A new lower bound for the on-line coloring of intervals with bandwidth
aa r X i v : . [ m a t h . C O ] A p r A NEW LOWER BOUND FOR THE ON-LINE COLORING OFINTERVALS WITH BANDWIDTH
PATRYK MIKOS
Abstract.
The on-line interval coloring and its variants are important com-binatorial problems with many applications in network multiplexing, resourceallocation and job scheduling. In this paper we present a new lower bound of4 . . For the on-line coloringof unit intervals with bandwidth we improve the lower bound of 1 .
831 to 2. Introduction An on-line coloring of intervals with bandwidth is a two-person game, played inrounds by Presenter and Algorithm. In each round Presenter introduces a newinterval on the real line and its bandwidth - a real number from [0 , γ andany point p on the real line, the sum of bandwidths of intervals containing p andcolored γ does not exceed 1. The color of the new interval is assigned beforePresenter introduces the next interval and the assignment is irrevocable. The goalof Algorithm is to minimize the number of different colors used during the game,while the goal of Presenter is to maximize it.An on-line coloring of unit intervals with bandwidth is a variant of on-line col-oring of intervals with bandwidth game in which all introduced intervals are oflength exactly 1.In the context of various on-line coloring games, the measure of quality of astrategy for Algorithm is given by the competitive analysis. A coloring strategyfor Algorithm is r -competitive if it uses at most r · c colors for any c -colorable set ofintervals. The absolute 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.Let χ A ( I ) be the number of colors used by Algorithm A on the set I of intervalswith bandwidth, and OP T ( I ) be the minimum number of colors required to colorintervals in the set I .The asymptotic competitive ratio for Algorithm A , denoted by R ∞ A , is defined asfollows: R ∞ A = lim inf k →∞ { χ A ( I ) k : OP T ( I ) = k } Research partially supported by NCN grant number 2014/14/A/ST6/00138. he asymptotic competitive ratio for a problem is the infimum of all values R ∞ A such that A is an Algorithm for this problem.In this paper we give lower bounds on competitive ratios for on-line coloring ofintervals with bandwidth and for unit version of this problem. We obtain theseresults by presenting explicit strategies for Presenter that force Algorithm to usemany colors while the presented set of intervals is colorable with a smaller numberof colors.1.1. Previous work.
A variant of on-line coloring of intervals with bandwidthin which all intervals introduced by Presenter have bandwidth 1 is known as anon-line interval coloring. The competitive ratio for this problem was established byKierstead and Trotter [6]. They constructed a strategy for Algorithm that uses atmost 3 ω − ω -colorable set of intervals. They also presented a matchinglower bound – a strategy for Presenter that forces Algorithm to use at least 3 ω − (cid:4) ω (cid:5) colors. Moreover, they showed that a natural greedy algorithm uses atmost 2 ω − for the asymp-totic competitive ratio in this problem. On-line coloring of unit intervals withbandwidth was studied by Epstein and Levy [5]. They presented a lower boundof 2 and upper bound of for the absolute competitive ratio in this problem. Forthe asymptotic competitive ratio, they showed a 3 . . Our result.
For the on-line coloring of intervals with bandwidth, we provethat the asymptotic competitive ratio is at least 4 . k − k -colorable. . Interval coloring
At first we recall a strategy proposed by Kierstead and Trotter for Presenter inthe on-line interval coloring game. We use this strategy as a substrategy in ourmain result.
Theorem 1 (Kierstead, Trotter [6]) . For every ω ∈ N + , there is a strategy forPresenter that forces Algorithm to use at least ω − different colors in the on-lineinterval coloring game played on a ω -colorable set of intervals. Moreover, Presentercan play in such a way that every introduced interval is contained in a fixed realinterval [ L, R ] . Below we present a strategy for Presenter in the on-line coloring of intervals withbandwidth. For a fixed k ∈ N + , we ensure that at any point of the game, the setof intervals introduced by Presenter is k -colorable. Definition 2.
A pair of sequences ([ j , . . . , j n ] , [ x , . . . , x n ]) such that x i ∈ N + , j i | k and ∀ q
Intervals introduced in the i -th subphase in relation tothe intervals introduced in the ( i − L R P M L R P M L i − R i − P i − M i − ✻ k ✻ Γ ✻ Γ ✻ Γ i − Figure 2.
