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Lower Bounds for On-line Interval Coloring with Vector and Cardinality Constraints

Lower Bounds for On-line Interval Coloring with Vector and Cardinality Constraints LOWER BOUNDS FOR ON-LINE INTERVAL COLORING WITH VECTOR AND CARDINALITY CONSTRAINTS GRZEGORZ GUTOWSKI AND PATRYK MIKOS Abstract. We propose two strategies for Presenter in the on-line interval graph coloring games. Specifically, we consider a setting in which each interval is associated with a d-dimensional vector of weights and the coloring needs to satisfy the d-dimensional bandwidth constraint, and the k-cardinality constraint. Such a variant was first introduced by Epstein and Levy and it is a natural model for resource-aware task scheduling with d different shared resources where at most k tasks can be scheduled simultaneously on a single machine. The first strategy forces any on-line interval coloring algorithm to use at least d d (5m − 3) different colors on an m + log d + 3 -colorable set of intervals. log d+3 k The second strategy forces any on-line interval coloring algorithm to use at least 5m d d different colors on an m + log d + 3 -colorable set of unit intervals. 2 log d+3 k 1. Introduction A proper coloring of a graph G is an assignment of colors to the vertices of the graph such that adjacent vertices receive distinct colors. A k-bounded coloring of G is a proper coloring of G such that the number of vertices that receive any single color is at most k. For a graph G, let χ(G) denote the chromatic number of G, that is the minimum number of colors in a proper coloring of G, and let ω(G) denote the clique number of G, that is, the maximum size of a subset of vertices such that any two vertices in the subset are adjacent. An on-line graph coloring game is a two-person game, played by Presenter and Algorithm. In each round Presenter introduces a new vertex of a graph with its adjacency status to all vertices presented earlier. Algorithm assigns a color to the incoming vertex in such a way that the coloring of the presented graph is proper. The color of the new vertex is assigned before Presenter introduces the next vertex and the assignment is irrevocable. The goal of Algorithm is to minimize the number of different colors used during the game. In the k-bounded variant of the game, the coloring constructed by Algorithm needs to be a k-bounded coloring of the presented graph. For an interval I = [l, r] on the real line, we say that l is the left endpoint, and r is the right endpoint of I. The length of interval I is the difference between its right endpoint and its left endpoint. A set of intervals on the real line represents a graph in the following way. Each interval represents a vertex and any two vertices arXiv:1608.03078v1 [math.CO] 10 Aug 2016 are joined by an edge whenever the corresponding intervals intersect. A graph which admits such a representation is an interval graph. An on-line interval coloring game is a two-person game, again played by Pre- senter and Algorithm. In each round Presenter introduces a new interval on the real line. Algorithm assigns a color to the incoming interval in such a way that the coloring of the presented interval graph is proper, i.e. all intervals of the same color are pairwise disjoint. The color of the new interval is assigned before Pre- senter introduces the next interval and the assignment is irrevocable. The goal of Algorithm is to minimize the number of different colors used during the game. We consider a few variants of the on-line interval coloring game. In the unit variant of the game, all intervals presented by Presenter are of length exactly 1. In the d-dimensional variant of the game, Presenter associates a d-dimensional vector of weights from [0, 1] with each presented interval. Moreover, the coloring constructed by Algorithm needs to satisfy a different condition. The condition is that for each color γ and any point p on the real line, the sum of weights of intervals containing p and colored γ does not exceed 1 in any of the coordinates. In the k-cardinality variant of the game, the coloring constructed by Algorithm needs to satisfy that for each color γ and any point p on the real line, the number of intervals containing p and colored γ does not exceed k. We are most interested in the on-line (k, d) interval coloring, a variant in which each interval has a d-dimensional vector of weights and the coloring must satisfy constraints of both d-dimensional and k-cardinality variant. That is, for each color γ and any point p, the number of intervals containing p and colored γ does not exceed k, and the sum of weights of those intervals does not exceed 1 in any coordinate. In the context of various on-line coloring games, the measure of the quality of a strategy for Algorithm is given by competitive analysis. A coloring strategy for Algorithm is r-competitive if it uses at most r · c colors for any c-colorable graph, or set of intervals, presented by Presenter. The competitive ratio for a problem is the infimum of all values r such that there exists an r-competitive strategy for Algorithm for this problem. In this paper we give lower bounds on competitive ratios for different problems. We obtain these results by presenting explicit strategies for Presenter that force any Algorithm strategy to use many colors while the presented graph, or set of intervals, is colorable with a smaller number of colors. We say that a strategy for Presenter in an on-line coloring problem is transparent if after each time Algorithm assigns a color to a vertex, or interval, Presenter colors the vertex with his own color and reveals that color to Algorithm. The coloring constructed by Presenter must satisfy the same conditions as the coloring constructed by Algorithm. The number of colors used by a transparent strategy for Presenter gives an upper bound on the minimum number of colors that can be used in a coloring. 2 1.1. Previous work. There is a simple strategy for Presenter in on-line graph col- oring game that forces Algorithm to use any number of colors while the constructed graph is 2-colorable. Thus, the competitive ratio for this problem is unbounded. Nevertheless, it is an interesting question what is the competitive ratio when the on-line game is played only for at most n rounds. Halld´orsson and Szegedy [4] presented a transparent strategy for Presenter that forces Algorithm to use at least 2 (1 + o(1)) different colors in n rounds of the game while the constructed log n graph is log n(1 + o(1))-colorable. The best known upper bound of O on log n the competitive ratio for the n-round on-line graph coloring problem was given by Lovasz, Saks and Trotter [6]. The competitive ratio for the on-line interval coloring problem was established by Kierstead and Trotter [5]. They constructed a strategy for Algorithm that uses at most 3ω − 2 colors while the clique size of the constructed graph is ω. They also presented a matching lower bound – a strategy for Presenter that forces Algorithm to use at least 3ω−2 colors. Unit variant of the on-line interval coloring problem was studied by Epstein and Levy [2]. They presented a strategy for 3ω Presenter that forces Algorithm to use at least colors while the clique size of the constructed graph is ω. Moreover, they showed that First-Fit algorithm uses at most 2ω−1 colors. Epstein and Levy [3] introduced many variants of the on-line interval coloring problem. The best known lower bound on the competitive ratio for the on-line (k, d) interval coloring is 3 for small k and for large k. For unit variant of this problem the best known lower bound is . Halld´orsson and Szegedy ideas were adopted by Azar et al. [1] to show lower bounds on competitive ratio for on-line d-vector bin packing. This problem is equivalent to a variant of d-dimensional on-line interval coloring where all presented intervals are the interval [0, 1] with different vectors of weights. Their strategy for Presenter shows that the competitive ratio for the on-line d-dimensional unit interval coloring problem is at least 2 (1 + o(1)). log d 1.2. Our results. We generalize Halld´orsson and Szegedy [4] strategy into the setting of the k-bounded coloring, and using the technique similar to the one by Azar et al. [1] we adopt it to the on-line (k, d) interval coloring problem. We present how to combine this technique with classical results by Kierstead and Trotter [5], and by Epstein and Levy [2, 3] to obtain a new lower bound of 5 (1 + o(1)) on the competitive ratio for the on-line (k, d) interval color- log d +log d ( ) 5 d ing, and a lower bound of (1 + o(1)) for unit variant of this problem. log d +log d ( ) 2. Graph coloring Theorem 1. For every n > 2 and k ∈ N , there is a transparent strategy for Presenter that forces Algorithm to use at least 2 different colors in the n- log n+3 round, k-bounded on-line graph coloring game and uses + log n + 3 colors. 3 Proof. Let b = ⌊log n⌋+3. The state of a k-bounded on-line graph coloring game is represented by a progress matrix M. Each cell M[i, j] is either empty or it contains exactly one vertex. At the beginning of the game, all cells are empty. A vertex in M[i, j] is colored by Algorithm with color j and by Presenter with color i. Each player can use a single color γ to color at most k vertices, so there are at most k vertices in any column, and in any row of the progress matrix. We say that a row with k vertices is depleted. Presenter can no longer use colors corresponding to depleted rows. Presenter maintains a set of exactly b active rows, denoted A, that contains all nonempty non-depleted rows and additionally some empty rows ({i : 1 6 |∪ M[i, j]| < k} ⊂ A and |A| = b). At the beginning of the game there are no depleted rows and A = {1, . . . , b}. When some row becomes depleted then it is removed from A and a new empty row is added to A. A pattern is a subset of rows. We say that a pattern p represents a column j if ∀i : i ∈ p ⇐⇒ M[i, j] 6= ∅. A pattern p is active if it is a nonempty subset of A such that |p| 6 . An active pattern p is present in M if at least one column of M is represented by p. 1 2 3 4 5 6 7 8 9 10 . . . 1 v v v v 1 3 5 7 2 v v v 2 10 12 3 v v v v 15 4 9 13 4 v v 6 11 5 v v 8 14 . . . Table 1. Example of a progress matrix after 15 rounds Table 1 shows a possible state of the progress matrix M after 15 rounds of the 4-bounded on-line graph coloring game with n = 4 and b = 4. In this example, the last introduced vertex v is colored by Algorithm with color 2 and by Presenter with color 3. Rows 1 and 3 are depleted and the set of active rows is A = {2, 4, 5, 6}. There are 10 different active patterns, but only 2 of them are present in M: pattern {2, 5} in column 6, and pattern {5} in column 9. The transparent strategy for Presenter for round t is as follows: (1) Choose an active pattern p that is not present in M. (2) Introduce a new vertex v that is adjacent to all vertices colored by Presenter with colors not in p . (3) Algorithm colors v with color γ. (4) Color v with any color ̺ such that ̺ ∈ p and M[̺, γ] = ∅. t t We claim that Presenter can follow this strategy as long as there is an active pattern not present in the progress matrix. To prove that, we need to show that in 4 step (4) Presenter always can choose an appropriate color ̺, and that the coloring constructed by Presenter is a k-bounded coloring of the constructed graph. Let q be a pattern that represents column M[∗, γ] of the progress matrix before round t. We claim that q ( p . Assume to the contrary that i ∈ q r p . It follows, t t that there is a vertex v in cell M[i, γ] and by rule (2) v is adjacent to v . Thus, Algorithm cannot color vertex v with color γ. Pattern q is present in M before round t and by rule (1) pattern p is not present in M before round t. It follows that q is a strict subset of p and Presenter has at least one choice for color ̺ in step (4). When Presenter assigns color ̺ to vertex v , we have that ̺ ∈ p ; p is an active t t t pattern; ̺ is an active row, and there are less than k vertices colored by Presenter with ̺. Rule (2) asserts that none of the vertices adjacent to v is colored with any of the colors in p . Thus, we have that Presenter can follow the strategy as long as there is a choice of an appropriate pattern in step (1). We claim that the game can be played for at least n rounds. Indeed, there are different patterns of size x and each one of them must represent a column of the progress matrix with exactly x vertices. Thus, when all active patterns represent some column of the progress matrix, the game has been played for at least x > n rounds. 16x6 x ⌊ ⌋ After n rounds, Presenter used colors corresponding to depleted and active rows. There are at most depleted rows and exactly ⌊log n⌋ + 3 active rows. Thus, Presenter uses at most + log n + 3 colors in the first n rounds. Let q be a pattern representing column M[∗, j] after n rounds. Let t be the last round when a vertex was added to column j. We have that q is a subset of pattern p which was an active pattern before round t, and the size of q is at most t j b n . Thus, there are at least 2 nonempty columns after n rounds. 2 log n+3 For fixed parameters n and k, denote a generalized Halld´orsson and Szegedy strategy by HS . Note that for k = +∞, there are no depleted rows in matrix k,n M and k-bounded coloring is simply a proper coloring. In this case we get the original Halld´orsson and Szegedy result for the on-line graph coloring problem. Theorem 2 (Halld´orsson, Szegedy [4]). For every integer n > 2, there is a trans- parent strategy for Presenter that forces Algorithm to use at least 2 colors in log n+3 the n-round on-line graph coloring game and uses log n + 3 colors. 3. Interval coloring In the proof of the following theorem, we use strategy HS to show a lower k,d bound on the competitive ratio for the on-line (k, d) interval coloring problem. Theorem 3. For every d > 2 and k, m ∈ N , there is a strategy for Presenter that forces Algorithm to use at least (5m − 3) different colors in the on-line (k, d) log d+3 interval coloring game while the constructed set of intervals is m + log d + 3 - colorable. Proof. For any fixed parameters k ∈ N , d > 2, L < R, ε ∈ (0, ) we describe an 1 1 auxiliary strategy HS (ε, L, R). Let α = 1− ε, δ = ε. In the t-th round of the k,d 2 2d on-line (k, d) interval coloring game, Presenter uses HS strategy to get a new k,d vertex v . Then, presents an interval [L, R] with weights w , where w = (x , . . . , x ) t t t 1 d is a d-dimensional vector with x = α, x = ε for all i < t such that v is adjacent t i i to v , and x = δ in every other coordinate. Figure 1 shows an example of w for t i 6 a vertex v that is adjacent to v and v . 