An Improved Approximation for Packing Big Two-Bar Charts
aa r X i v : . [ c s . C G ] J a n An Improved Approximation for Packing BigTwo-Bar Charts ⋆ Adil Erzin − − − X ] and Vladimir Shenmaier − − − Sobolev Institute of Mathematics, SB RAS, Novosibirsk 630090, Russia a [email protected], [email protected] Abstract.
Recently, we presented a new Two-Bar Charts Packing Prob-lem (2-BCPP), in which it is necessary to pack two-bar charts (2-BCs)in a unit-height strip of minimum length. The problem is a general-ization of the Bin Packing Problem and 2-D Vector Packing Problem.Earlier, we have proposed several polynomial approximation algorithms.In particular, when each 2-BC has at least one bar of height more than1/2, we have proposed a 3/2–approximation polynomial algorithm. Thispaper proposes an O ( n )–time 16/11–approximation algorithm for pack-ing 2-BCs when at least one bar of each BC has a height not less than1/2 and an O ( n . )–time 5/4–approximation algorithm for packing non-increasing or non-decreasing 2-BCs when each 2-BC has at least one barwhich height is more than 1/2, where n is the number of 2-BCs. Keywords:
Two-Bar Charts · Strip Packing · MaxTSP · Approximation
Where the problem in question came from can be found in our papers [9,10]. Inits refined form, it is formulated as follows. We have a set of bar charts (BCs)consisting of two bars each. Any bar has a length equal to 1, and its heightdoes not exceed 1. Let us denote such charts as 2-BCs. It is required to finda feasible min-length packing of all 2-BCs in a unit-height strip. If we dividethe strip into equal unit-length cells, then the packing length is the number ofcells in which there is at least one bar. In a feasible packing, each BC’s bars donot change order and must occupy adjacent cells, but they can move verticallyindependently of each other.This problem was first examined in [10], where we described similar prob-lems that have been well studied. Similar problems that were studied reasonablywell are the bin packing problem (BPP) [2,8,16,17,22,29,30], the strip packingproblem (SPP)[1,6,13,14,24,26], and the two-dimensional vector packing problem(2-DVPP) [3,5,19,28].In the BPP, a set of items L , each item’s size, and a set of identical containers(bins) are given. All items must be placed in a minimum number of bins. BPP is ⋆ The research is carried out within the framework of the state contract of the SobolevInstitute of Mathematics (project 0314–2019–0014). A. Erzin, V. Shenmaier a strongly NP-hard problem. However, many approximate algorithms have beenproposed for it. One of the well-known algorithms is First Fit Decreasing (FFD).As part of this algorithm, objects are numbered in non-increasing order. All itemsare scanned in order, and the current item is placed in the first suitable bin. In1973 was proved that the FFD algorithm uses no more than 11 / OP T ( L ) + 4containers [16], where OP T ( L ) is the minimal number of bins to pack the itemsfrom the set L . Then in 1985, the additive constant was reduced to 3 [2], in1991 it was reduced to 1 [29], in 1997 it was reduced to 7/9 [22], and finally in2007 was found the tight boundary of the additive constant equals 6/9 [8]. AModified First Fit Decreasing (MFFD) algorithm improves FFD. It was shownthat M F F D ( L ) ≤ / OP T ( L ) + 31 / M F F D ( L ) ≤ / OP T ( L ) + 1 [30].In the Strip Packing Problem (SPP) for each rectangle i ∈ L , we know itslength and height. It is required to pack all rectangles (without rotation) ina minimum length strip. The Bottom-Left algorithm [1] arranges rectangles indescending order of height and yields a 3–approximate solution. In 1980 wasproposed algorithms with ratio 2.7 [6]. Sleator [25] proposed a 2.5–approximatealgorithm, and this ratio was reduced by Schiermeyer [24] and Steinberg [26] to2. The smallest estimate for the ratio known to date is (5 / ε ) OP T ( L ), forany ε > / ε )–approximate solution, for any ε > O ( n )–time algorithm to build a packing for n OP T + 1, where
