcat:urgg2siehvadtoolqcmnkuvolm
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