6 2 5 ε ε δ δ δ δ δ δ 1 2 3 4 5 6 7 8 . . . d Figure 1. Encoding of v in a d-dimensional vector of weights ([L, R], w ) is colored by Algorithm with color γ . Then, γ is forwarded to HS t t t k,d as the color of v . HS colors v with ̺ , but Presenter discards that information. t k,d t t See Figure 2 for a diagram of the strategy HS (ε, L, R), and Figure 3 for an k,d example encoding of a graph. [L, R], w HS k,d Figure 2. Encoding of HS strategy k,d We claim that the encoding strategy ensures that any intervals I and I can get i j the same color iff vertices v and v are not adjacent. First, assume that i < j and i j v is adjacent to v . Vector w has α in the i-th coordinate, vector w has ε in the i j i j i-th coordinate, and α + ε > 1. Thus, intervals I and I must be colored with i j different colors. Let J ⊂ {I , ..., I } be the set of intervals colored with γ before 1 t−1 round t 6 d. Assume that v is not adjacent to any of the vertices in J . Denote the l-th coordinate of the sum of vectors of weights of intervals in J by W . For any 1 6 l 6 d, if I ∈ J then W = α + δ(|J | − 1) < 1 − δ. In this case we have l l that v is not adjacent to v , and that the l-th coordinate of the vector of weights t l Algorithm encode of v is δ. If I ∈/ J then W 6 ε|J | < 1− ε. For l = t, we have W 6 δ|J | 6 1− α. t l l l Thus, the sum of vector of weights of the intervals in the set J ∪ {I } does not exceed 1 in any coordinate and I can be colored with γ. w α δ δ δ δ δ w ε α δ δ δ δ ❞ 2 ✏ ◗ ❦ w δ ε α δ δ δ ✏ ◗ v ✏ v v 1 3 6 ❞✏ ✮ ❞ ◗❞ w ε δ ε α δ δ ❍ ❨ ❅ ■ ❍ w δ ε ε δ α δ ❅ ❍ 2 ✁ ❍ 5 ✠ ✁☛ ❅❞✛ ❍❞ w δ δ δ ε ε α Figure 3. Example of a graph and vectors of weights corresponding to the vertices Consider a sequence of parameters {ε } defined as ε = . See that for i i i∈N + 2d every i ∈ N we have ε ∈ (0, ) and we can use HS (ε , L, R) strategy. Let + i k,d i 1 1 α = 1 − ε and δ = ε = ε . i i i i i+1 2 2d Let J be a set of intervals constructed by HS (ε , L , R ) strategy and J be i k,d i i i j a set of intervals constructed by HS (ε , L , R ) strategy. Assume that i < j, k,d j j j [L , R ] ∩ [L , R ] 6= ∅ and that the construction of J is finished before the con- i i j j i struction of J starts. Any interval I ∈ J has weight α in one of the coordinates j j j and every interval in J has weight either α , ε or δ in that coordinate. In any i i i i case, sum of those weights exceeds 1 and no two intervals, one in J , other in J i j can be colored with the same color. The rest of the proof uses a technique similar to the one by Kierstead and Trot- d d ter [5]. For m ∈ N , let c = (5m − 3) , and o = m + log d + 3 . By + m m log d+3 k induction on m, we show a strategy S for Presenter such that: all introduced intervals are contained in a fixed region [A, B]; all intervals come from calls of strategies HS (ε, L, R) with ε in {ε , . . . , ε }; Algorithm uses at least c differ- k,d 1 3m m ent colors; constructed set of intervals is o -colorable. For m = 1 and a fixed region [A, B], Presenter uses strategy HS (ε , A, B). This strategy forces Algorithm to k,d 1 use at least c different colors, and the constructed set of intervals is o -colorable. 1 1 Thus, in this case we are done. m+1 Let c¯ = 3 + 1. Presenter splits the fixed region [A, B] into c¯ disjoint regions [l , r ], . . . , [l , r ]. By induction, in each region Presenter can use strategy 1 1 c¯ c¯ S independently. As a result, in each region [l , r ], we get a set of intervals m i i J . If during the construction Algorithm uses at least c colors, we are done. i m+1 Otherwise, let C be a c -element subset of colors used by Algorithm to color J . i m i Some c -element set of colors C appears on the list (C , . . . , C ) at least 4 times. m 1 c¯ Let a, b, c, d ∈ {1, . . . , c¯}, a < b < c < d be indices such that C = C = C = C = a b c d C . Define p = (r + l ) for i = a, b, c. i i i+1 Presenter uses strategy HS (ε , l , p ) to get a set of intervals K and then k,d 3m+1 a a 1 r +p c c strategy HS (ε , , r ) to get a set of intervals K . Let D be a c -element k,d 3m+1 d 2 1 1 7 subset of colors used by Algorithm to color K , and D be a c -element subset 1 2 1 of colors used to color K . The construction of J and J is finished before the 2 a d construction of K and K started and region [l , p ] covers [l , r ], and region 1 2 a a a a r +p c c ∗ , r covers [l , r ]. Thus, none of the colors in C can be used to color any d d d interval in K or K . Now, the strategy splits into two cases: we either have 1 2 1 1 |D ∩ D | 6 c or |D ∩ D | > c . 1 2 1 1 2 1 2 2 1 r +p a a Case 1. |D ∩ D | 6 c . Presenter uses strategy HS (ε , , p ) to get 1 2 1 k,d 3m+2 c 2 2 a set of intervals K . Let D be the set of colors used to color K . See Figure 4 3 3 3 r +p a a for a diagram of the construction. Region , p covers [l , r ] and we get c b b C ∩ D = ∅. Moreover, any interval in K intersects any interval in K , and any 3 3 1 interval in K . Thus, D ∩ D = ∅, D ∩ D = ∅, and Algorithm uses at least 2 3 1 3 2 |C ∪ D ∪ D ∪ D | > c + c + c + c = c colors. Each set of intervals 1 2 3 m 1 1 1 m+1 J , . . . , J intersects with intervals in at most one of the sets K , K , or K . Thus, 1 c¯ 1 2 3 all presented intervals can be colored with max{2, m + 1}( + log d + 3) = o m+1 colors and in this case we are done. p p p a b c l r l r l r l r a a b b c c d d C C C C a b c d ❄ ❄ ❄ D D 1 2 Figure 4. Construction in case |D ∩ D | 6 c 1 2 1 1 r +p b b Case 2. |D ∩ D | > c . Presenter uses strategy HS (ε , , p ) to get a 1 2 1 k,d 3m+2 c 2 2 r +p a a set of intervals K and then strategy HS (ε , , p ) to get K . Let D be 4 k,d 3m+3 b 5 4 a c -element subset of colors used by Algorithm to color K , and D be the set of 1 4 5 colors used to color K . See Figure 5 for a diagram of the construction. Similar ∗ ∗ argument as in the previous case gives C ∩ D = ∅, C ∩ D = ∅, D ∩ D = ∅, 4 5 2 4 D ∩D = ∅, and D ∩D = ∅. Set D contains at least c elements from the set D . 