OP T is the minimum packinglength for 2-BCPP. Then in [11], we presented the polynomial algorithms to solvethe particular cases of the 2-BCPP when all BCs are “big” (at least one bar’sheight of each BC is more than 1/2). For the case of big non-increasing or non-decreasing BCs (when either all BCs have the first bar not less than the second,or all BCs have the second bar not less than the first), an O ( n . )–time 3/2–approximate algorithm was proposed. For arbitrary big BCs, the O ( n )–time3/2–approximate algorithm has been proposed. The indicated time complexitycharacterizes the developed algorithms’ time execution based on the constructionof O ( n ) matchings. To achieve the specified accuracy, one can construct only one(first) matching. Therefore, the complexity of getting the specified accuracy is O ( n . ) and O ( n ), respectively. This paper updates the estimates for the packing length of big 2-BCs, keepingthe time complexity. First, we give a 5 / O ( n . )–time algorithmfor the version of 2-BCPP which we call 2-BCPP | n Improved Approximation for Packing Big Two-Bar Charts 3 instances of 2-BCPP with big non-increasing or non-decreasing 2-BCs. In 2-BCPP |
1, we are given arbitrary (not necessarily big and not necessarily non-increasing or non-decreasing) 2-BCs, and the goal is to find a min-length packingin which two neighboring 2-BCs intersect by at most one cell on the strip. Theproposed algorithm is based on an approximation-preserving reduction of 2-BCPP | | / O ( n )–time algorithm for 2-BCPP with big BCs. This estimate is valid for the case of “non-strictly big” BCs,in which at least one bar is of height at least 1/2. In obtaining this estimate,we use the known algorithm for finding a max-cardinality matching [12] and theproposed approximation algorithm for 2-BCPP | / |
1. In Section 4, we describe a 16 / Let a semi-infinite strip of unit height be given on a plane in the first quadrant,the lower boundary of which coincides with the abscissa. A set S , | S | = n of2-BCs is also given. Each chart, i ∈ S , consists of two unit-length bars. Theheight of the first (left) bar is a i ∈ (0 ,
1] and of the second (right) b i ∈ (0 , , , . . . . Definition 1.
Packing is a function p : S → Z + , which associates with eachBC i the cell number of the strip p ( i ) into which the first bar of BC i falls andthe sum of the bar’s heights that fall into any cell does not exceed 1. As a result of packing p , bars from 2-BC i occupy the cells p ( i ) and p ( i ) + 1. Definition 2.
The packing length L ( p ) is the number of strip cells in which atleast one bar falls. We assume that any packing p begins from the first cell, and in each cell1 , . . . , L ( p ), there is at least one bar. If this is not the case, then the wholepacking or a part of it can be moved to the left.In [10], we formulated 2-BCPP in the form of BLP, which we do not need inthis paper. Here the problem can be formulated as follows. A. Erzin, V. Shenmaier n -element set S of 2-BCs, it is required toconstruct a packing of S into a strip of minimum length, i.e., usingthe strip’s minimum number of cells. The 2-BCPP is strongly NP-hard as a generalization of the BPP [16]. More-over, the problem is (3 / − ε )–inapproximable unless P=NP [27]. In [10], weproposed an O ( n )–time algorithm, which packs the 2-BCs in the strip of lengthat most 2 OP T + 1, where
OP T is the minimum packing length. In [11], we pro-posed two packing algorithms based on sequential matchings. Using only the firstmatching, one can construct a 3/2–approximate solution with time complexity O ( n ) for the case when all BCs are big and with O ( n . ) time complexity whenadditionally the BCs are non-increasing or non-decreasing.This paper proposes two new packing algorithms based on matching andapproximation solutions to the max-weight Hamiltonian tour in the completedigraph with arcs’ weight 0 or 1. If all 2-BCs are non-strictly big, one algo-rithm constructs a 16/11–approximate solution with time complexity O ( n ). Ifall 2-BCs are big non-increasing or non-decreasing, then the other algorithmconstructs a 5/4–approximate solution with time complexity O ( n . ). Definition 3.
