# Single machine parallel batch scheduling with unbounded capacity

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Workshop on Combinatorics and Graph Theory 21th, April, 2006 Nankai University Single machine parallel batch scheduling with unbounded capacity Yuan Jinjiang Department of mathematics, Zhengzhou University Zhengzhou, Henan Page 1 of 30

2 1 Introduction and Problem Formulation Let n jobs J 1, J 2,..., J n and a single machine that can handle batch jobs at the same time be given, where a batch is a subset of jobs and distinct batches cannot have a job in common. We assume in this report that each batch can contain arbitrary many jobs, i.e., the capacity of batches is unbounded. There is also another model called bounded batch scheduling in which each batch contains at most c jobs. But in this report we only consider the unbounded version. Each job J j has a processing time p j > 0. The processing time of a batch B is defined by p B = max{p j : J j B}. Page 2 of 30 That is, the jobs in a common batch are processed in a parallel form simultaneously.

3 Hence, the studied scheduling model is called parallel batch scheduling. (If the processing time of a batch B is defined by p B = J j B p j, then the corresponding model is called serial batch scheduling.) Each job J j has a release date r j. This means that job J j cannot start being processing before the time moment r j. The release date of a batch B is defined by r B = max{r j : J j B}. Clearly, batch B cannot be processed before r B. There are precedence relations between the jobs, i.e., J i J j means that J i must be completed before J j starts; furthermore, J i J j and J j J k imply J i J k. Page 3 of 30

4 Especially, we say the precedence relations between the jobs are chains if the set of all jobs can be partitioned into m job subsets {J (i,j) : 1 j n i }, 1 i m, such that the precedence relations are given by m chains (total orders): J (i,1) J (i,2)... J (i,ni ), 1 i m. If J i and J j are two jobs such that J i J j, we require that J j is processed at or after the completion time of J i, and so J i and J j cannot be processed in the same batch. In fact, if J i B x and J j B y, then batch B x must be processed before batch B y. Following Brucker (2001) and Lenstra et al. scheduling problem is denoted by 1 r j ; prec; p-batch f, (1977), the parallel batch Page 4 of 30 where f is the objective function to be minimized, and p-batch means parallel batch.

5 A feasible schedule is given by a batch sequence BS = (B 1, B 2,..., B N ), such that, for any two jobs J i and J j with J i J j, if J i B x and J j B y, then x < y. Whence a feasible batch sequence BS = (B 1, B 2,..., B N ) is given, we assume that each batch starts at the earliest possible starting time such that every two distinct batches do not overlap each other. Hence, the starting time S Bj and completion time C BI can be defined recursively in the following way: S B1 = r B1 and C B1 = S B1 + p B1, and for i 2, Page 5 of 30 S Bi = max{r Bi, C Bi 1 } and C Bi = S Bi + p Bi.

6 If r j = 0 for all jobs, then the completion time of any batch B x is naturally C Bx = x i=1 Suppose that each job J j has a due date d j. max p j. J j B i L j (π) = C j (π) d j : the lateness of job J j under schedule π. T j (π) = max{0, L j (π)}: the tardiness of job J j under schedule π. U j (π) = 1 if T j (π) > 0 and U j (π) = 0 if T j (π) 0. Page 6 of 30

7 The common objective functions has the following form: C max : C max (π) = max{c j (π) : 1 j n}, the makespan; L max : Cj : Tj : Uj : wj C j : wj T j : wj U j : L max (π) = max{l j (π) : 1 j n}, the maximum lateness; 1 i n C j(π), the total completion time; 1 i n T j(π), the total tardiness; 1 i n U j(π), the number of tardy jobs; 1 i n w jc j (π), the total completion time; 1 i n w jt j (π), the total tardiness; 1 i n w ju j (π), the weighted number of tardy jobs. All of the above objective functions are regular, where a scheduling objective function f(c 1, C 2,..., C n ) is called regular if it is nondecreasing for each job s completion time C j. Occasionally, some results or algorithms may be applied for all regular functions. Page 7 of 30

8 2 The existing complexity results and open problems Polynomially solved problems: 1 p-batch C max, O(n), Brucker et al. (JOS, 1998) 1 r j ; p-batch C max, O(n 2 ), Lee and Uzsoy (IJPR, 1999) 1 p-batch f max, polynomial, Brucker et al. (JOS, 1998) 1 p-batch L max, O(n 2 ), Brucker et al. (JOS, 1998) 1 p-batch C j, O(n log n), Brucker et al. (JOS, 1998) 1 p-batch w j C j, O(n log n), Brucker et al. (JOS, 1998) 1 p-batch U j, O(n 3 ), Brucker et al. (JOS, 1998) 1 r j ; p j = p; p-batch f, polynomial, Baptiste (MMOR, 2000) 1 prec; p j = p; p-batch f, O(n 2 ), Brucker and Knust (2000) 1 r j ; prec; p j = p; p-batch C max, O(n 2 ), Cheng, Yuan and Yang (CAOR, 2005), posed as open by Brucker and Knust (2000) Page 8 of 30