1 5 4 5 2 1 1 Thus, we have |D ∩ D | 6 c . Algorithm uses at least |C ∪ D ∪ D ∪ D | > 4 1 1 1 4 5 c + c + c + c = c colors. Each set of intervals J , . . . , J intersects with m 1 1 1 m+1 1 c¯ intervals in at most one of the sets K , K , K , or K . Thus, all intervals can be 1 2 4 5 colored with max{2, m + 1}( + log d + 3) = o colors. m+1 p p p a b c l r l r l r l r a a b b c c d d C C C C a b c d ❄ ❄ ❄ D D 1 2 Figure 5. Construction in case |D ∩ D | ≥ c 1 2 1 4. Unit interval coloring Theorem 4. For every d > 2 and k, m ∈ N , there is a strategy for Presenter that 5m d forces Algorithm to use at least different colors in the on-line (k, d) unit 2 log d+3 interval coloring game while the constructed set of intervals is m + log d + 3 - colorable. Proof. The proof combines strategy HS (ε, L, R) introduced in the proof of The- k,d orem 3 with technique similar to the one by Epstein and Levy [2, 3]. Assume that the sequence of encoding parameters {ε } is defined the same way as in the i i∈N proof of Theorem 3. The strategy consists of 3 phases. In the initial phase Presenter uses strategy HS (ε , 0, 1) for i = 1, . . . , sequentially. There is a coloring of all intervals k,d i m d introduced in the initial phase using + log d + 3 colors, but Algorithm uses 2 k m 2d m 2d at least colors. Let C be a -element subset of colors used init 2 log d+3 2 log d+3 by Algorithm in the initial phase. For L < R < L + 1, let Sep(L, R) be the separation strategy that introduces d unit intervals in the following way. Initialize l = L, and r = R. To get next interval, calculate p = (l + r) and introduce interval I = [p, p + 1]. If Algorithm colors I with color in C , then update r = p. Otherwise, mark interval I and init update l = p. Observe that to the left of p there are only left endpoints of marked intervals. Moreover, all introduced intervals have nonempty intersection. m 3 The separation phase consists of 2 subphases. We fix L = and R = 2. 1 1 2 2 For i = 2, . . . , 2 , points L and R are established after the (i − 1)-th subphase. i i Denote by Sub , the strategy for the i-th subphase being a combination of the HS (ε , 0, 1) strategy and the Sep(L , R ) strategy. Strategy Sub introduces d k,d i i i i intervals. The position of each interval is determined using Sep(L , R ) strategy, i i and the d-dimensional vector of weights associated with each interval is determined according to HS (ε , 0, 1). See Figure 6 for a diagram of the strategy Sub . k,d i i At the end of each subphase, Presenter decides whether the subphase is marked or not. The set of marked subphases is denoted by M. Let C be the set of colors used by Algorithm in the i-th subphase and not present in the set C . Subphase init i is marked if and only if one of the following conditions holds: (1) the number of remaining subphases including the i-th is − |M|. d m (2) |C | > and |M| < , log d+3 2 Observe that at the end of the separation phase we have exactly marked subphases. 9 [0, 1], w I , w t t Figure 6. Strategy Sub for a single subphase Let L be the left endpoint of the leftmost interval introduced in the i-th sub- phase. Let L be the left endpoint of the rightmost interval introduced in the i-th subphase and colored by Algorithm with a color c ∈/ C . Set L = L if such an init i interval does not exist. Let R be the left endpoint of the leftmost interval intro- duced in the i-th subphase and colored by Algorithm with a color c ∈ C . Set init R = R if such an interval does not exist. If subphase i is marked then L = L i i+1 and R = R. Otherwise, L = L and R = L . This completes the definition i+1 i+1 i i+1 of the separation phase. m 1 ′ ′ Let m = 2 and P = (L + R ). Observe that every interval intro- m +1 m +1 2 2 duced in the separation phase with the left endpoint to the left of P belongs to a marked subphase and is colored with a color c ∈/ C . Let C be the set of colors init sep used in the separation phase to color intervals with the left endpoint to the left of P . 2d In each subphase Algorithm uses at least different colors, so in the sepa- log d+3 m 2d ration phase Algorithm uses at least 2 colors in total. Because |C | = init 2 log d+3 m 2d m 2d , Algorithm, in the separation phase, uses at least colors not 2 log d+3 2 log d+3 in C . The set of marked subphases M contains x subphases in which Algorithm init d m m used at least such colors and −x last subphases. From the first +x log d+3 2 2 subphases only x subphases are marked. By the definition, in an unmarked sub- m d phase i for i 6 + x, Algorithm uses less than colors not in C . Thus, init 2 log d+3 m d m at most such colors from subphases 1 up to + x are not in the set 2 log d+3 2 C . All colors not in C used in the subphase i for i > + x are in the set sep init m d C . Thus, |C | > . sep sep 2 log d+3 In the final phase, Presenter uses strategy HS (ε , P − 1, P ) for i = m + k,d i 1, . . . , m + sequentially. Every interval introduced in the final phase intersects with every interval from the initial phase and every interval from the separation phase with the left endpoint to the left of P . Thus, each color used in the final phase belongs neither to C nor to C . In the final phase, Algorithm uses at init sep m 2d least colors. 2 log d+3 Algorithm combine Sep(L , R ) i i HS (ε , 0, 1) k,d i m m m d 5m d In total, Algorithm uses at least 2 + + 2 = dif- 2 2 2 log d+3 2 log d+3 ferent colors. On the other hand, the presented set of intervals can be easily colored m m d d using + + log d + 3 = m + log d + 3 colors. 2 2 k k Note that for k = +∞, the strategy HS becomes independent of k-cardinality ∞,d constraint. This gives two new bounds on the competitive ratio for on-line d- dimensional interval coloring problems. Theorem 5. For every d > 2 and n ∈ N , there is a strategy for Presenter that forces Algorithm to use at least (5m − 3) different colors in the on- log d+3 line d-dimensional interval coloring game while the constructed set of intervals is m(log d + 3)-colorable. Theorem 6. For every d > 2 and n ∈ N , there is a strategy for Presenter 5m d that forces Algorithm to use at least different colors in the on-line d- 2 log d+3 dimensional unit interval coloring game while the constructed set of intervals is m(log d + 3)-colorable. References [1] Yossi Azar, Ilan Reuven Cohen, Seny Kamara, and Bruce Shepherd. Tight bounds for online vector bin packing. In STOC 2013: 45th Annual ACM Symposium on Theory of Computing, Palo Alto, CA, USA, June 2013. Proceedings, pages 961–970, 2013. [2] Leah Epstein and Meital Levy. Online interval coloring and variants. In ICALP 2005: 32nd International Colloquim on Automata, Languages and Programming, Lisbon, Portugal, July 2005. Proceedings, volume 3580 of Lecture Notes in Computer Science, pages 602–613, 2005. [3] Leah Epstein and Meital Levy. Online interval coloring with packing constraints. In MFCS 2005: 30th International Symposium on Mathematical Foundations of Computer Science, Gdan´sk, Poland, August 2005. Proceedings, volume 3618 of Lecture Notes in Computer Sci- ence, pages 295–307, 2005. [4] Magnu´s M. Halldo´rsson and Mario Szegedy. Lower bounds for on-line graph coloring. Theo- retical Computer Science, 130(1):163 – 174, 1994. [5] Henry A. Kierstead and William T. Trotter. An extremal problem in recursive combinatorics. In 12th Southeastern Conference on Combinatorics, Graph Theory and Computing, Baton Rouge, LA, USA, March 1981. Proceedings, vol. II, volume 33 of Congressus Numerantium, pages 143–153, 1981. [6] Laszlo Lovasz, Michael Saks, and W.T. Trotter. An on-line graph coloring algorithm with sublinear performance ratio. Discrete Mathematics, 75(1):319 – 325, 1989. Theoretical Computer Science Department,, Faculty of Mathematics and Com- puter Science,, Jagiellonian University, Krakow, Poland E-mail address: {gutowski,mikos}@tcs.uj.edu.pl http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Computing Research Repository arXiv (Cornell University)

