Liu, H

Wong, PW

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Complexity and online algorithmsfor a coloring problem on a lineThomas Erlebach ∗ Fu-Hong Liu † Hsiang-Hsuan Liu † ‡ (Speaker)Mordechai Shalom § Prudence Wong ‡ Shmuel Zaks ¶1 IntroductionWe consider a coloring problem that arises in the context of wavelength assignment onan optical line network [1, 6]. Related coloring problems on a line network with differentobjective functions include [2, 3, 7, 8].We are given a set of n intervals I = {I1, I2, · · · , In}. Each Ij is an interval [sj , ej),where sj and ej denote the start and end time of the interval Ij , respectively. Twointervals Ii and Ij are said to be overlapping if Ii ∩ Ij 6= ∅. We say that interval Ijcontains time t if t ∈ Ij . The load at time t, denoted by load(t) is the number of intervalscontaining t. The length of Ij , denoted by `(Ij), is defined as ej − sj . The maximumand minimum length over all intervals in I is denoted by `max and `min, respectively.The length of any set of intervals S is defined as the sum of the length of all intervals inS, i.e., `(S) = ∑I∈S `(I). Without loss of generality, we can assume that the intervalsin I form a contiguous interval, otherwise, the scheduling of each contiguous interval isindependent of the others, so we can focus on just one such contiguous interval.We are also given an infinite set of colors Λ = {1, 2, 3, · · · }, each color i is associatedwith a cost λ(i) ≥ 1 which is an increasing function such that λ(i) < λ(i′) if i < i′.Unless specified otherwise we assume λ(i) = i. A coloring ω : I → Λ is valid if forany pair of overlapping intervals Ii 6= Ij , ω(Ii) 6= ω(Ij). The cost of ω at any time t,denoted by cost(ω, t), is the maximum cost of the color over all intervals containing t,i.e., cost(ω, t) = maxI:t∈I λ(ω(I)). The total cost of ω, denoted as cost(ω), is defined asthe sum of the cost over all times t, i.e., cost(ω) =∫t cost(ω, t). The objective of theMinSumMax problem is to find a valid coloring ω such that cost(ω) is minimized.For any algorithm A we also denote its coloring by A, and its cost is cost(A). Wedenote by O the optimal oﬄine algorithm.∗t.erlebach@leicester.ac.uk Department of Informatics, University of Leicester, UK†{fhliu,hhliu}@cs.nthu.edu.tw Department of Computer Science, National Tsing Hua University,Taiwan‡{hhliu,pwong}@liverpool.ac.uk Department of Computer Science, University of Liverpool, UK§cmshalom@telhai.ac.il TelHai College, Upper Galilee, 12210, Israel¶zaks@cs.technion.ac.il Department of Computer Science, Technion, Haifa, Israel12 Our contribution2.1 The MinSumMax problemA more general problem has been studied in [1] which was proved to be NP-complete.On the other hand, their 2-approximation algorithm applies to our problem.We first show by a reduction from circular arc coloring [5] that, our problemMinSumMax which is simpler remains NP-complete.Theorem 1. MinSumMax is NP-complete.We then propose online algorithms which give O(1)-competitiveness when the ratioof maximum and minimum length of intervals is bounded by a constant and O(log `max`min )-competitiveness for arbitrary length intervals where `max and `min denote the maximumand minimum length of intervals.Bounded length intervals. Consider bounded length intervals for which we assumethere is a constant k such that for any interval I in the input, we have `(I) ∈ [`min, k·`min).When an interval Ij ∈ I arrives, G assigns the minimum color that is valid for it, i.e., Gassigns the minimum color i such that for all j′ < j and Ij′ ∩ Ij 6= ∅, we have G(Ij′) 6= i.Our analysis identifies intervals that “contribute” to the cost of a coloring and usestheir length to bound the total cost. The skyline of a coloring ω is the function of themaximum color used at any time t. We denote the set of intervals on the skyline by ISwhich contains intervals whose color is on the skyline for some time t, i.e., an interval Iis in IS if there exists t ∈ I such that cost(ω, t) = λ(ω(I)). We further select a subset I∗Sof intervals on the skyline of G from IS, partition the time horizon into segments basedon I∗S , and show that we can “charge” the cost of G and O to this subset, thus allowing