Two 2-BCs form a t -union if they can be placed in the strip’s − t cells. In what follows, we will need a problem of constructing a max-weight Hamil-tonian tour in a complete digraph with arc’s weights 0 and 1. Let us denote thisproblem as MaxATSP(0,1). / In this section, we describe a 5 / |
1: Given an n -element set S of 2-BCs, it is required toconstruct a min-length packing of S into a strip in which every pairof successive BCs forms a t -union, where t ≤ In this version, the left and the right bars in each BC i ∈ S may have arbitraryvalues a i , b i ∈ (0 , | | | n Improved Approximation for Packing Big Two-Bar Charts 5 | , Consider the complete weighted directed graph G ( S ) = ( S, S ) in which theweight of each arc ( i, j ) ∈ S is defined as (cid:26) b i + a j ≤ G ( S ) describe which pairs ofBCs can form a 1-union and which cannot. Note that, in the general case, theseweights are asymmetric since b i + a j may differ from b j + a i .Denote by P ( S ) the set of S packings which consist of 0- and 1-unions. Then,for each packing P ∈ P ( S ), define the Hamiltonian cycle H ( P ) in G ( S ) withthe sequence of vertices i , . . . , i n , i , where i , . . . , i n is the sequence of 2-BCsin P in order left to right. Obviously, we have w ( H ( P )) ≥ k ( P ) , (1)where w ( . ) is the total weight of a Hamiltonian cycle and k ( . ) denotes thenumber of 1-unions in a packing.Now, let H be an arbitrary Hamiltonian cycle in G ( S ). If w ( H ) < n , selecta BC i ∈ S such that the arc in H incoming to i is of zero weight; otherwise,select any chart i ∈ S . Let i , . . . , i n , i be the sequence of vertices in H startingwith i = i and define the packing P ( H ) whose sequence of charts in order leftto right is i , . . . , i n and each pair ( i t , i t +1 ), 1 ≤ t < n , forms a 1-union wheneverit is possible, i.e., when b i t + a i t +1 ≤
1. Then it is easy to see that k ( P ( H )) = min { w ( H ) , n − } . (2)Immediate corollaries of (1), (2) are the following simple statements, whichgive the desired reduction to MaxATSP(0,1). Lemma 1. If H ∗ is a max-weight Hamiltonian cycle in G ( S ) , then P ( H ∗ ) isa min-length packing in P ( S ) .Proof. Indeed, by (2), we have k ( P ( H ∗ )) = min { w ( H ∗ ) , n − } . Suppose that k ( P ) > min { w ( H ∗ ) , n − } for some P ∈ P ( S ). Then, since the number of1-unions in any packing is at most n −
1, we have k ( P ) > w ( H ∗ ). So, by (1),we obtain that w ( H ( P )) ≥ k ( P ) > w ( H ∗ ), which contradicts the choice of H ∗ .The lemma is proved. (cid:3) Lemma 2.
Suppose that H is an α –approximate solution to MaxATSP(0,1) in G ( S ) for some α ∈ (0 , . Then the number of 1-unions in the packing P ( H ) isat least α of that in an optimum packing in P ( S ) .Proof. By Lemma 1, if H ∗ is a max-weight Hamiltonian cycle in G ( S ), then P ( H ∗ ) is a min-length packing in P ( S ), so the number of 1-unions in optimumpackings is k ( P ( H ∗ )). On the other hand, by (2), we have k ( P ( H )) = min { w ( H ) , n − } ≥ α min { w ( H ∗ ) , n − } = αk ( P ( H ∗ )) . The lemma is proved. (cid:3)
A. Erzin, V. Shenmaier
By Lemma 2, if H is an α –approximate solution to MaxATSP(0,1) in G ( S )and k ∗ is the number of 1-unions in an optimum packing in P ( S ), then theapproximation ratio of the packing P ( H ) is at most2 n − αk ∗ n − k ∗ ≤ n − α ( n − n − ( n −
1) = (2 − α ) n + αn + 1 < − α. (3)Thus, we obtain an approximation-preserving reduction of 2-BCPP | , α –approximate so-lutions of the corresponding instances of MaxATSP(0 ,
1) to (2 − α )–approximatesolutions of 2-BCPP | ,
1) where the input graphs are of even size. Indeed, if n is odd, wewill consider the set S ′ = S ∪ { τ } , where τ is the dummy chart with bars a τ = 1and b τ = 1. Lemma 3.