Lower Bounds for On-line Interval Coloring with Vector and Cardinality Constraints

Gutowski, Grzegorz; Mikos, Patryk
Computing Research Repository , Volume 2023 (1608) – Aug 10, 2016
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LOWER BOUNDS FOR ON-LINE INTERVAL COLORING WITH VECTOR AND CARDINALITY CONSTRAINTS GRZEGORZ GUTOWSKI AND PATRYK MIKOS Abstract. We propose two strategies for Presenter in the on-line interval graph coloring games. Specifically, we consider a setting in which each interval is associated with a d-dimensional vector of weights and the coloring needs to satisfy the d-dimensional bandwidth constraint, and the k-cardinality constraint. Such a variant was first introduced by Epstein and Levy and it is a natural model for resource-aware task scheduling with d different shared resources where at most k tasks can be scheduled simultaneously on a single machine. The first strategy forces any on-line interval coloring algorithm to use at least d d (5m − 3) different colors on an m + log d + 3 -colorable set of intervals. log d+3 k The second strategy forces any on-line interval coloring algorithm to use at least 5m d d different colors on an m + log d + 3 -colorable set of unit intervals. 2 log d+3 k 1. Introduction A proper coloring of a graph G is an assignment of colors to the vertices of the graph such that adjacent vertices receive distinct colors. A k-bounded coloring of G is a proper coloring of G such that the number of vertices that receive any single color is at most k. For a graph G, let χ(G) denote the chromatic number of G, that is the minimum number of colors in a proper coloring of G, and let ω(G) denote the clique number of G, that is, the maximum size of a subset of vertices such that any two vertices in the subset are adjacent. An on-line graph coloring game is a two-person game, played by Presenter and Algorithm. In each round Presenter introduces a new vertex of a graph with its adjacency status to all vertices presented earlier. Algorithm assigns a color to the incoming vertex in such a way that the coloring of the presented graph is proper. The color of the new vertex is assigned before Presenter introduces the next vertex and the assignment is irrevocable. The goal of Algorithm is to minimize the number of different colors used during the game. In the k-bounded variant of the game, the coloring constructed by Algorithm needs to be a k-bounded coloring of the presented graph. For an interval I = [l, r] on the real line, we say that l is the left endpoint, and r is the right endpoint of I. The length of interval I is the difference between its right endpoint and its left endpoint. A set of intervals on the real line represents a graph in the following way. Each interval represents a vertex and any two vertices arXiv:1608.03078v1 [math.CO] 10 Aug 2016 are joined by an edge whenever the corresponding intervals intersect. A graph which admits such a representation is an interval graph. An on-line interval coloring game is a two-person game, again played by Pre- senter and Algorithm. In each round Presenter introduces a new interval on the real line. Algorithm assigns a color to the incoming interval in such a way that the coloring of the presented interval graph is proper, i.e. all intervals of the same color are pairwise disjoint. The color of the new interval is assigned before Pre- senter introduces the next interval and the assignment is irrevocable. The goal of Algorithm is to minimize the number of different colors used during the game. We consider a few variants of the on-line interval coloring game. In the unit variant of the game, all intervals presented by Presenter are of length exactly 1. In the d-dimensional variant of the game, Presenter associates a d-dimensional vector of weights from [0, 1] with each presented interval. Moreover, the coloring constructed by Algorithm needs to satisfy a different condition. The condition is that for each color γ and any point p on the real line, the sum of weights of intervals containing p and colored γ does not exceed 1 in any of the coordinates. In the k-cardinality variant of the game, the coloring constructed by Algorithm needs to satisfy that for each color γ and any point p on the real line, the number of intervals containing p and colored γ does not exceed k. We are most interested in the on-line (k, d) interval coloring, a variant in which each interval has a d-dimensional vector of weights and the coloring must satisfy constraints of both d-dimensional and k-cardinality variant. That is, for each color γ and any point p, the number of intervals containing p and colored γ does not exceed k, and the sum of weights of those intervals does not exceed 1 in any coordinate. In the context of various on-line coloring games, the measure of the quality of a strategy for Algorithm is given by competitive analysis. A coloring strategy for Algorithm is r-competitive if it uses at most r · c colors for any c-colorable graph, or set of intervals, presented by Presenter. The competitive ratio for a problem is the infimum of all values r such that there exists an r-competitive strategy for Algorithm for this problem. In this paper we give lower bounds on competitive ratios for different problems. We obtain these results by presenting explicit strategies for Presenter that force any Algorithm strategy to use many colors while the presented graph, or set of intervals, is colorable with a smaller number of colors. We say that a strategy for Presenter in an on-line coloring problem is transparent if after each time Algorithm assigns a color to a vertex, or interval, Presenter colors the vertex with his own color and reveals that color to Algorithm. The coloring constructed by Presenter must satisfy the same conditions as the coloring constructed by Algorithm. The number of colors used by a transparent strategy for Presenter gives an upper bound on the minimum number of colors that can be used in a coloring. 2 1.1. Previous work. There is a simple strategy for Presenter in on-line graph col- oring game that forces Algorithm to use any number of colors while the constructed graph is 2-colorable. Thus, the competitive ratio for this problem is unbounded. Nevertheless, it is an interesting question what is the competitive ratio when the on-line game is played only for at most n rounds. Halld´orsson and Szegedy [4] presented a transparent strategy for Presenter that forces Algorithm to use at least 2 (1 + o(1)) different colors in n rounds of the game while the constructed log n graph is log n(1 + o(1))-colorable. The best known upper bound of O on log n the competitive ratio for the n-round on-line graph coloring problem was given by Lovasz, Saks and Trotter [6]. The competitive ratio for the on-line interval coloring problem was established by Kierstead and Trotter [5]. They constructed a strategy for Algorithm that uses at most 3ω − 2 colors while the clique size of the constructed graph is ω. They also presented a matching lower bound – a strategy for Presenter that forces Algorithm to use at least 3ω−2 colors. Unit variant of the on-line interval coloring problem was studied by Epstein and Levy [2]. They presented a strategy for 3ω Presenter that forces Algorithm to use at least colors while the clique size of the constructed graph is ω. Moreover, they showed that First-Fit algorithm uses at most 2ω−1 colors. Epstein and Levy [3] introduced many variants of the on-line interval coloring problem. The best known lower bound on the competitive ratio for the on-line (k, d) interval coloring is 3 for small k and for large k. For unit variant of this problem the best known lower bound is . Halld´orsson and Szegedy ideas were adopted by Azar et al. [1] to show lower bounds on competitive ratio for on-line d-vector bin packing. This problem is equivalent to a variant of d-dimensional on-line interval coloring where all presented intervals are the interval [0, 1] with different vectors of weights. Their strategy for Presenter shows that the competitive ratio for the on-line d-dimensional unit interval coloring problem is at least 2 (1 + o(1)). log d 1.2. Our results. We generalize Halld´orsson and Szegedy [4] strategy into the setting of the k-bounded coloring, and using the technique similar to the one by Azar et al. [1] we adopt it to the on-line (k, d) interval coloring problem. We present how to combine this technique with classical results by Kierstead and Trotter [5], and by Epstein and Levy [2, 3] to obtain a new lower bound of 5 (1 + o(1)) on the competitive ratio for the on-line (k, d) interval color- log d +log d ( ) 5 d ing, and a lower bound of (1 + o(1)) for unit variant of this problem. log d +log d ( ) 2. Graph coloring Theorem 1. For every n > 2 and k ∈ N , there is a transparent strategy for Presenter that forces Algorithm to use at least 2 different colors in the n- log n+3 round, k-bounded on-line graph coloring game and uses + log n + 3 colors. 3 Proof. Let b = ⌊log n⌋+3. The state of a k-bounded on-line graph coloring game is represented by a progress matrix M. Each cell M[i, j] is either empty or it contains exactly one vertex. At the beginning of the game, all cells are empty. A vertex in M[i, j] is colored by Algorithm with color j and by Presenter with color i. Each player can use a single color γ to color at most k vertices, so there are at most k vertices in any column, and in any row of the progress matrix. We say that a row with k vertices is depleted. Presenter can no longer use colors corresponding to depleted rows. Presenter maintains a set of exactly b active rows, denoted A, that contains all nonempty non-depleted rows and additionally some empty rows ({i : 1 6 |∪ M[i, j]| < k} ⊂ A and |A| = b). At the beginning of the game there are no depleted rows and A = {1, . . . , b}. When some row becomes depleted then it is removed from A and a new empty row is added to A. A pattern is a subset of rows. We say that a pattern p represents a column j if ∀i : i ∈ p ⇐⇒ M[i, j] 6= ∅. A pattern p is active if it is a nonempty subset of A such that |p| 6 . An active pattern p is present in M if at least one column of M is represented by p. 1 2 3 4 5 6 7 8 9 10 . . . 1 v v v v 1 3 5 7 2 v v v 2 10 12 3 v v v v 15 4 9 13 4 v v 6 11 5 v v 8 14 . . . Table 1. Example of a progress matrix after 15 rounds Table 1 shows a possible state of the progress matrix M after 15 rounds of the 4-bounded on-line graph coloring game with n = 4 and b = 4. In this example, the last introduced vertex v is colored by Algorithm with color 2 and by Presenter with color 3. Rows 1 and 3 are depleted and the set of active rows is A = {2, 4, 5, 6}. There are 10 different active patterns, but only 2 of them are present in M: pattern {2, 5} in column 6, and pattern {5} in column 9. The transparent strategy for Presenter for round t is as follows: (1) Choose an active pattern p that is not present in M. (2) Introduce a new vertex v that is adjacent to all vertices colored by Presenter with colors not in p . (3) Algorithm colors v with color γ. (4) Color v with any color ̺ such that ̺ ∈ p and M[̺, γ] = ∅. t t We claim that Presenter can follow this strategy as long as there is an active pattern not present in the progress matrix. To prove that, we need to show that in 4 step (4) Presenter always can choose an appropriate color ̺, and that the coloring constructed by Presenter is a k-bounded coloring of the constructed graph. Let q be a pattern that represents column M[∗, γ] of the progress matrix before round t. We claim that q ( p . Assume to the contrary that i ∈ q r p . It follows, t t that there is a vertex v in cell M[i, γ] and by rule (2) v is adjacent to v . Thus, Algorithm cannot color vertex v with color γ. Pattern q is present in M before round t and by rule (1) pattern p is not present in M before round t. It follows that q is a strict subset of p and Presenter has at least one choice for color ̺ in step (4). When Presenter assigns color ̺ to vertex v , we have that ̺ ∈ p ; p is an active t t t pattern; ̺ is an active row, and there are less than k vertices colored by Presenter with ̺. Rule (2) asserts that none of the vertices adjacent to v is colored with any of the colors in p . Thus, we have that Presenter can follow the strategy as long as there is a choice of an appropriate pattern in step (1). We claim that the game can be played for at least n rounds. Indeed, there are different patterns of size x and each one of them must represent a column of the progress matrix with exactly x vertices. Thus, when all active patterns represent some column of the progress matrix, the game has been played for at least x > n rounds. 16x6 x ⌊ ⌋ After n rounds, Presenter used colors corresponding to depleted and active rows. There are at most depleted rows and exactly ⌊log n⌋ + 3 active rows. Thus, Presenter uses at most + log n + 3 colors in the first n rounds. Let q be a pattern representing column M[∗, j] after n rounds. Let t be the last round when a vertex was added to column j. We have that q is a subset of pattern p which was an active pattern before round t, and the size of q is at most t j b n . Thus, there are at least 2 nonempty columns after n rounds. 2 log n+3 For fixed parameters n and k, denote a generalized Halld´orsson and Szegedy strategy by HS . Note that for k = +∞, there are no depleted rows in matrix k,n M and k-bounded coloring is simply a proper coloring. In this case we get the original Halld´orsson and Szegedy result for the on-line graph coloring problem. Theorem 2 (Halld´orsson, Szegedy [4]). For every integer n > 2, there is a trans- parent strategy for Presenter that forces Algorithm to use at least 2 colors in log n+3 the n-round on-line graph coloring game and uses log n + 3 colors. 3. Interval coloring In the proof of the following theorem, we use strategy HS to show a lower k,d bound on the competitive ratio for the on-line (k, d) interval coloring problem. Theorem 3. For every d > 2 and k, m ∈ N , there is a strategy for Presenter that forces Algorithm to use at least (5m − 3) different colors in the on-line (k, d) log d+3 interval coloring game while the constructed set of intervals is m + log d + 3 - colorable. Proof. For any fixed parameters k ∈ N , d > 2, L < R, ε ∈ (0, ) we describe an 1 1 auxiliary strategy HS (ε, L, R). Let α = 1− ε, δ = ε. In the t-th round of the k,d 2 2d on-line (k, d) interval coloring game, Presenter uses HS strategy to get a new k,d vertex v . Then, presents an interval [L, R] with weights w , where w = (x , . . . , x ) t t t 1 d is a d-dimensional vector with x = α, x = ε for all i < t such that v is adjacent t i i to v , and x = δ in every other coordinate. Figure 1 shows an example of w for t i 6 a vertex v that is adjacent to v and v . 