9 NP-hard problems: 1 p-batch w j T j, ordinary NP-hard, O(n 2 P ), Brucker et al. (JOS, 1998) 1 p-batch T j, ordinary NP-hard, Liu, Yuan and Cheng (ORL, 2003), Posed as open by Brucker et al. (JOS, 1998) 1 p-batch w j U j, ordinary NP-hard, O(n 2 P ), Brucker et al. (JOS, 1998) 1 r j ; p-batch L max, ordinary NP-hard, O(n 2 P ), Cheng, Liu and Yu (IIE Trans. 2001) 1 r j ; p-batch w j C j, ordinary NP-hard, O(n 2 P ), Deng and Zhang (LNCS. 1999) 1 r j ; p-batch f, ordinary NP-hard, O(n 4 P 3 ), 1 chains; p-batch C max, Liu, Yuan and Cheng (ORL, 2003) strongly NP-hard, 1 chains; p-batch C j, strongly NP-hard, Cheng, Ng, Yuan and Liu (NRL, 2004) Cheng, Ng, Yuan and Liu (NRL, 2004) Page 9 of 30

10 The last two results implies that 1 chains; p-batch f is strongly NP-hard for every f {C max, L max, C j, T j, U j, w j C j, w j T j, w j U j }. The exact complexity of these problems are posed as open (Brucker and Knust, 2000) in the from 1 prec; p-batch f. Page 10 of 30

11 Open problems: (1) 1 r j, p-batch C j. See Liu, Yuan and Cheng (ORL, 2003). (2) 1 prec; r j ; p j = p; p-batch f for every f {L max, C j, T j, U j, w j C j, w j T j, w j U j }. (3) 1 chains; r j ; p j = p; p-batch f for every f {L max, C j, T j, U j, w j C j, w j T j, w j U j }. Page 11 of 30

12 3 We will show that 1 chains; p-batch C max is strongly NP-hard. The reduction will use the NP-complete vertex cover problem of graphs. For a graph G, V = V (G) and E = E(G) denote its sets of vertices and edges, respectively. An edge e with end vertices u and v will be denoted by e = uv = vu. For e = uv E, we say that e is incident to u and u is incident to e. A vertex subset S V (G) is said to be a vertex cover of G, if for every edge e E(G) there is u S such that e is incident to u. Page 12 of 30 A path P of G is a sequence of vertices (v 1, v 2,..., v k ) such that v i v j for i j and v i v i+1 E(G) for 1 i k 1. A path P of G can be regarded as a subgraph of G.

13 A Hamiltonian path of a graph G is a path P of G such that V (P ) = V (G). For two graphs G and H, the joint of G and H, denoted by G + H, is obtained from G and H by joining each vertex in G with each vertex in H with edges. Vertex cover problem: For a given graph G and a positive integer k with k V (G) 1, is there a vertex cover S of G such that S k? By Garey (1976 and 1979), it is known that the vertex cover problem is NPcomplete in the strong sense. For the vertex cover problem, if a Hamiltonian path of the considered graph is given as input, the special vertex cover problem is called restricted vertex cover problem in this paper. Page 13 of 30

14 Lemma 1 The restricted vertex cover problem of graphs is NP-complete in the strong sense. Proof The restricted vertex cover problem of graphs is a subproblem of the vertex cover problem, and so in the class NP. To prove the NP-completeness, we establish a polynomial reduction from the vertex cover problem of graphs to the restricted one. Let an instance of the vertex cover problem be given, which inputs a graph G with V (G) = {v 1, v 2,..., v n } and an integer k with 1 k n 1 and asks whether or not there is a vertex cover S of G such that S k. We construct an instance of the restricted vertex cover problem (i.e., a graph H together with a Hamiltonian path P and an integer k with 1 k V (H) 1) as follows. Set H = G+K n, where K n is the complete graph with vertex set V (K n ) = {u 1, u 2,..., u n } such that V (K n ) V (G) =. Set P = (u 1, v 1, u 2, v 2,..., u n, v n ) and k = n + k. Page 14 of 30

15 The above construction takes a polynomial time. Moreover, P is clearly a Hamiltonian path of H. It is routine to check that S is a vertex cover of G with S k if and only if S = S {u 1, u 2,..., u n } is a vertex cover of H with S k = n + k. This means that the vertex cover problem can be polynomially reduced to the restricted vertex cover problem. By the fact that the vertex cover problem is strongly NP-complete, we conclude that the restricted vertex cover problem is strongly NP-complete. Page 15 of 30

