A new point of NP-hardness for unique games

@inproceedings{ODonnell2012ANP,
  title={A new point of NP-hardness for unique games},
  author={Ryan O'Donnell and John Wright},
  booktitle={STOC '12},
  year={2012}
}
We show that distinguishing 1/2-satisfiable Unique-Games instances from (3/8 + ε)-satisfiable instances is NP-hard (for all ε > 0). A consequence is that we match or improve the best known c vs. s NP-hardness result for Unique-Games for all values of c (except for c very close to 0). For these c, ours is the first hardness result showing that it helps to take the alphabet size larger than 2. Our NP-hardness reductions are quasilinear-size and thus show nearly full exponential time is required… Expand
Hardness of Maximum Constraint Satisfaction
We show optimal (up to a constant factor) NP-hardness for maximum constraint satisfaction problem with k variables per constraint (Max-k-CSP), whenever k is larger than the domain size. This followsExpand
Approximation resistance from pairwise independent subgroups
  • S. Chan
  • Computer Science, Mathematics
  • STOC '13
  • 2013
TLDR
The main ingredient is a new gap-amplification technique inspired by XOR-lemmas that improves the NP-hardness of approximating Independent-Set on bounded-degree graphs, Almost-Coloring, Two-Prover-One-Round-Game, and various other problems. Expand
Approximation resistance on satisfiable instances for predicates with few accepting inputs
TLDR
It is proved that for all integer k ≥ 3, there is a predicate P on k Boolean variables with 2~O(k1/3) accepting assignments that is approximation resistant even on satisfiable instances, which improves the best previously known result by Hastad and Khot. Expand
New NP-Hardness Results for 3-Coloring and 2-to-1 Label Cover
We show that given a 3-colorable graph, it is NP-hard to find a 3-coloring with (16/17 + ε) of the edges bichromatic. In a related result, we show that given a satisfiable instance of the 2-to-1Expand
Circumventing d-to-1 for Approximation Resistance of Satisfiable Predicates Strictly Containing Parity of Width Four - (Extended Abstract)
  • C. Wenner
  • Mathematics, Computer Science
  • APPROX-RANDOM
  • 2012
TLDR
This work extends modern hardness of approximation techniques by Mossel et al. to projection games, eliminating dependencies on the degree of projections via Smooth Label Cover, and proves unconditionally the same approximation resistance result for predicates of width four. Expand
A New Point of NP-Hardness for 2-to-1 Label Cover
We show that given a satisfiable instance of the 2-to-1 Label Cover problem, it is NP-hard to find a \((\frac{23}{24} + \epsilon)\)-satisfying assignment.
Strengthened Hardness for Approximating Minimum Unique Game and Small Set Expansion
  • Peng Cui
  • Computer Science, Mathematics
  • Electron. Colloquium Comput. Complex.
  • 2015
TLDR
A variation of Feige's Hypothesis, which claims that it is hard on average refuting Unbalanced Max 3-XOR under biased assignments on a natural distribution, is put forward. Expand
Label Cover Reductions for Unconditional Approximation Hardness of Constraint Satisfaction
We give the first examples of non-trivially positively-useless predicates subject only to P != NP. In particular, for every constraint function Q : {-1,1}^4 -> R, we constructExpand
Circumventing d-to-1 for Approximation Resistance of Satisfiable Predicates Strictly Containing Parity of Width at Least Four
  • C. Wenner
  • Computer Science, Mathematics
  • Theory Comput.
  • 2012
TLDR
Improved approximation hardness results for two classical problems: it is NP-hard to approximate Maximum Acyclic Subgraph with a factor better than 14/15 and Maximum Betweenness with a factors better than 1/2 and Gaussian elimination can efficiently find exact solutions for satisfiable collections of so-called parity constraints. Expand
On non-optimally expanding sets in Grassmann graphs
We study the structure of non-expanding sets in the Grassmann graph. We put forth a hypothesis stating that every small set whose expansion is smaller than 1– δ must be correlated with one of aExpand
...
1
2
3
...

References

SHOWING 1-10 OF 62 REFERENCES
Conditional hardness for satisfiable 3-CSPs
TLDR
It is proved that if Khot's d-to-1 Conjecture holds for any finite constant integer d, then NP naPCP1,5/8+ µ[O(log n),3] for any constant µ > 0.1 is NP-hard. Expand
Subexponential Algorithms for dto-1 Two-Prover Games and for Certifying Almost Perfect Expansion
A question raised by the recent subexponential algorithm for Unique Games (Arora, Barak, Steurer, FOCS 2010) is what other “hard-looking” problems admit good approximation algorithms withExpand
On Approximation Hardness of the Minimum 2SAT-DELETION Problem
TLDR
A lower approximation bound of \(8\sqrt 5-15\approx 2.88854\) is proved and a lower bound of \(\frac32\) is provided for highly restricted instances with exactly 4 occurrences of every variable. Expand
Conditional hardness for approximate coloring
TLDR
It is proved that the problem ALMOST-3-COLORINGε is hard for any constant ε>0, assuming Khot's Unique Games conjecture, and the result is based on bounding various generalized noise-stability quantities using the invariance principle of Mossel et al. Expand
Some optimal inapproximability results
We prove optimal, up to an arbitrary ε > 0, inapproximability results for Max-E k-Sat for k ≥ 3, maximizing the number of satisfied linear equations in an over-determined system of linear equationsExpand
Two Query PCP with Sub-Constant Error
  • Dana Moshkovitz, R. Raz
  • Computer Science, Mathematics
  • 2008 49th Annual IEEE Symposium on Foundations of Computer Science
  • 2008
TLDR
This work shows that the NP-Complete language 3Sat has a PCPverifier that makes two queries to a proof of almost-linear size and achieves sub-constant probability of error o(1), and improves many of the hardness of approximation results that are proved using the parallel repetition theorem. Expand
Spectral Algorithms for Unique Games
  • A. Kolla
  • Computer Science, Mathematics
  • 2010 IEEE 25th Annual Conference on Computational Complexity
  • 2010
TLDR
A new algorithm for Unique Games is given which is based on purely spectral techniques, in contrast to previous work in the area, which relies heavily on semidefinite programming (SDP), and is able to recover a good assignment given a highly satisfiable instance of Unique Games. Expand
Clique is hard to approximate within n/sup 1-/spl epsiv//
  • J. Håstad
  • Mathematics, Computer Science
  • Proceedings of 37th Conference on Foundations of Computer Science
  • 1996
TLDR
The author proves that unless NP=coR, Max Clique is hard to approximate in polynomial time within a factor n/sup 1-/spl epsiv// for any /spl delta/>0, constructing a proof system for NP which uses /splDelta/ amortized free bits. Expand
Approximating np-hard problems efficient algorithms and their limits
Most combinatorial optimization problems are NP-hard to solve optimally. A natural approach to cope with this intractability is to design an “approximation algorithm”—an efficient algorithm that isExpand
Subexponential Algorithms for Unique Games and Related Problems
TLDR
A sub exponential time approximation algorithm for the Unique Games problem that is exponential in an arbitrarily small polynomial of the input size, n, and shows that for every $\epsilon>0$ and every regular $n$-vertex graph~$G, one can break into disjoint parts so that the stochastic adjacency matrix of the induced graph on each part has at most n eigenvalues larger than $1-\eta. Expand
...
1
2
3
4
5
...