Improved bounds and algorithms for hypergraph 2‐coloring

Radhakrishnan, Jaikumar; Srinivasan, Aravind

doi:10.1002/(sici)1098-2418(200001)16:1<4::aid-rsa2>3.3.co;2-u

Cited by 21 publications

(24 citation statements)

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“…This property, also called Property B, has been studied in the extremal combinatorics literature for long. Much work has been done on determining sufficient conditions under which a hypergraph family is 2-colorable and on solving the corresponding algorithmic questions [11,6,7,25,26,29,27].…”

Section: Introductioncontrasting

confidence: 99%

Hardness of Approximate Hypergraph Coloring

Guruswami¹,

Håstad²,

Sudan³

2002

SIAM J. Comput.

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We introduce the notion of covering complexity of a verifier for probabilistically checkable proofs (PCP). Such a verifier is given an input, a claimed theorem, and an oracle, representing a purported proof of the theorem. The verifier is also given a random string and decides whether to accept the proof or not, based on the given random string. We define the covering complexity of such a verifier, on a given input, to be the minimum number of proofs needed to "satisfy" the verifier on every random string, i.e., on every random string, at least one of the given proofs must be accepted by the verifier. 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. We present a PCP verifier for NP statements that queries only four bits and yet has a covering complexity of one for true statements and a super-constant covering complexity for statements not in the language. Moreover, the acceptance predicate of this verifier is a simple Not-all-Equal check on the four bits it reads. This enables us 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 hardness assumption.

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Section: Introductioncontrasting

confidence: 99%

Hardness of Approximate Hypergraph Coloring

Guruswami¹,

Håstad²,

Sudan³

2002

SIAM J. Comput.

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“…Upper bound was obtained by Erdős in [5] and has not been improved since. The most recent improvement on the lower bound has been made by Radhakrishnan and Srinivasan in [12]. Many results on similar extermal parameters of hypergraph coloring can be found in surveys by Kostochka [9] and Raigorodskii and Shabanov [13].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…In particular the number of red vertices in P R and blue vertices in P B can only decrease. Moreover at least one of these values decreases in each iteration of the main loop (lines [8][9][10][11][12][13][14]. Therefore the procedure always stops.…”

Section: Multipass Greedy Coloringmentioning

confidence: 99%

Multipass greedy coloring of simple uniform hypergraphs

Kozik

2015

Random Struct. Alg.

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Abstract. Let m * (n) be the minimum number of edges in an n-uniform simple hypergraph that is not two colorable. We prove that m * (n) = Ω(4 n / ln 2 (n)). Our result generalizes to r-coloring of b-simple uniform hypergraphs. For fixed r and b we prove that a maximum vertex degree in b-simple n-uniform hypergraph that is not r-colorable must be Ω(r n / ln(n)). By trimming arguments it implies that every such graph has Ω((r n / ln(n)) b+1/b ) edges. For any fixed r 2 our techniques yield also a lower bound Ω(r n / ln(n)) for van der Waerden numbers W (n, r).

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“…The first is wellknown and regards 2-colorable hypergraphs, also said to possess Property B. Several papers have presented bounds on m(k), the minimum number of edges in a k-uniform hypergraph that does not have Property B (see [1], [2], [5] and [6]). The second comes from Ramsey theory, where appropriate properties of graphs containing a given graph with a fixed order can be used to prove negative partition relations for unordered graphs (see [3] and [4] for early papers on this topic).…”

Section: Introductionmentioning

confidence: 99%