16 Theorem 2 The scheduling problem 1 chains; p-batch C max is strongly NP-hard. Proof The decision version of the considered scheduling problem asks, for a given instance of the problem and a positive integer Y, whether there is a feasible batch sequence BS such that C max (BS) Y. It can be easily seen that the decision problem is in NP. To prove the strong NP-completeness, we use the strongly NP-complete restricted vertex cover problem for the reduction. Let an instance of the restricted vertex cover problem be given, which inputs a graph G with V (G) = {v 1, v 2,..., v n }, a Hamiltonian path P = (v 1, v 2,..., v n ) of G and an integer k with 1 k n 1 and asks whether or not there is a vertex cover S of G such that S k. Write m = E(G). We construct an instance of the decision version of the scheduling problem 1 chains; p-batch C max as follows. Page 16 of 30

17 We have m(n 1)+n jobs. Each vertex v i V (G) corresponds to a vertex job J i, 1 i n. Each edge v i v j E(G) with i < j corresponds to n 1 edge jobs J (1;i,j), J (2;i,j), J (3;i,j),..., J (n 1;i,j). The processing time of each job is defined in the following way. For 1 i n, the processing time p i of the vertex job J i is n. For 1 x n 1 and v i v j E(G) with i < j, the processing time p (x;i,j) of the edge job J (x;i,j) is n + 1, if either i = x or j = x + 1; p (x;i,j) = n, otherwise. (We should note that, for every edge v i v j E(G) with i < j, among the jobs J (x;i,j), 1 x n 1, J (i;i,j) and J (j 1;i,j) are the only jobs with processing time n + 1.) Page 17 of 30

18 The immediate precedence relations between the jobs are defined by the following m + 1 chains: J 1 J 2... J n ; J (1;i,j) J (2;i,j)... J (n 1;i,j), for v i v j E(G) with i < j. The threshold value Y is defined as Y = n 2 + k. We ask whether there is a feasible batch sequence BS such that the makespan under BS is at most Y. The above reduction takes a polynomial time. In the following, we will prove that the instance of the restricted vertex cover problem has a vertex cover S V (G) such that S k if and only if the instance of the problem 1 chains; p-batch C max has a feasible batch sequence with makespan at most Y. Page 18 of 30

19 If the instance of the restricted vertex cover problem has a vertex cover S V (G) such that S k, then for each edge uv E(G) at least one of u and v is in S. For the reason that P = (v 1, v 2,..., v n ) is a Hamiltonian path of G, for each i with 1 i n 1, either v i S or v i+1 S. We define the batch sequence BS = (B 1, B 2,..., B n ) in the following way. For 1 i n, we set J i B i. For v i v j E(G) with i < j, if {v i, v j } S, we set J (x;i,j) B x, when 1 x j 2, J (x;i,j) B x+1, when j 1 x n 1; if v i S and v j S, we set J (x;i,j) B x, for 1 x n 1; Page 19 of 30 if v i S and v j S, we set J (x;i,j) B x+1, for 1 x n 1. It is not hard to see that BS = (B 1, B 2,..., B n ) is a feasible schedule, and that the processing time of each batch is either n or n + 1.

20 Now for a job J, let σ(j) be the vertex v x such that J B x. Claim 1 For every job J with processing time n + 1, σ(j) S. Let J be a job with processing time n + 1. Then there must be an edge v i v j E(G) with i < j, such that either J = J (i;i,j) or J = J (j 1;i,j). By the definition of vertex cover, at least one of v i and v j is in S. We distinguish the following three cases. Case 1 J = J (i;i,i+1). In this case, by the definition of the batch sequence BS, if v i+1 S, then σ(j) = v i+1 S; if v i S and v i+1 S, then σ(j) = v i S. Case 2 J = J (i;i,j) and i j 2. In this case, by the definition of the batch sequence BS, if v i S, then σ(j) = v i S; if v i S, then by the fact v i v i+1 E(G), σ(j) = v i+1 S. Case 3 J = J (j 1;i,j) and i j 2. In this case, again by the definition of the batch sequence BS, if v j S, then σ(j) = v j S; if v j S, then by the fact v j 1 v j E(G), σ(j) = v j 1 S. This completes the proof of Claim 1. Page 20 of 30