6 2 5 ε ε δ δ δ δ δ δ 1 2 3 4 5 6 7 8 . . . d Figure 1. Encoding of v in a d-dimensional vector of weights ([L, R], w ) is colored by Algorithm with color γ . Then, γ is forwarded to HS t t t k,d as the color of v . HS colors v with ̺ , but Presenter discards that information. t k,d t t See Figure 2 for a diagram of the strategy HS (ε, L, R), and Figure 3 for an k,d example encoding of a graph. [L, R], w HS k,d Figure 2. Encoding of HS strategy k,d We claim that the encoding strategy ensures that any intervals I and I can get i j the same color iff vertices v and v are not adjacent. First, assume that i < j and i j v is adjacent to v . Vector w has α in the i-th coordinate, vector w has ε in the i j i j i-th coordinate, and α + ε > 1. Thus, intervals I and I must be colored with i j different colors. Let J ⊂ {I , ..., I } be the set of intervals colored with γ before 1 t−1 round t 6 d. Assume that v is not adjacent to any of the vertices in J . Denote the l-th coordinate of the sum of vectors of weights of intervals in J by W . For any 1 6 l 6 d, if I ∈ J then W = α + δ(|J | − 1) < 1 − δ. In this case we have l l that v is not adjacent to v , and that the l-th coordinate of the vector of weights t l Algorithm encode of v is δ. If I ∈/ J then W 6 ε|J | < 1− ε. For l = t, we have W 6 δ|J | 6 1− α. t l l l Thus, the sum of vector of weights of the intervals in the set J ∪ {I } does not exceed 1 in any coordinate and I can be colored with γ. w α δ δ δ δ δ w ε α δ δ δ δ ❞ 2 ✏ ◗ ❦ w δ ε α δ δ δ ✏ ◗ v ✏ v v 1 3 6 ❞✏ ✮ ❞ ◗❞ w ε δ ε α δ δ ❍ ❨ ❅ ■ ❍ w δ ε ε δ α δ ❅ ❍ 2 ✁ ❍ 5 ✠ ✁☛ ❅❞✛ ❍❞ w δ δ δ ε ε α Figure 3. Example of a graph and vectors of weights corresponding to the vertices Consider a sequence of parameters {ε } defined as ε = . See that for i i i∈N + 2d every i ∈ N we have ε ∈ (0, ) and we can use HS (ε , L, R) strategy. Let + i k,d i 1 1 α = 1 − ε and δ = ε = ε . i i i i i+1 2 2d Let J be a set of intervals constructed by HS (ε , L , R ) strategy and J be i k,d i i i j a set of intervals constructed by HS (ε , L , R ) strategy. Assume that i < j, k,d j j j [L , R ] ∩ [L , R ] 6= ∅ and that the construction of J is finished before the con- i i j j i struction of J starts. Any interval I ∈ J has weight α in one of the coordinates j j j and every interval in J has weight either α , ε or δ in that coordinate. In any i i i i case, sum of those weights exceeds 1 and no two intervals, one in J , other in J i j can be colored with the same color. The rest of the proof uses a technique similar to the one by Kierstead and Trot- d d ter [5]. For m ∈ N , let c = (5m − 3) , and o = m + log d + 3 . By + m m log d+3 k induction on m, we show a strategy S for Presenter such that: all introduced intervals are contained in a fixed region [A, B]; all intervals come from calls of strategies HS (ε, L, R) with ε in {ε , . . . , ε }; Algorithm uses at least c differ- k,d 1 3m m ent colors; constructed set of intervals is o -colorable. For m = 1 and a fixed region [A, B], Presenter uses strategy HS (ε , A, B). This strategy forces Algorithm to k,d 1 use at least c different colors, and the constructed set of intervals is o -colorable. 1 1 Thus, in this case we are done. m+1 Let c¯ = 3 + 1. Presenter splits the fixed region [A, B] into c¯ disjoint regions [l , r ], . . . , [l , r ]. By induction, in each region Presenter can use strategy 1 1 c¯ c¯ S independently. As a result, in each region [l , r ], we get a set of intervals m i i J . If during the construction Algorithm uses at least c colors, we are done. i m+1 Otherwise, let C be a c -element subset of colors used by Algorithm to color J . i m i Some c -element set of colors C appears on the list (C , . . . , C ) at least 4 times. m 1 c¯ Let a, b, c, d ∈ {1, . . . , c¯}, a < b < c < d be indices such that C = C = C = C = a b c d C . Define p = (r + l ) for i = a, b, c. i i i+1 Presenter uses strategy HS (ε , l , p ) to get a set of intervals K and then k,d 3m+1 a a 1 r +p c c strategy HS (ε , , r ) to get a set of intervals K . Let D be a c -element k,d 3m+1 d 2 1 1 7 subset of colors used by Algorithm to color K , and D be a c -element subset 1 2 1 of colors used to color K . The construction of J and J is finished before the 2 a d construction of K and K started and region [l , p ] covers [l , r ], and region 1 2 a a a a r +p c c ∗ , r covers [l , r ]. Thus, none of the colors in C can be used to color any d d d interval in K or K . Now, the strategy splits into two cases: we either have 1 2 1 1 |D ∩ D | 6 c or |D ∩ D | > c . 1 2 1 1 2 1 2 2 1 r +p a a Case 1. |D ∩ D | 6 c . Presenter uses strategy HS (ε , , p ) to get 1 2 1 k,d 3m+2 c 2 2 a set of intervals K . Let D be the set of colors used to color K . See Figure 4 3 3 3 r +p a a for a diagram of the construction. Region , p covers [l , r ] and we get c b b C ∩ D = ∅. Moreover, any interval in K intersects any interval in K , and any 3 3 1 interval in K . Thus, D ∩ D = ∅, D ∩ D = ∅, and Algorithm uses at least 2 3 1 3 2 |C ∪ D ∪ D ∪ D | > c + c + c + c = c colors. Each set of intervals 1 2 3 m 1 1 1 m+1 J , . . . , J intersects with intervals in at most one of the sets K , K , or K . Thus, 1 c¯ 1 2 3 all presented intervals can be colored with max{2, m + 1}( + log d + 3) = o m+1 colors and in this case we are done. p p p a b c l r l r l r l r a a b b c c d d C C C C a b c d ❄ ❄ ❄ D D 1 2 Figure 4. Construction in case |D ∩ D | 6 c 1 2 1 1 r +p b b Case 2. |D ∩ D | > c . Presenter uses strategy HS (ε , , p ) to get a 1 2 1 k,d 3m+2 c 2 2 r +p a a set of intervals K and then strategy HS (ε , , p ) to get K . Let D be 4 k,d 3m+3 b 5 4 a c -element subset of colors used by Algorithm to color K , and D be the set of 1 4 5 colors used to color K . See Figure 5 for a diagram of the construction. Similar ∗ ∗ argument as in the previous case gives C ∩ D = ∅, C ∩ D = ∅, D ∩ D = ∅, 4 5 2 4 D ∩D = ∅, and D ∩D = ∅. Set D contains at least c elements from the set D . 