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

2012
*
arXiv
*
pre-print

In a related

arXiv:1210.5648v2
fatcat:qjxagczemfdppbflroqazi6geq
*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. ... Interestingly,*to*a certain extent the reverse is also true: it is "folklore" that*hardness**results**for*graph*3*-*Coloring*immediately imply*hardness**results**for*the*2*-*to*-*1**Label**Cover*problem with*label*...##
###
NP-Hardness of Coloring 2-Colorable Hypergraph with Poly-Logarithmically Many Colors

2018
*
International Colloquium on Automata, Languages and Programming
*

Our

doi:10.4230/lipics.icalp.2018.15
dblp:conf/icalp/Bhangale18
fatcat:mgbf47aqozgf5dbcg6a33wvnba
*result*is the first*to*show the*NP*-*hardness*of*coloring*a c-*colorable*k-uniform hypergraph with poly-logarithmically many*colors*,*for*any constants c ≥*2**and*k ≥*3*. ... 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*. ... 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*? ...##
###
Improved Hardness of Approximating Chromatic Number
[article]

2013
*
arXiv
*
pre-print

We prove that

arXiv:1301.5216v3
fatcat:akcrrznhqjhynm3puyq7feoocu
*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 . ... It is known that*coloring**3*-*colorable*graph with 5*colors*is*NP*-*hard*,*and**for*general K-*colorable*graph it is*NP*-*hard**to**color*with K +*2*⌈ K*3*⌉*colors*[13, 9] . ...##
###
Improved Hardness of Approximating Chromatic Number
[chapter]

2013
*
Lecture Notes in Computer Science
*

We prove that

doi:10.1007/978-3-642-40328-6_17
fatcat:nqgtmn7hrbbfbisaxzzhpsozjy
*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 . ... It is known that*coloring**3*-*colorable*graph with 5*colors*is*NP*-*hard*,*and**for*general K-*colorable*graph it is*NP*-*hard**to**color*with K +*2*⌈ K*3*⌉*colors*[13, 9] . ...##
###
d-To-1 Hardness of Coloring 3-Colorable Graphs with O(1) Colors

2020
*
International Colloquium on Automata, Languages and Programming
*

The d-

doi:10.4230/lipics.icalp.2020.62
dblp:conf/icalp/GuruswamiS20
fatcat:jlb2a77e25cvpcjwi7tbdklmja
*to*-*1*conjecture of Khot asserts that it is*NP*-*hard**to*satisfy an ε fraction of constraints of a satisfiable d-*to*-*1**Label**Cover*instance,*for*arbitrarily small ε > 0. ... Earlier, the*hardness*of O(*1*)-*coloring*a 4-*colorable*graphs is known under the*2*-*to*-*1*conjecture, which is the strongest in the family of d-*to*-*1*conjectures,*and*the*hardness**for**3*-*colorable*graphs is ... Since b(n) is increasing*and*b(n) > n*for*all n ≥ 4, it follows that a*NP*-*hardness**result**for*O(*1*)-*coloring*q-*colorable*graphs also implies*NP*-*hardness*of O(*1*)-*coloring*4-*colorable*graphs. ...##
###
A New Point of NP-Hardness for 2-to-1 Label Cover
[chapter]

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. ... The first

*result*along these lines was due

*to*Dinur, Mossel,

*and*Regev [DMR09] who showed that the

*2*-

*to*-

*1*Conjecture implies that it is

*NP*-

*hard*

*to*C-

*color*a 4-

*colorable*graph

*for*any constant C. ... Our

*results*In this paper, we focus on proving

*NP*-

*hardness*

*for*the

*2*-

*to*-

*1*

*Label*

*Cover*problem. ...

##
###
Rainbow Coloring Hardness via Low Sensitivity Polymorphisms

2020
*
SIAM Journal on Discrete Mathematics
*

Given as input such a hypergraph, finding a r-rainbow

doi:10.1137/19m127731x
fatcat:ji3puh626fb43psqamoqrhjjky
*coloring*of it is*NP*-*hard**for*all k ≥*3**and*r ≥*2*. ... In this work we consider the next smaller value of r = k −*1*,*and*prove that in this case it is*NP*-*hard**to*rainbow*color*the hypergraph with q := k−*2**2**colors*. ... This is the basis of the*hardness**results**for*(*2*+ )-SAT [4]*and**3*-*coloring*graphs that admit a homomorphism*to*C k*for*any fixed odd integer k [24] . ...##
###
Inapproximability Results for Combinatorial Auctions with Submodular Utility Functions

2007
*
Algorithmica
*

Our

doi:10.1007/s00453-007-9105-7
fatcat:jo7odecozrbb7ikluyqsz2se4u
*result*is based on a reduction from a multi-prover proof system*for*MAX-*3*-*COLORING*. ... −*1*/e 0.632, unless P=*NP*. ... Acknowledgements We would like*to*thank Amin Saberi*and*Vahab Mirrokni*for*useful discussions*and*Shahar Dobzinski*for*comments. We would also like*to*thank the anonymous referees*for*their comments. ...##
###
Inapproximability Results for Combinatorial Auctions with Submodular Utility Functions
[chapter]