usto bound the two costs. The partition of time is based on a notion of extended interval.For any interval Ij , we define its hat interval as Ihj = [sj−k`min, ej +k`min) and extendedhat interval as Iej = [sj − 3k`min, ej + 3k`min).We show that we can select I∗S satisfying the following properties. We denote by q(j)the indexes of intervals chosen, i.e., I∗S = {Iq(1), Iq(2), · · · } where q(j) < q(j + 1).Lemma 2. We can select I∗S such that for any j ≥ 1, we have (i) Ihq(j) ∩ Ihq(j+1) = ∅;(ii) Ieq(j) ∪ Ieq(j+1) forms a contiguous interval; (iii) sq(j+1)− sq(j) ≤ 7k`min and eq(j+1)−eq(j) ≤ 7k`min. (iv) For any interval I ∈ IS, there exists j ≥ 1 such that I ⊆ Ieq(j) andG(I) ≤ G(Iq(j)).The above properties then lead to the following theorem.Theorem 3. The greedy algorithm G is 7k-competitive when k = `max/`min is a constant.Arbitrary length intervals. One can show that the above competitive ratio ofthe greedy algorithm is tight up to a constant factor, meaning that for arbitrary lengthintervals, the competitive ratio can be very large. To have a better competitive ratio,we use a standard technique of classification to partition the input intervals I intoO(log `max`min ) classes where in each class the length of the intervals is bounded to bewithin a ratio of 2. This means that if we apply the greedy algorithm for individualclass, then we obtain a competitive ratio of 14 = 2 × 7. Let L = 1 + dlog `max`min e. Wedenote the classes by C1, C2, · · · , CL. We assume that the classified-greedy algorithm,denoted by C, knows about the ratio of the maximum to minimum length of intervals.2C divides the colors into bands of size L, the i-th color in each band is reserved forintervals in class Ci. Precisely, the color set Λi = {i, i+L, i+ 2L, i+ 3L, · · · } is reservedfor intervals in Ci. Intervals in each class Ci are colored independently using the greedyalgorithm G over the color set Λi.At first glance, the competitive ratio of C is O(L2). A more detailed analysis revealsthat the competitive ratio is indeed O(L).Theorem 4. For arbitrary length intervals, C is O(log `max`min )-competitive.Lower bound for arbitrary length intervals. There is a lower bound for anydeterministic online algorithm:Theorem 5. There is an adversary such that for any deterministic online algorithm A,cost(A)/cost(O) ≥ 12 log `max`min .2.2 Variant — permutation problemIn the MinSumMax problem the algorithm has to assign a color to each interval. Weconsider a variant of the problem where the input is sets of intervals each of whichcontains intervals that are pairwise non-overlapping and all such intervals in a set aregiven the same color. The problem is to find a permutation of the colors such that thecost of the coloring (as defined above) is minimized. This problem may sound easiersince we do not need to assign color but only need to find a permutation of colors. Itturns out that this variant is still NP-complete via a reduction from the directed optimallinear arrangement problem [4].References[1] M. Alicherry and R. Bhatia. Line system design and a generalized coloring problem.In ESA, pages 19–30, 2003.[2] J. Chang, S. Khuller, and K. Mukherjee. LP rounding and combinatorial algorithmsfor minimizing active and busy time. In SPAA, pages 118–127, 2014.[3] L. Epstein, T. Erlebach, and A. Levin. Variable sized online interval coloring withbandwidth. Algorithmica, 53(3):385–401, 2009.[4] M.R. Garey and D.S. Johnson. Computers and Intractability: A Guide to the Theoryof NP-Completeness. W. H. Freeman, 1979.[5] M.R. Garey, D.S. Johnson, G.L. Miller, and C.H. Papadimitriou. The complexity ofcoloring circular arcs and chords. SIAM J. Algebraic Discrete Methods, 1:216–227,1980.[6] V. Kumar and A. Rudra. Approximation algorithms for wavelength assignment. InFSTTCS, pages 152–163, 2005.[7] G.B. Mertzios, M. Shalom, A. Voloshin, P.W.H. Wong, and S. Zaks. Optimizingbusy time on parallel machines. TCS, 562:524–541, 2015.[8] M. Shalom, A. Voloshin, P.W.H. Wong, F.C.C. Yung, and S. Zaks. Online optimiza-tion of busy time on parallel machines. TCS, 560:190–206, 2014.3

Complexity and Online Algorithms for a Coloring Problem on a Line

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