1 5 4 5 2 1 1 Thus, we have |D ∩ D | 6 c . Algorithm uses at least |C ∪ D ∪ D ∪ D | > 4 1 1 1 4 5 c + c + c + c = c colors. Each set of intervals J , . . . , J intersects with m 1 1 1 m+1 1 c¯ intervals in at most one of the sets K , K , K , or K . Thus, all intervals can be 1 2 4 5 colored with max{2, m + 1}( + log d + 3) = o colors. m+1 p p p a b c l r l r l r l r a a b b c c d d C C C C a b c d ❄ ❄ ❄ D D 1 2 Figure 5. Construction in case |D ∩ D | ≥ c 1 2 1 4. Unit interval coloring Theorem 4. For every d > 2 and k, m ∈ N , there is a strategy for Presenter that 5m d forces Algorithm to use at least different colors in the on-line (k, d) unit 2 log d+3 interval coloring game while the constructed set of intervals is m + log d + 3 - colorable. Proof. The proof combines strategy HS (ε, L, R) introduced in the proof of The- k,d orem 3 with technique similar to the one by Epstein and Levy [2, 3]. Assume that the sequence of encoding parameters {ε } is defined the same way as in the i i∈N proof of Theorem 3. The strategy consists of 3 phases. In the initial phase Presenter uses strategy HS (ε , 0, 1) for i = 1, . . . , sequentially. There is a coloring of all intervals k,d i m d introduced in the initial phase using + log d + 3 colors, but Algorithm uses 2 k m 2d m 2d at least colors. Let C be a -element subset of colors used init 2 log d+3 2 log d+3 by Algorithm in the initial phase. For L < R < L + 1, let Sep(L, R) be the separation strategy that introduces d unit intervals in the following way. Initialize l = L, and r = R. To get next interval, calculate p = (l + r) and introduce interval I = [p, p + 1]. If Algorithm colors I with color in C , then update r = p. Otherwise, mark interval I and init update l = p. Observe that to the left of p there are only left endpoints of marked intervals. Moreover, all introduced intervals have nonempty intersection. m 3 The separation phase consists of 2 subphases. We fix L = and R = 2. 1 1 2 2 For i = 2, . . . , 2 , points L and R are established after the (i − 1)-th subphase. i i Denote by Sub , the strategy for the i-th subphase being a combination of the HS (ε , 0, 1) strategy and the Sep(L , R ) strategy. Strategy Sub introduces d k,d i i i i intervals. The position of each interval is determined using Sep(L , R ) strategy, i i and the d-dimensional vector of weights associated with each interval is determined according to HS (ε , 0, 1). See Figure 6 for a diagram of the strategy Sub . k,d i i At the end of each subphase, Presenter decides whether the subphase is marked or not. The set of marked subphases is denoted by M. Let C be the set of colors used by Algorithm in the i-th subphase and not present in the set C . Subphase init i is marked if and only if one of the following conditions holds: (1) the number of remaining subphases including the i-th is − |M|. d m (2) |C | > and |M| < , log d+3 2 Observe that at the end of the separation phase we have exactly marked subphases. 9 [0, 1], w I , w t t Figure 6. Strategy Sub for a single subphase Let L be the left endpoint of the leftmost interval introduced in the i-th sub- phase. Let L be the left endpoint of the rightmost interval introduced in the i-th subphase and colored by Algorithm with a color c ∈/ C . Set L = L if such an init i interval does not exist. Let R be the left endpoint of the leftmost interval intro- duced in the i-th subphase and colored by Algorithm with a color c ∈ C . Set init R = R if such an interval does not exist. If subphase i is marked then L = L i i+1 and R = R. Otherwise, L = L and R = L . This completes the definition i+1 i+1 i i+1 of the separation phase. m 1 ′ ′ Let m = 2 and P = (L + R ). Observe that every interval intro- m +1 m +1 2 2 duced in the separation phase with the left endpoint to the left of P belongs to a marked subphase and is colored with a color c ∈/ C . Let C be the set of colors init sep used in the separation phase to color intervals with the left endpoint to the left of P . 2d In each subphase Algorithm uses at least different colors, so in the sepa- log d+3 m 2d ration phase Algorithm uses at least 2 colors in total. Because |C | = init 2 log d+3 m 2d m 2d , Algorithm, in the separation phase, uses at least colors not 2 log d+3 2 log d+3 in C . The set of marked subphases M contains x subphases in which Algorithm init d m m used at least such colors and −x last subphases. From the first +x log d+3 2 2 subphases only x subphases are marked. By the definition, in an unmarked sub- m d phase i for i 6 + x, Algorithm uses less than colors not in C . Thus, init 2 log d+3 m d m at most such colors from subphases 1 up to + x are not in the set 2 log d+3 2 C . All colors not in C used in the subphase i for i > + x are in the set sep init m d C . Thus, |C | > . sep sep 2 log d+3 In the final phase, Presenter uses strategy HS (ε , P − 1, P ) for i = m + k,d i 1, . . . , m + sequentially. Every interval introduced in the final phase intersects with every interval from the initial phase and every interval from the separation phase with the left endpoint to the left of P . Thus, each color used in the final phase belongs neither to C nor to C . In the final phase, Algorithm uses at init sep m 2d least colors. 2 log d+3 Algorithm combine Sep(L , R ) i i HS (ε , 0, 1) k,d i m m m d 5m d In total, Algorithm uses at least 2 + + 2 = dif- 2 2 2 log d+3 2 log d+3 ferent colors. On the other hand, the presented set of intervals can be easily colored m m d d using + + log d + 3 = m + log d + 3 colors. 2 2 k k Note that for k = +∞, the strategy HS becomes independent of k-cardinality ∞,d constraint. This gives two new bounds on the competitive ratio for on-line d- dimensional interval coloring problems. Theorem 5. For every d > 2 and n ∈ N , there is a strategy for Presenter that forces Algorithm to use at least (5m − 3) different colors in the on- log d+3 line d-dimensional interval coloring game while the constructed set of intervals is m(log d + 3)-colorable. Theorem 6. For every d > 2 and n ∈ N , there is a strategy for Presenter 5m d that forces Algorithm to use at least different colors in the on-line d- 2 log d+3 dimensional unit interval coloring game while the constructed set of intervals is m(log d + 3)-colorable. References [1] Yossi Azar, Ilan Reuven Cohen, Seny Kamara, and Bruce Shepherd. Tight bounds for online vector bin packing. In STOC 2013: 45th Annual ACM Symposium on Theory of Computing, Palo Alto, CA, USA, June 2013. Proceedings, pages 961–970, 2013. [2] Leah Epstein and Meital Levy. Online interval coloring and variants. In ICALP 2005: 32nd International Colloquim on Automata, Languages and Programming, Lisbon, Portugal, July 2005. Proceedings, volume 3580 of Lecture Notes in Computer Science, pages 602–613, 2005. [3] Leah Epstein and Meital Levy. Online interval coloring with packing constraints. In MFCS 2005: 30th International Symposium on Mathematical Foundations of Computer Science, Gdan´sk, Poland, August 2005. Proceedings, volume 3618 of Lecture Notes in Computer Sci- ence, pages 295–307, 2005. [4] Magnu´s M. Halldo´rsson and Mario Szegedy. Lower bounds for on-line graph coloring. Theo- retical Computer Science, 130(1):163 – 174, 1994. [5] Henry A. Kierstead and William T. Trotter. An extremal problem in recursive combinatorics. In 12th Southeastern Conference on Combinatorics, Graph Theory and Computing, Baton Rouge, LA, USA, March 1981. Proceedings, vol. II, volume 33 of Congressus Numerantium, pages 143–153, 1981. [6] Laszlo Lovasz, Michael Saks, and W.T. Trotter. An on-line graph coloring algorithm with sublinear performance ratio. Discrete Mathematics, 75(1):319 – 325, 1989. Theoretical Computer Science Department,, Faculty of Mathematics and Com- puter Science,, Jagiellonian University, Krakow, Poland E-mail address: {gutowski,mikos}@tcs.uj.edu.pl

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Published: Aug 10, 2016

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