2005
*
Lecture Notes in Computer Science
*

Our

doi:10.1007/11600930_10
fatcat:67sbsrg5nbglrhalhceiovw4m4
*result*is based on a reduction from a multi-prover proof system*for*MAX-*3*-*COLORING*. ... −*1*/e 0.632, unless P=*NP*. ... Acknowledgements We would like*to*thank Amin Saberi*and*Vahab Mirrokni*for*useful discussions*and*Shahar Dobzinski*for*comments. We would also like*to*thank the anonymous referees*for*their comments. ...##
###
On Complexity and Approximability of the Labeled Maximum/Perfect Matching Problems
[chapter]

2005
*
Lecture Notes in Computer Science
*

In this paper, we deal with both the complexity

doi:10.1007/11602613_93
fatcat:2ojn36muwbgetldpkvty35n7te
*and*the approximability of the*labeled*perfect matching problem in bipartite graphs. ... Given a simple graph G = (V, E) with n vertices with a*color*(or*label*) function L : E → {c1, . . . , cq}, the*labeled*maximum matching problem consists in finding a maximum matching on G that uses a minimum ... We conjecture that*Labeled*M in P M is not O(n ε )-approximable in complete bipartite graphs. Thus, a challenge will be*to*give better approximate algorithms or*to*improve the lower bound. ...##
###
A new point of NP-hardness for 2-to-1 Label Cover
[article]

2012
*
arXiv
*
pre-print

We show that given a satisfiable instance of the

arXiv:1204.5666v1
fatcat:a5b4kdi6ojc7blmjtrbuqmpdei
*2*-*to*-*1**Label**Cover*problem, it is*NP*-*hard**to*find a (23/24 + )-satisfying assignment. ... Our*results*In this paper, we focus on proving*NP*-*hardness**for*the*2*-*to*-*1**Label**Cover*problem. ... The main*result*of our paper gives an improved*hardness**result*: Theorem 1.2.*For*all > 0, (*1*, 23 24 + )-deciding the*2*-*to*-*1**Label**Cover*problem with*label*set sizes*3*& 6 is*NP*-*hard*. ...##
###
The labeled perfect matching in bipartite graphs

2005
*
Information Processing Letters
*

Given a simple graph G = (V, E) with |V | = 2n vertices such that E contains a perfect matching (of size n), together with a

doi:10.1016/j.ipl.2005.06.009
fatcat:ygk54vxvxzeo5hnipt3hbe3vpi
*color*(or*label*) function L : E → {c*1*, . . . , c q }, the*labeled*perfect ... In this paper, we deal with both the complexity*and*the approximability of the*labeled*perfect matching problem in bipartite graphs. ... Thus, we deduce from this*result*that*Labeled*M ax P M r is*NP*-*hard**for*any r ≥*2*. We strengthen this*result*using a reduction from Max balanced*2*-Sat. ...##
###
Fixed-parameter complexity of λ-labelings

2001
*
Discrete Applied Mathematics
*

We show several

doi:10.1016/s0166-218x(00)00387-5
fatcat:5f6hb3oln5bnri66hgf6i5rnza
*hardness**results**for*L(G; p; q) including that*for*any p ¿ q¿*1*there is a = (p; q) such that deciding if L(G; p; q)6 is*NP*-complete,*and*that*for*p¿2q, this decision is*NP*-complete*for*... We study the minimum value = (G) such that G admits a -*labeling*. We show that*for*every ÿxed value k¿4 it is*NP*-complete*to*determine whether (G)6k. ... Most of our*NP*-*hardness**results*are obtained via graph*covers*. Deÿnition 2.3. Let G be a graph*and*H be a multigraph. ...##
###
Almost Optimal Inapproximability of Multidimensional Packing Problems
[article]

2021
*
arXiv
*
pre-print

We also show that the problem is

arXiv:2101.02854v2
fatcat:s3nxsafvurbmxmmtdrdubumoz4
*NP*-*hard**to*approximate within (loglog d)^ω(*1*).*3*. ... Previously, no*hardness**results*that grow with d were known*for*Vector Scheduling*and*Vector Bin*Covering*when d is part of the input*and**for*Vector Bin Packing when d is a fixed constant. ... Acknowledgements I am greatly indebted*to*Venkatesan Guruswami*for*helpful discussions,*for*his detailed feedback on the manuscript which significantly improved the presentation,*and**for*his encouragement ...##
###
The Hardness of 3-Uniform Hypergraph Coloring

2005
*
Combinatorica
*

This enables us

doi:10.1007/s00493-005-0032-4
fatcat:455einszcfcgncrd4ld62ukeyi
*to*prove that*for*any constant c, it is*NP*-*hard**to**color*a*2*-*colorable*4-uniform hypergraph using just c*colors*,*and*also yields a super-constant inapproximability*result*under a stronger ... The*covering*complexity of PCP verifiers offers a promising route*to*getting stronger inapproximability*results**for*some minimization problems,*and*in particular, (hyper)-graph*coloring*problems. ... No non-trivial*hardness**results*seem*to*be known,*and*in fact it was not known prior*to*our work if*3*-*coloring*a*2*-*colorable*4-uniform hypergraph is*NP*-*hard*. ...
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