21 Now by the result of Claim 1, for each v x S, the processing time of B x is n under the batch sequence BS; for each v y S, the processing time of B y is at most n + 1 under the batch sequence BS. Hence, the makespan under BS is at most n(n S ) + (n + 1) S = n 2 + S n 2 + k = Y. On the other hand, suppose that BS = (A 1, A 2,..., A N ) is a feasible batch sequence such that the makespan under BS is at most Y = n 2 + k. By the fact that the processing time of each job is either n or n + 1, we know that the processing time of each batch A i (1 i N) is either n or n + 1. This means that N n. Because J 1 J 2... J n, there are at least n different batches under any feasible batch sequence. Hence, we must have N = n. Set S = {v x : the processing time of the batch A x is n + 1}. By the fact that the makespan under BS is at most Y = n 2 + k, S k. Hence, we only need to show that S is a vertex cover of G in the following. Page 21 of 30

22 Let v i v j with i < j be an edge of G. Then J (i;i,j) and J (j 1;i,j) are the only jobs with processing time n + 1 among the jobs J (x;i,j), 1 i n 1. By the fact that the chain J (1;i,j) J (2;i,j) J (3;i,j)... J (n 1;i,j) contains n 1 jobs, one of the following cases must occur: either J (i;i,j) A i and J (j 1;i,j) A j 1, or J (i;i,j) A i and J (j 1;i,j) A j, or J (i;i,j) A i+1 and J (j 1;i,j) A j. In the first case, v i S ; in the second case, {v i, v j } S ; and in the third case, v j A j. Hence, at least one of v i and v j is in S. This means that S is a vertex cover of G with S k. The proof is completed. Page 22 of 30

23 Theorem 3 The scheduling problem 1 chains; p-batch C j is strongly NP-hard. Proof The decision version of the considered scheduling problem asks, for a given instance of the problem and a positive integer Y, whether there is a feasible batch sequence BS such that C j (BS) Y. It can easily be seen that the decision problem is in NP. To prove the strong NP-completeness, we again use the strongly NP-complete restricted vertex cover problem for the reduction. Let an instance of the restricted vertex cover problem be given, which inputs a graph G with V (G) = {v 1, v 2,..., v n }, a Hamiltonian path P = (v 1, v 2,..., v n ) of G and an integer k with 1 k n 1 and asks whether or not there is a vertex cover S of G such that S k. Write m = E(G). We construct an instance of the decision version of the scheduling problem 1 chains; p-batch C j as follows. Page 23 of 30

24 We have m(n 1) + n + M jobs, where M = (mn m + n)(n 2 + k). Each vertex v i V (G) corresponds to a vertex job J i, 1 i n. Each edge v i v j E(G) with i < j corresponds to n 1 edge jobs J (1;i,j), J (2;i,j), J (3;i,j),..., J (n 1;i,j). In addition, we have M large jobs J n+1, J n+2,..., J n+m. The processing time of each job is defined in the following way. For 1 i n, the processing time p i of the vertex job J i is n. For 1 i M, the processing time p n+i of the large job J n+i is nm. For 1 x n 1 and v i v j E(G) with i < j, the processing time p (x;i,j) of the edge job J (x;i,j) is n + 1, if either i = x or j = x + 1; p (x;i,j) = n, otherwise. Page 24 of 30

25 The immediate precedence relations between jobs are defined by the following m + 1 chains: J 1 J 2... J n J n+1 J n+2... J n+m ; J (1;i,j) J (2;i,j)... J (n 1;i,j), for v i v j E(G) with i < j. The threshold value Y is defined as Y = (n 2 + k + 1)M nm 2 (M + 1). We ask whether or not there is a feasible batch sequence BS such that Cj (BS) Y. The above reduction takes a polynomial time. In the following, we will prove that the instance of the restricted vertex cover problem has a vertex cover S V (G) such that S k if and only if the instance of the problem 1 chains; p-batch C j has a feasible batch sequence BS such that Cj (BS) Y. Page 25 of 30

26 If the instance of the restricted vertex cover problem has a vertex cover S V (G) such that S k, we define the batch sequence BS = (B 1, B 2,..., B n+m ) in the following way. For 1 i n + M, we set J i B i. For v i v j E(G) with i < j, if {v i, v j } S, we set J (x;i,j) B x, when 1 x j 2, J (x;i,j) B x+1, when j 1 x n 1; if v i S and v j S, we set J (x;i,j) B x, for 1 x n 1; Page 26 of 30 if v i S and v j S, we set J (x;i,j) B x+1, for 1 x n 1.