Distribution of intervals in the first i − k -schema ([ j , . . . , j n ] , [ x , . . . , x n ]) actually describesa valid strategy for Presenter, i.e. when Presenter is able to force Algorithm to useat least x i new colors in the i -th subphase.During the i -th subphase, we have L i + s i p i R i . Thus, the distance betweenrightmost right endpoint of an interval introduced in the i -th subphase and leftmostright endpoint of interval introduced in this phase is at most s i , hence all intervalsintroduced by Presenter in the i -th subphase form a clique. See Figure 1.Note that to the right of l i − there are only marked intervals from subphases1 , . . . , i −
1. Each interval introduced in the i -th subphase intersects with everyinterval previously introduced in the i -th subphase and all marked intervals fromsubphases 1 , . . . , i −
1. Thus, if Presenter introduces at most kj i ( k − Γ i − ) intervalsin the i -th subphase, then all intervals can be colored with k colors. ntervals introduced in the i -th subphase intersect with exactly χ i − markedintervals from subphases 1 , . . . , i − M each of themhas a different color. Thus, in the real interval [ L i , R i ], each color marked in asubphase 1 q < i has accumulated bandwidth j q k . We assumed that ∀ q
For a fixed k ∈ N + and a k -strategy ([1] , [ k ]) we have Γ = 1 . Pre-senter using this strategy forces Algorithm to use at least k + 3( k − − k − colors, while the set of introduced intervals is k -colorable. Thus, the asymptoticcompetitive ratio in the on-line coloring of intervals with bandwidth is at least . j i x i
120 1 1 1 1 1 2 2 2 4 5 4 8Γ i Table 1.
Example of a strategy S . Example 5.
Consider the -strategy given by the values j i , x i and Γ i from theTable 1. Presenter using this strategy forces Algorithm to use − − colors, while the set of introduced intervals is -colorable. Thus, the absolutecompetitive ratio for the on-line coloring of intervals with bandwidth is at least . Example 5 is an example of a k -strategy for Presenter for a fixed k , and gives alower bound for the absolute competitive ratio. In order to give lower bounds forthe asymptotic competitive ratio, we introduce a notion of a scalable strategy . efinition 6. For a k -strategy S k = ([ j , . . . , j n ] , [ x , . . . , x n ]) , an ak -schema S ak =([ aj , . . . , aj n ] , [ ax , . . . , ax n ]) for a ∈ N + is called an a -scaled S k schema. Note that a -scaled k -strategy might not be an ak -strategy. For example S isa 120-strategy but S is not a 360-strategy. To see this, observe that in S wehave 12 = x > ∆ = 11.Consider a zk -scaled S k schema for z ∈ N + . We would like to introduce addi-tional constraints on a k -strategy S k that will ensure that S zkk is a zk -strategy.In the i -th subphase of the game played using a schema S zkk Presenter forcesAlgorithm to use zkx i new colors. Each new marked interval has bandwidth zkj i zk = j i k for some j i such that j i | k . Thus, these new intervals after i -th sub-phase are colored by Presenter with j i k zkx i = zj i x i colors. By induction, for the S zkk schema we have Γ i = z Σ iq =1 j q x q and χ i = zk Σ iq =1 x q . Presenter using thisschema in the i -th subphase can introduce kj i ( zk − Γ i − ) = zkj i (cid:0) k − Σ i − q =1 j q x q (cid:1) in-tervals. Moreover, at most kj i − c . Thus, in the i -th subphase Algorithm is forced to use atleast ∆ i = l j i k (cid:16) zkj i (cid:0) k − Σ i − q =1 j q x q (cid:1) − zk Σ i − q =1 x q (cid:16) kj i − (cid:17)(cid:17)m new colors, which aftersimplifying is ∆ i = z (cid:0) k + Σ i − q =1 ( j i − j q − k ) x q (cid:1) . The S zkk schema is a zk -strategyif ∀ i : zkx i ∆ i . Thus, we have a condition(1) x i k + 1 k Σ i − q =1 ( j i − j q − k ) x q Definition 7. A k -strategy S k that satisfies Equation (1) is called a scalable strat-egy . Note that Equation (1) does not depend on z . This leads to the following lemma. Lemma 8.
If a k -strategy S k satisfies Equation (1) , then for every z ∈ N + a zk -scaled S k schema is a zk -strategy. Lemma 9.
