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Recursive Randomized Coloring Beats Fair Dice Random Colorings
, 2001
"... We investigate a re ned recursive coloring approach to construct balanced colorings for hypergraphs. A coloring is called balanced if each hyperedge has (roughly) the same number of vertices in each color. ..."
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We investigate a re ned recursive coloring approach to construct balanced colorings for hypergraphs. A coloring is called balanced if each hyperedge has (roughly) the same number of vertices in each color.
Balanced kColorings
"... While discrepancy theory is normally only studied in the context of 2colorings, we explore the problem of kcoloring, for k 2, a set of vertices to minimize imbalance among a family of subsets of vertices. The imbalance is the maximum, over all subsets in the family, of the largest difference b ..."
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While discrepancy theory is normally only studied in the context of 2colorings, we explore the problem of kcoloring, for k 2, a set of vertices to minimize imbalance among a family of subsets of vertices. The imbalance is the maximum, over all subsets in the family, of the largest difference between the size of any two color classes in that subset. The discrepancy is the minimum possible imbalance. We show that the discrepancy is always at most 4d \Gamma 3, where d (the "dimension") is the maximum number of subsets containing a common vertex. For 2colorings, the bound on the discrepancy is at most maxf2d \Gamma 3; 2g. Finally, we prove that several restricted versions of computing the discrepancy are NPcomplete. 1
Discrepancy in Different Numbers of Colors
, 2001
"... In this article we investigate the interrelation between the discrepancies of a given hypergraph in dierent numbers of colors. Being an extreme example we determine the multicolor discrepancies of the k{balanced hypergraph H nk on partition classes of (equal) size n. Let c; k; n 2 N. Set k 0 := k ..."
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In this article we investigate the interrelation between the discrepancies of a given hypergraph in dierent numbers of colors. Being an extreme example we determine the multicolor discrepancies of the k{balanced hypergraph H nk on partition classes of (equal) size n. Let c; k; n 2 N. Set k 0 := k mod c and b nkc := n n d c k e k c . For the discrepancy in c colors we show b nk 0 c disc(H nk ; c) < b nk 0 c + 1; if k 0 6= 0, and disc(H nk ; c) = 0, if c divides k. This shows that in general there is little correlation between the discrepancies of H nk in dierent numbers of colors. If c divides k though, disc(H; c) k c disc(H; k) holds for any hypergraph H. Keywords: Discrepancy, hypergraph coloring. AMS Subject Classication: Primary 11K38, Secondary 05C15. supported by the graduate school `Eziente Algorithmen und Multiskalenmethoden', Deutsche Forschungsgemeinschaft 1 1
Coloring tDimensional mBoxes
, 2001
"... . Call the set S1 S t t{dimensional m{box if jS i j = m for every i = 1; : : : ; t. Let R t (m; r) be the smallest integer R such that for every r{coloring of t{fold cartesian product of [R] one can nd a monochromatic t{dimensional m{box. We give a lower and an upper bound for R t (m; r). We a ..."
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. Call the set S1 S t t{dimensional m{box if jS i j = m for every i = 1; : : : ; t. Let R t (m; r) be the smallest integer R such that for every r{coloring of t{fold cartesian product of [R] one can nd a monochromatic t{dimensional m{box. We give a lower and an upper bound for R t (m; r). We also consider the discrepancy problem connected to this setsystem. Among other bounds we prove that the discrepancy of the hypergraph of all 2{dimensional m{boxes in [R] [R] is equal to (R 3 2 ) for m a constant fraction (less than 1 2 ) of R. 1. Introduction and results In this article we investigate the problem of coloring a t{dimensional grid. This raises two types of questions: Ramseytheory asks for conditions that imply the existence of a monochromatic box, discrepancy theory asks for the maximal deviation that occurs in one box. To be more precise: Let R; m; t 2 N . Denote by [R] the set f1; 2; : : : ; Rg and by [R] t the t{fold cartesian product of [R] with itself (no...
THE HEREDITARY DISCREPANCY IS NEARLY INDEPENDENT OF THE NUMBER OF COLORS
"... Abstract. We investigate the discrepancy (or balanced coloring) problem for hypergraphs and matrices in arbitrary numbers of colors. We show that the hereditary discrepancy in two different numbers a, b ∈ N≥2 of colors is the same apart from constant factors, i.e., herdisc(·,b)=Θ(herdisc(·,a)). This ..."
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Abstract. We investigate the discrepancy (or balanced coloring) problem for hypergraphs and matrices in arbitrary numbers of colors. We show that the hereditary discrepancy in two different numbers a, b ∈ N≥2 of colors is the same apart from constant factors, i.e., herdisc(·,b)=Θ(herdisc(·,a)). This contrasts the ordinary discrepancy problem, where no correlation exists in many cases. 1. Introduction and
Discrepancy of Products of Hypergraphs
"... For a hypergraph H = (V, E), its d–fold symmetric product is ∆ d H = (V d, {E d E ∈ E}). We give several upper and lower bounds for the ccolor discrepancy of such products. In particular, we show that the bound disc( ∆ d H, 2) ≤ disc(H, 2) proven for all d in [B. Doerr, A. Srivastav, and P. Wehr, ..."
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For a hypergraph H = (V, E), its d–fold symmetric product is ∆ d H = (V d, {E d E ∈ E}). We give several upper and lower bounds for the ccolor discrepancy of such products. In particular, we show that the bound disc( ∆ d H, 2) ≤ disc(H, 2) proven for all d in [B. Doerr, A. Srivastav, and P. Wehr, Discrepancy of Cartesian products of arithmetic progressions, Electron. J. Combin. 11(2004), Research Paper 5, 16 pp.] cannot be extended to more than c = 2 colors. In fact, for any c and d such that c does not divide d!, there are hypergraphs having arbitrary large discrepancy and disc( ∆ d H, c) = Ωd(disc(H, c) d). Apart from constant factors (depending on c and d), in these cases the symmetric product behaves no better than the general direct product H d, which satisfies disc(H d, c) = Oc,d(disc(H, c) d).
Discrepancy of Symmetric Products of Hypergraphs
"... For a hypergraph H =(V,E), its d–fold symmetric product is defined to be ∆ d H =(V d, {E d E ∈E}). We give several upper and lower bounds for the ccolor discrepancy of such products. In particular, we show that the bound disc( ∆ d H, 2) ≤ disc(H, 2) proven for all d in [B. Doerr, A. Srivastav, and ..."
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For a hypergraph H =(V,E), its d–fold symmetric product is defined to be ∆ d H =(V d, {E d E ∈E}). We give several upper and lower bounds for the ccolor discrepancy of such products. In particular, we show that the bound disc( ∆ d H, 2) ≤ disc(H, 2) proven for all d in [B. Doerr, A. Srivastav, and P. Wehr, Discrepancy of Cartesian products of arithmetic progressions, Electron. J. Combin. 11(2004), Research Paper 5, 16 pp.] cannot be extended to more than c = 2 colors. In fact, for any c and d such that c does not divide d!, there are hypergraphs having arbitrary large discrepancy and disc( ∆ d H,c)=Ωd(disc(H,c) d). Apart from constant factors (depending on c and d), in these cases the symmetric product behaves no better than the general direct product H d, which satisfies disc(H d,c)=Oc,d(disc(H,c) d). 1