27 It is not hard to see that BS = (B 1, B 2,..., B n+m ) is a feasible schedule. By the discussion of the if part of Theorem 2, the completion time of the batch B i, 1 i n, is at most n 2 + k. Then the completion time of the batch B n+i = {J n+i }, 1 i M, is at most (n 2 + k) + inm. For the reason that the first n batches contain mn m + n jobs, the total completion time of jobs can be roughly estimated as M Cj (BS) (mn m + n)(n 2 + k) + (n 2 + k + inm) = Y. This implies that the instance of the scheduling problem has a required batch sequence. i=1 Page 27 of 30

28 On the other hand, suppose that BS = (A 1, A 2,..., A N ) is a feasible batch sequence such that the total completion time of jobs under BS is at most Y. By the fact that J 1 J 2... J n J n+1 J n+2... J n+m, the n vertex jobs must be processed before the large jobs with each vertex job being processed in one batch. Because each vertex job has processing time n, the starting time of any batch that contains a large job must be at least n 2. If there is an edge job J that is processed either in the same batch as a large job or after a large job, then by the fact that the large job has processing time nm, the completion time of job J is at least n 2 + nm. Because the completion time of the large job J n+i is at least n 2 + inm, the total completion time of the jobs under BS is greater than Page 28 of 30 n 2 + nm + M (n 2 + inm) = Y + (n k 1)M + n 2 > Y. i=i

29 This contradicts our assumption. Hence, each edge job must be processed before every large job. Denote by the maximum completion time of vertex jobs and edges jobs. If n 2 + k + 1, then the completion time of the large job J n+i is at least n 2 + k inm. It follows that the total completion time of the jobs under BS is greater than M (n 2 + k inm) = Y. i=1 This contradicts our assumption again. Hence, we must have n 2 + k. The above discussion means that there is a feasible batch sequence BS for the vertex jobs and edge jobs such that the makespan is at most n 2 + k. By the discussion of the only if part of Theorem 2, there is a vertex cover S of G such that S k. This completes the proof. Page 29 of 30

30 Thank You! Page 30 of 30

### An improved on-line algorithm for scheduling on two unrestrictive parallel batch processing machines

This is the Pre-Published Version. An improved on-line algorithm for scheduling on two unrestrictive parallel batch processing machines Q.Q. Nong, T.C.E. Cheng, C.T. Ng Department of Mathematics, Ocean

### Classification - Examples

Lecture 2 Scheduling 1 Classification - Examples 1 r j C max given: n jobs with processing times p 1,...,p n and release dates r 1,...,r n jobs have to be scheduled without preemption on one machine taking

### 2.3 Scheduling jobs on identical parallel machines

2.3 Scheduling jobs on identical parallel machines There are jobs to be processed, and there are identical machines (running in parallel) to which each job may be assigned Each job = 1,,, must be processed

### JUST-IN-TIME SCHEDULING WITH PERIODIC TIME SLOTS. Received December May 12, 2003; revised February 5, 2004

Scientiae Mathematicae Japonicae Online, Vol. 10, (2004), 431 437 431 JUST-IN-TIME SCHEDULING WITH PERIODIC TIME SLOTS Ondřej Čepeka and Shao Chin Sung b Received December May 12, 2003; revised February

### Classification - Examples -1- 1 r j C max given: n jobs with processing times p 1,..., p n and release dates

Lecture 2 Scheduling 1 Classification - Examples -1-1 r j C max given: n jobs with processing times p 1,..., p n and release dates r 1,..., r n jobs have to be scheduled without preemption on one machine

### Scheduling Single Machine Scheduling. Tim Nieberg

Scheduling Single Machine Scheduling Tim Nieberg Single machine models Observation: for non-preemptive problems and regular objectives, a sequence in which the jobs are processed is sufficient to describe

### Scheduling Shop Scheduling. Tim Nieberg

Scheduling Shop Scheduling Tim Nieberg Shop models: General Introduction Remark: Consider non preemptive problems with regular objectives Notation Shop Problems: m machines, n jobs 1,..., n operations

### Minimum Makespan Scheduling

Minimum Makespan Scheduling Minimum makespan scheduling: Definition and variants Factor 2 algorithm for identical machines PTAS for identical machines Factor 2 algorithm for unrelated machines Martin Zachariasen,

### Duplicating and its Applications in Batch Scheduling

Duplicating and its Applications in Batch Scheduling Yuzhong Zhang 1 Chunsong Bai 1 Shouyang Wang 2 1 College of Operations Research and Management Sciences Qufu Normal University, Shandong 276826, China

### Approximation Algorithms. Scheduling. Approximation algorithms. Scheduling jobs on a single machine

Approximation algorithms Approximation Algorithms Fast. Cheap. Reliable. Choose two. NP-hard problems: choose 2 of optimal polynomial time all instances Approximation algorithms. Trade-off between time

### Complexity Theory. IE 661: Scheduling Theory Fall 2003 Satyaki Ghosh Dastidar

Complexity Theory IE 661: Scheduling Theory Fall 2003 Satyaki Ghosh Dastidar Outline Goals Computation of Problems Concepts and Definitions Complexity Classes and Problems Polynomial Time Reductions Examples

### Answers to some of the exercises.

Answers to some of the exercises. Chapter 2. Ex.2.1 (a) There are several ways to do this. Here is one possibility. The idea is to apply the k-center algorithm first to D and then for each center in D

### Batch Scheduling of Deteriorating Products

Decision Making in Manufacturing and Services Vol. 1 2007 No. 1 2 pp. 25 34 Batch Scheduling of Deteriorating Products Maksim S. Barketau, T.C. Edwin Cheng, Mikhail Y. Kovalyov, C.T. Daniel Ng Abstract.