For every k, z ∈ N + and a scalable k -strategy S k the competitive ratioguaranteed by the S zkk strategy is not less than the competitive ratio guaranteed bythe S k strategy.Proof. Let S k = ([ j , . . . , j n ] , [ x , . . . , x n ]). Presenter using a strategy S k forcesAlgorithm to use X = Σ ni =1 x i + 3( k − Γ n ) − k -colorable. Presenter using a strategy S zkk forces Al-gorithm to use ¯ X = Σ ni =1 zkx i + 3 (cid:0) zk − ¯Γ n (cid:1) − zk -colorable. Observe that the number of colors re-quired in greedy coloring of all intervals marked by S zkk strategy is at most zk timesbigger than the number of colors required in greedy coloring of all intervals markedby S k strategy, i.e. ¯Γ n zk Γ n . Thus, we have zk ¯ X > k X + k − zk > k X . (cid:3) i x i
120 1 1 1 1 1 2 2 2 3 6 4 8Γ i Table 2.
Example of a scalable strategy ¯ S Example 10.
Table 2 is a description of a scalable strategy with competitive ratio . This strategy implies a lower bound of + = 4 for the asymptoticcompetitive ratio in the on-line coloring of intervals with bandwidth. In order to obtain the best lower bound for the asymptotic competitive ratiowe can chose k to be a highly composed number . As a sequence j , . . . , j n wechose consecutive divisors of k and as a sequence x , . . . , x n we greedily choose themaximum numbers x i such that the resulting strategy is a scalable strategy.Table 3 contains the list of lower bounds for the asymptotic competitive ratiowe got for some values of k using this method. k ratio60 4 . . . . . . . . . . . . k ratio2162160 4 . . . . . . . . . . . . Table 3.
A table of asymptotic competitive ratios for different val-ues of k Theorem 11.
Asymptotic competitive ratio for the on-line coloring of intervalswith bandwidth is at least . . . Unit intervals coloring
Theorem 12.
For every k ∈ N + , there is a strategy for Presenter that forcesAlgorithm to use at least k − different colors in the on-line coloring of unitintervals with bandwidth played on a k -colorable set of intervals.Proof. For a given k ∈ N + , Presenter at first plays only the separation phase of a k -strategy ([1] , [ k ]). Because L = 0, R = 2 and s = ( R − L ), every introducedinterval has length 1. Moreover, there is a point p = ( l + r ) such that everymarked interval has its right endpoint to the right of p and every non-markedinterval has its right endpoint to the left of p .Now, Presenter introduces k − p , p + 1] of bandwidth 1 each. Everyinterval introduced in this phase gets a new color. Thus, Algorithm uses |M| + k − k − k -colorable. (cid:3) References [1] Udo Adamy and Thomas Erlebach. Online coloring of intervals with bandwidth. In
WAOA2003: 1st International Workshop on Approximation and Online Algorithms, Budapest, Hun-gary, September 2003. Proceedings , volume 2909 of
Lecture Notes in Computer Science , pages1–12, 2004.[2] Yossi Azar, Amos Fiat, Meital Levy, and NS Narayanaswamy. An improved algorithm for on-line coloring of intervals with bandwidth.
Theoretical Computer Science , 363(1):18–27, 2006.[3] J´anos Csirik and Gerhard J. Woeginger.
On-line packing and covering problems , pages 147–177. Springer Berlin Heidelberg, Berlin, Heidelberg, 1998.[4] Leah Epstein and Meital Levy. Online interval coloring and variants. In
ICALP 2005: 32ndInternational Colloquim on Automata, Languages and Programming, Lisbon, Portugal, July2005. Proceedings , volume 3580 of
Lecture Notes in Computer Science , pages 602–613, 2005.[5] Leah Epstein and Meital Levy. Online interval coloring with packing constraints. In
MFCS2005: 30th International Symposium on Mathematical Foundations of Computer Science,Gda´nsk, Poland, August 2005. Proceedings , volume 3618 of
Lecture Notes in Computer Sci-ence , pages 295–307, 2005.[6] Henry A. Kierstead and William T. Trotter. An extremal problem in recursive combinatorics.In , volume 33 of
Congressus Numerantium ,pages 143–153, 1981.[7] NS Narayanaswamy. Dynamic storage allocation and on-line colouring interval graphs. In
COCOON 2004: 10th Annual International Conference on Computing and Combinatorics,Jeju Island, Korea, August 2004. Proceedings , volume 3106 of
Lecture Notes in ComputerScience , pages 329–338, 2004.[8] Sriram V. Pemmaraju, Rajiv Raman, and Kasturi R. Varadarajan. Max-coloring and onlinecoloring with bandwidths on interval graphs.
ACM Transactions on Algorithms , 7(3):35:1–35:21, 2011.
Theoretical Computer Science Department,, Faculty of Mathematics and Com-puter Science,, Jagiellonian University, Krak´ow, Poland
E-mail address : [email protected]@tcs.uj.edu.pl