Hardness of Approximate Hypergraph Coloring

Guruswami, Venkatesan; Håstad, Johan; Sudan, Madhu

doi:10.1137/s0097539700377165

Cited by 43 publications

(40 citation statements)

References 29 publications

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“…Recently, several researchers [11,14,20] have shown that it is hard to color uniform hypergraphs, i.e., hypergraphs where each hyperedge is of equal size.…”

Section: Hardness Results For Smaller Concept Classesmentioning

confidence: 99%

The complexity of properly learning simple concept classes

Alekhnovich¹,

Braverman

Feldman

et al. 2008

Journal of Computer and System Sciences

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We consider the complexity of properly learning concept classes, i.e. when the learner must output a hypothesis of the same form as the unknown concept. We present the following new upper and lower bounds on well-known concept classes:• We show that unless NP = RP, there is no polynomial-time PAC learning algorithm for DNF formulas where the hypothesis is an OR-of-thresholds. Note that as special cases, we show that neither DNF nor OR-of-thresholds are properly learnable unless NP = RP. Previous hardness results have required strong restrictions on the size of the output DNF formula. We also prove that it is NP-hard to learn the intersection of 2 halfspaces by the intersection of k halfspaces for any constant k 0. Previous work held for the case when k = .• Assuming that NP DTIME(2 n ) for a certain constant < 1 we show that it is not possible to learn size s decision trees by size s k decision trees for any k 0. Previous hardness results for learning decision trees held for k 2. • We present the first non-trivial upper bounds on properly learning DNF formulas. More specifically, we show how to learn size s DNF by DNF in time 2Õ ( √ n log s) .The hardness results for DNF formulas and intersections of halfspaces are obtained via specialized graph products for amplifying the hardness of approximating the chromatic number as well as applying recent work on the hardness of approximate hypergraph coloring. The hardness results for decision trees, as well as the new upper bounds, are obtained by developing a connection between automatizability in proof complexity and learnability, which may have other applications.

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“…Recently, several researchers [11,14,20] have shown that it is hard to color uniform hypergraphs, i.e., hypergraphs where each hyperedge is of equal size.…”

Section: Hardness Results For Smaller Concept Classesmentioning

confidence: 99%

The complexity of properly learning simple concept classes

Alekhnovich¹,

Braverman

Feldman

et al. 2008

Journal of Computer and System Sciences

View full text Add to dashboard Cite

show abstract

“…For each i, 1 ≤ i ≤ m, at most six equations can be simultaneously satisfied: at most one of (5)-(7) and at most five of (8)- (15). If any component of x is neither 1 nor −1, we can set it to 1; if w i is not 0 and y i +z i is not 0, we can set them to 0; for old values of y i , z i , since y i +z i is not 0, at most one of (14) and (15) is satisfied.…”

Section: The Hardness Of Approximating Eq 1 Z [3]mentioning

confidence: 99%

“…There has been much work on this study. Many strong inapproximability results for problems such as Max Cut, Max Di-Cut, Exact Satisfiability, and Vertex Cover [7][8][9]15,17,18,20,23] can be obtained from the connection. In [17], Håstad has proved that it is NP-hard to approximate maximum simultaneously satisfiable equations over a finite Abelian group G. Later, the result is extended to all finite groups [10].…”

Section: Introductionmentioning

confidence: 99%

Inapproximability results for equations over infinite groups

Chen

Yin

Chen

2010

Theoretical Computer Science

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a b s t r a c tAn equation over a group G is an expression of form w 1 . . . w k = 1 G , where each w i is either a variable, an inverted variable, or a group constant and 1 G denotes the identity element; such an equation is satisfiable if there is a setting of the variables to values in G such that the equality is realized (Engebretsen et al. (2002) [10]).In this paper, we study the problem of simultaneously satisfying a family of equationsthe problem of determining the maximum number of simultaneously satisfiable equations in which each equation has occurrences of exactly k different variables. When G is an infinite cyclic group, we show that it is NPhard to approximate EQ 1 G [3] to within 48/47 − , where EQ 1 G [3] denotes the special case of EQ G [3] in which a variable may only appear once in each equation; it is NP-hard to approximate EQ 1 G [2] to within 30/29 − ; it is NP-hard to approximate the maximum number of simultaneously satisfiable equations of degree at most d to within d− for any ; for any k ≥ 4, it is NP-hard to approximate EQ G [k] within any constant factor. These results extend Håstad's results (Håstad (2001) [17]) and results of (Engebretsen et al. (2002) [10]), who established the inapproximability results for equations over finite Abelian groups and any finite groups respectively.

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“…Many of the recent hardness results are shown via constructions of new Probabilistically Checkable Proof systems (PCPs) (see, e.g., [4,15,17,18]). These constructions typically involve two modules, the so-called Outer PCP and the so-called Inner PCP.…”

Section: Main Techniquesmentioning

confidence: 99%

Vertex cover might be hard to approximate to within 2−ε

Khot

Regev

2008

Journal of Computer and System Sciences

516

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Based on a conjecture regarding the power of unique 2-prover-1-round games presented in [S. Khot, On the power of unique 2-Prover 1-Round games, in: Proc. 34th ACM Symp. on Theory of Computing, STOC, May 2002, pp. 767-775], we show that vertex cover is hard to approximate within any constant factor better than 2. We actually show a stronger result, namely, based on the same conjecture, vertex cover on k-uniform hypergraphs is hard to approximate within any constant factor better than k.

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Hardness of Approximate Hypergraph Coloring

Cited by 43 publications

References 29 publications

The complexity of properly learning simple concept classes

The complexity of properly learning simple concept classes

Inapproximability results for equations over infinite groups

Vertex cover might be hard to approximate to within 2−ε

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