### Algorithm Design and Analysis

Algorithm Design and Analysis LECTURE 27 Approximation Algorithms Load Balancing Weighted Vertex Cover Reminder: Fill out SRTEs online Don t forget to click submit Sofya Raskhodnikova 12/6/2011 S. Raskhodnikova;

### Lecture 6: Approximation via LP Rounding

Lecture 6: Approximation via LP Rounding Let G = (V, E) be an (undirected) graph. A subset C V is called a vertex cover for G if for every edge (v i, v j ) E we have v i C or v j C (or both). In other

### Fairness in Routing and Load Balancing

Fairness in Routing and Load Balancing Jon Kleinberg Yuval Rabani Éva Tardos Abstract We consider the issue of network routing subject to explicit fairness conditions. The optimization of fairness criteria

### Handout #Ch7 San Skulrattanakulchai Gustavus Adolphus College Dec 6, 2010. Chapter 7: Digraphs

MCS-236: Graph Theory Handout #Ch7 San Skulrattanakulchai Gustavus Adolphus College Dec 6, 2010 Chapter 7: Digraphs Strong Digraphs Definitions. A digraph is an ordered pair (V, E), where V is the set

### 5 INTEGER LINEAR PROGRAMMING (ILP) E. Amaldi Fondamenti di R.O. Politecnico di Milano 1

5 INTEGER LINEAR PROGRAMMING (ILP) E. Amaldi Fondamenti di R.O. Politecnico di Milano 1 General Integer Linear Program: (ILP) min c T x Ax b x 0 integer Assumption: A, b integer The integrality condition

### Partitioning edge-coloured complete graphs into monochromatic cycles and paths

arxiv:1205.5492v1 [math.co] 24 May 2012 Partitioning edge-coloured complete graphs into monochromatic cycles and paths Alexey Pokrovskiy Departement of Mathematics, London School of Economics and Political

### Divisor graphs have arbitrary order and size

Divisor graphs have arbitrary order and size arxiv:math/0606483v1 [math.co] 20 Jun 2006 Le Anh Vinh School of Mathematics University of New South Wales Sydney 2052 Australia Abstract A divisor graph G

### Minimizing the Number of Machines in a Unit-Time Scheduling Problem

Minimizing the Number of Machines in a Unit-Time Scheduling Problem Svetlana A. Kravchenko 1 United Institute of Informatics Problems, Surganova St. 6, 220012 Minsk, Belarus kravch@newman.bas-net.by Frank

### Solutions to Homework 6

Solutions to Homework 6 Debasish Das EECS Department, Northwestern University ddas@northwestern.edu 1 Problem 5.24 We want to find light spanning trees with certain special properties. Given is one example

### Introduction to Scheduling Theory

Introduction to Scheduling Theory Arnaud Legrand Laboratoire Informatique et Distribution IMAG CNRS, France arnaud.legrand@imag.fr November 8, 2004 1/ 26 Outline 1 Task graphs from outer space 2 Scheduling

### Approximability of Two-Machine No-Wait Flowshop Scheduling with Availability Constraints

Approximability of Two-Machine No-Wait Flowshop Scheduling with Availability Constraints T.C. Edwin Cheng 1, and Zhaohui Liu 1,2 1 Department of Management, The Hong Kong Polytechnic University Kowloon,

### Approximation Algorithms: LP Relaxation, Rounding, and Randomized Rounding Techniques. My T. Thai

Approximation Algorithms: LP Relaxation, Rounding, and Randomized Rounding Techniques My T. Thai 1 Overview An overview of LP relaxation and rounding method is as follows: 1. Formulate an optimization

### 8.1 Min Degree Spanning Tree

CS880: Approximations Algorithms Scribe: Siddharth Barman Lecturer: Shuchi Chawla Topic: Min Degree Spanning Tree Date: 02/15/07 In this lecture we give a local search based algorithm for the Min Degree

### 10.1 Integer Programming and LP relaxation

CS787: Advanced Algorithms Lecture 10: LP Relaxation and Rounding In this lecture we will design approximation algorithms using linear programming. The key insight behind this approach is that the closely