Suppose that H is an α –approximate solution to MaxATSP(0,1)in G ( S ′ ) for some α ∈ (0 , and P is the packing obtained from P ( H ) byremoving τ . Then the number of -unions in the packing P is at least α of thatin an optimum packing in P ( S ) .Proof. Let H ∗ be an optimum solution of MaxATSP(0,1) on G ( S ′ ). Then, by(2), we have k ( P ( H )) = min { w ( H ) , n } ≥ min { αw ( H ∗ ) , n } . But the BC τ cannot be in any 1-union, so k ( P ) = k ( P ( H )). At the same time, if P ∗ is anoptimum packing in P ( S ) and the sequence of charts in P ∗ in order left to rightis i , . . . , i n , then the weight of the Hamiltonian cycle H τ ( P ∗ ) = ( i , . . . , i n , τ, i )in G ( S ′ ) is exactly k ( P ∗ ). So we have k ( P ) = k ( P ( H )) ≥ min { αw ( H ∗ ) , n } ≥ min { αw ( H τ ( P ∗ )) , n } = αk ( P ∗ ) . The lemma is proved. (cid:3)
Thus, if H is an α –approximate solution to MaxATSP(0,1) in G ( S ′ ) and, asbefore, k ∗ is the number of 1-unions in an optimum packing in P ( S ), then weobtain a solution of 2-BCPP | S with approximation ratio boundedby the same expressions as in (3). | It remains to recall the known algorithmic results for MaxATSP(0 , ,
1) are the 3 / / O ( n . ) if n is evenand O ( n . ) if n is odd. We suggest using the above reduction to MaxTSPwith an even number of vertices and applying Paluch’s algorithm. The resultingalgorithm for 2-BCPP | n Improved Approximation for Packing Big Two-Bar Charts 7 Algorithm A . Input: a set S of n Output: a packing P ∈ P ( S ). Step . If n is even, construct the graph G = G ( S ); otherwise, construct thegraph G = G ( S ′ ), where S ′ = S ∪ { τ } , a τ = b τ = 1. Step . By using the algorithm from [23], find a 3 / H toMaxATSP(0,1) in G . Step . If n is even, return P = P ( H ); otherwise, return the packing P obtainedfrom P ( H ) by removing τ .By Lemmas 2, 3 and estimate (3), the approximation ratio of the packingreturned by Algorithm A is less than 2 − / /
4. So we prove
Theorem 1.
Algorithm A finds a / –approximate solution to -BCPP | intime O ( n . ) . / This section describes a 16 / i withbars a i , b i ∈ (0 ,
1] is non-strictly big if max { a i , b i } ≥ /
2, i.e., the case whenmax { a i , b i } = 1 / | big: Given an n -element set S of non-strictly big 2-BCs,it is required to construct a min-length packing of S . Let us make some simple observations. First, it is easy to see that, since allthe charts in S are non-strictly big, then any two pairs of charts { i , i } and { i , i } which form 2-unions in any packing of S are disjoint. On the other hand,any set of disjoint pairs of charts forming 2-unions gives a feasible solution of 2-BCPP | big. In particular, we can get such a solution by finding a max-cardinalitymatching M ∗ in the graph G ( S ) whose vertices are the charts of S and the edgesare the unordered pairs i, j ∈ S admitting a 2-union, i.e., for which a i + a j ≤ b i + b j ≤
1. Let P ( M ∗ ) be the packing of S , which consists of 0- and 2-unionsand all 2-unions formed by endpoints of the edges in M ∗ .The second observation is that any feasible solution of 2-BCPP | S isalso that of 2-BCPP | big. On the other hand, any feasible solution to the latterproblem can be easily transformed to 2-BCPP | i and j form a 2-union and a i + b j ≥ a j + b i , then we shift the BC j and all thecharts lying to the right of it one cell right. As a result, we get one cell withcontent height b i + a j ≤ ( a i + b j + a j + b i ) / ≤
1, while the content of the othercells does not increase. So we get a feasible packing where the pair ( i, j ) of BCsforms a 1-union. Denote the packing constructed by the described processing ofall 2-unions in P as Γ ( P ). Then Γ ( P ) ∈ P ( S ) and the number of 1-unions in Γ ( P ) is exactly the total number of 1- and 2-unions in P . A. Erzin, V. Shenmaier
If an optimum solution of 2-BCPP | big contains a small number of 1-unionsand a significant number of 2-unions, this packing is not much shorter than P ( M ∗ ). If, on the contrary, an optimum solution of 2-BCPP | big contains a sig-nificant number of 1-unions and a small number of 2-unions, then it is not muchshorter than an optimum solution of 2-BCPP |
1. So the best of P ( M ∗ ) and anoptimum of 2-BCPP | | big. Basedon this hypothesis, we suggest the following algorithm: Algorithm A . Input: a set S of n big 2-BCs. Output: a packing P of S . Step . By using the algorithm from [12], construct a max-cardinality matching M ∗ in G ( S ). Step . By using Algorithm A , find an approximate solution P to 2-BCPP | S . Step . If the length of P ( M ∗ ) is less than that of P , return P = P ( M ∗ );otherwise, return P = P . Theorem 2.