### Computer Algorithms. NP-Complete Problems. CISC 4080 Yanjun Li

Computer Algorithms NP-Complete Problems NP-completeness The quest for efficient algorithms is about finding clever ways to bypass the process of exhaustive search, using clues from the input in order

### Good luck, veel succes!

Final exam Advanced Linear Programming, May 7, 13.00-16.00 Switch off your mobile phone, PDA and any other mobile device and put it far away. No books or other reading materials are allowed. This exam

### Determination of the normalization level of database schemas through equivalence classes of attributes

Computer Science Journal of Moldova, vol.17, no.2(50), 2009 Determination of the normalization level of database schemas through equivalence classes of attributes Cotelea Vitalie Abstract In this paper,

### Why? A central concept in Computer Science. Algorithms are ubiquitous.

Analysis of Algorithms: A Brief Introduction Why? A central concept in Computer Science. Algorithms are ubiquitous. Using the Internet (sending email, transferring files, use of search engines, online

### Approximation Algorithms

Approximation Algorithms or: How I Learned to Stop Worrying and Deal with NP-Completeness Ong Jit Sheng, Jonathan (A0073924B) March, 2012 Overview Key Results (I) General techniques: Greedy algorithms

### NP-complete? NP-hard? Some Foundations of Complexity. Prof. Sven Hartmann Clausthal University of Technology Department of Informatics

NP-complete? NP-hard? Some Foundations of Complexity Prof. Sven Hartmann Clausthal University of Technology Department of Informatics Tractability of Problems Some problems are undecidable: no computer

### Discrete Applied Mathematics. The firefighter problem with more than one firefighter on trees

Discrete Applied Mathematics 161 (2013) 899 908 Contents lists available at SciVerse ScienceDirect Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam The firefighter problem with

### Optimal Online-list Batch Scheduling

Optimal Online-list Batch Scheduling Jacob Jan Paulus a,, Deshi Ye b, Guochuan Zhang b a University of Twente, P.O. box 217, 7500AE Enschede, The Netherlands b Zhejiang University, Hangzhou 310027, China

### Approximated Distributed Minimum Vertex Cover Algorithms for Bounded Degree Graphs

Approximated Distributed Minimum Vertex Cover Algorithms for Bounded Degree Graphs Yong Zhang 1.2, Francis Y.L. Chin 2, and Hing-Fung Ting 2 1 College of Mathematics and Computer Science, Hebei University,

### A 2-factor in which each cycle has long length in claw-free graphs

A -factor in which each cycle has long length in claw-free graphs Roman Čada Shuya Chiba Kiyoshi Yoshimoto 3 Department of Mathematics University of West Bohemia and Institute of Theoretical Computer Science

### SCORE SETS IN ORIENTED GRAPHS

Applicable Analysis and Discrete Mathematics, 2 (2008), 107 113. Available electronically at http://pefmath.etf.bg.ac.yu SCORE SETS IN ORIENTED GRAPHS S. Pirzada, T. A. Naikoo The score of a vertex v in

### On the k-path cover problem for cacti

On the k-path cover problem for cacti Zemin Jin and Xueliang Li Center for Combinatorics and LPMC Nankai University Tianjin 300071, P.R. China zeminjin@eyou.com, x.li@eyou.com Abstract In this paper we

### Resource Allocation with Time Intervals

Resource Allocation with Time Intervals Andreas Darmann Ulrich Pferschy Joachim Schauer Abstract We study a resource allocation problem where jobs have the following characteristics: Each job consumes

### On-line machine scheduling with batch setups

On-line machine scheduling with batch setups Lele Zhang, Andrew Wirth Department of Mechanical Engineering The University of Melbourne, VIC 3010, Australia Abstract We study a class of scheduling problems

### princeton univ. F 13 cos 521: Advanced Algorithm Design Lecture 6: Provable Approximation via Linear Programming Lecturer: Sanjeev Arora

princeton univ. F 13 cos 521: Advanced Algorithm Design Lecture 6: Provable Approximation via Linear Programming Lecturer: Sanjeev Arora Scribe: One of the running themes in this course is the notion of

### On the independence number of graphs with maximum degree 3

On the independence number of graphs with maximum degree 3 Iyad A. Kanj Fenghui Zhang Abstract Let G be an undirected graph with maximum degree at most 3 such that G does not contain any of the three graphs

### 1 Digraphs. Definition 1

1 Digraphs Definition 1 Adigraphordirected graphgisatriplecomprisedofavertex set V(G), edge set E(G), and a function assigning each edge an ordered pair of vertices (tail, head); these vertices together

### ! Solve problem to optimality. ! Solve problem in poly-time. ! Solve arbitrary instances of the problem. !-approximation algorithm.