Algorithm A finds a / –approximate solution to -BCPP | bigin time O ( n ) .Proof. Let P ∗ be a min-length packing of S . Denote by k and k the numbersof 1- and 2-unions in P ∗ , respectively. Then, since the 2-unions in P ∗ form amatching in G ( S ), the cardinality of M ∗ is at least k . Therefore, the length of P ( M ∗ ) is at most 2 n − k .Next, we estimate the length of P . By the construction of the packing Γ ( P ∗ ),we have k ( Γ ( P ∗ )) = k + k . By the description of Algorithm A and Lemmas2 and 3, the number of 1-unions in P is at least 3 / P ( S ). In particular, we have k ( P ) ≥ (3 / k ( Γ ( P ∗ )) = (3 / k + k ) . So the length of the packing P is at most 2 n − (3 / k + k ).At the same time, it is easy to see that the length of P ∗ is 2 n − k − k . More-over, as shown in [11], this length is at least n . It follows that the approximationratio of the solution returned by Algorithm A is at mostmin { n − k , n − (3 / k + k ) } n − k − k , (4)where 2 n − k − k ≥ n or, equivalently, where k ≤ n − k . Obviously, themaximum value of expression (4) is attained at the maximum possible value of k , i.e., when k = n − k . Therefore, this expression is bounded bymin { n − k , n − (3 / n − k ) } n . (5) n Improved Approximation for Packing Big Two-Bar Charts 9 Next, the values 2 n − k and 2 n − (3 / n − k ) are decreasing and increasingfunctions of k respectively, while the denominator in the expression (5) does notdepend on k . It follows that the maximum of this expression is attained when2 n − k = 2 n − (3 / n − k ), i.e., when k = (3 / n . So the approximationratio of Algorithm A is at most2 n − / nn = 16 / . It remains to note that finding a max-cardinality matching at Step 1 by thealgorithm from [12] takes time O ( n ), while Step 2 is performed in time O ( n . )by Theorem 1. Thus, the running time of Algorithm A is O ( n ). The theoremis proved. (cid:3) Remark 1.
The packing returned by Algorithm A contains either 1- or 2-unions.So this packing is also an approximate solution of the version of 2-BCPP where,given arbitrary 2-BCs (not necessarily big), it is required to find a min-lengthpacking, in which every cell of the strip contains at most two bars of differentcharts. Note that, using a slightly modified algorithm based on the same ideas,easy to get a 19 / Remark 2.
It can be easily proved that, if we are given an oracle which returnsan optimum solution to 2-BCPP |
1, e.g., if we can solve MaxATSP(0 ,
1) exactly,then the best of P ( M ∗ ) and an optimum of 2-BCPP | / | big. We considered a problem in which it is necessary to pack n two-bar charts(2-BCs) in a unit-height strip of minimum length. The problem is a generaliza-tion of the bin packing problem and 2-D vector packing problem. Earlier, weproposed an O ( n )–time algorithm, which builds a packing of length at most2 OP T + 1 for arbitrary 2-BCs, where
OP T is the minimum length of the pack-ing. Then, we proposed an O ( n . )– and O ( n )–time packing algorithms basedon the sequential matchings. Using only the first matching, one can construct a3/2–approximate solution with time complexity O ( n ) for big BCs (when eachBC has at least one bar of height greater than 1/2) and with O ( n . ) timecomplexity when additionally the BCs are non-increasing or non-decreasing.This paper proposes two new packing algorithms based on matching andconstructing an approximate solution to the MaxATSP(0,1). We prove that forpacking arbitrary non-strictly big 2-BCs (the height of at least one bar of eachBC is not less than 1/2), one algorithm constructs a 16/11–approximate solutionwith time complexity O ( n ). If all 2-BCs are big (at least one bar of each BChas a height greater than 1/2) non-increasing or non-decreasing, then anotheralgorithm constructs a 5/4–approximate solution with time complexity O ( n . ).We plan to conduct a numerical experiment to compare the solutions con-structed by various approximation algorithms with the optimal solution yielded by the software package for BLP (for example, CPLEX) [10]. We are also plan-ning to obtain a new accuracy estimate for the 2-BCPP problem with arbitrary2-BCs. References
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