Approximation Algorithms Chapter Approximation Algorithms Q Suppose I need to solve an NP-hard problem What should I do? A Theory says you're unlikely to find a poly-time algorithm Must sacrifice one of

### ! Solve problem to optimality. ! Solve problem in poly-time. ! Solve arbitrary instances of the problem. #-approximation algorithm.

Approximation Algorithms 11 Approximation Algorithms Q Suppose I need to solve an NP-hard problem What should I do? A Theory says you're unlikely to find a poly-time algorithm Must sacrifice one of three

### 8.1 Makespan Scheduling

600.469 / 600.669 Approximation Algorithms Lecturer: Michael Dinitz Topic: Dynamic Programing: Min-Makespan and Bin Packing Date: 2/19/15 Scribe: Gabriel Kaptchuk 8.1 Makespan Scheduling Consider an instance

### Triangle deletion. Ernie Croot. February 3, 2010

Triangle deletion Ernie Croot February 3, 2010 1 Introduction The purpose of this note is to give an intuitive outline of the triangle deletion theorem of Ruzsa and Szemerédi, which says that if G = (V,

### Every tree contains a large induced subgraph with all degrees odd

Every tree contains a large induced subgraph with all degrees odd A.J. Radcliffe Carnegie Mellon University, Pittsburgh, PA A.D. Scott Department of Pure Mathematics and Mathematical Statistics University

### 11. APPROXIMATION ALGORITHMS

11. APPROXIMATION ALGORITHMS load balancing center selection pricing method: vertex cover LP rounding: vertex cover generalized load balancing knapsack problem Lecture slides by Kevin Wayne Copyright 2005

### Tenacity and rupture degree of permutation graphs of complete bipartite graphs

Tenacity and rupture degree of permutation graphs of complete bipartite graphs Fengwei Li, Qingfang Ye and Xueliang Li Department of mathematics, Shaoxing University, Shaoxing Zhejiang 312000, P.R. China

### 5.1 Bipartite Matching

CS787: Advanced Algorithms Lecture 5: Applications of Network Flow In the last lecture, we looked at the problem of finding the maximum flow in a graph, and how it can be efficiently solved using the Ford-Fulkerson

### Total colorings of planar graphs with small maximum degree

Total colorings of planar graphs with small maximum degree Bing Wang 1,, Jian-Liang Wu, Si-Feng Tian 1 Department of Mathematics, Zaozhuang University, Shandong, 77160, China School of Mathematics, Shandong

Approximation Algorithms Chapter Approximation Algorithms Q. Suppose I need to solve an NP-hard problem. What should I do? A. Theory says you're unlikely to find a poly-time algorithm. Must sacrifice one

### Section 3 Sequences and Limits, Continued.

Section 3 Sequences and Limits, Continued. Lemma 3.6 Let {a n } n N be a convergent sequence for which a n 0 for all n N and it α 0. Then there exists N N such that for all n N. α a n 3 α In particular

### A note on properties for a complementary graph and its tree graph

A note on properties for a complementary graph and its tree graph Abulimiti Yiming Department of Mathematics Institute of Mathematics & Informations Xinjiang Normal University China Masami Yasuda Department

### Weighted Sum Coloring in Batch Scheduling of Conflicting Jobs

Weighted Sum Coloring in Batch Scheduling of Conflicting Jobs Leah Epstein Magnús M. Halldórsson Asaf Levin Hadas Shachnai Abstract Motivated by applications in batch scheduling of jobs in manufacturing

### Week 5 Integral Polyhedra

Week 5 Integral Polyhedra We have seen some examples 1 of linear programming formulation that are integral, meaning that every basic feasible solution is an integral vector. This week we develop a theory

### Applied Algorithm Design Lecture 5

Applied Algorithm Design Lecture 5 Pietro Michiardi Eurecom Pietro Michiardi (Eurecom) Applied Algorithm Design Lecture 5 1 / 86 Approximation Algorithms Pietro Michiardi (Eurecom) Applied Algorithm Design

### Exponential time algorithms for graph coloring

Exponential time algorithms for graph coloring Uriel Feige Lecture notes, March 14, 2011 1 Introduction Let [n] denote the set {1,..., k}. A k-labeling of vertices of a graph G(V, E) is a function V [k].

### 1 Approximating Set Cover

CS 05: Algorithms (Grad) Feb 2-24, 2005 Approximating Set Cover. Definition An Instance (X, F ) of the set-covering problem consists of a finite set X and a family F of subset of X, such that every elemennt

### Finding the Shortest Move-Sequence in the Graph-Generalized 15-Puzzle is NP-Hard

Finding the Shortest Move-Sequence in the Graph-Generalized 15-Puzzle is NP-Hard Oded Goldreich Abstract. Following Wilson (J. Comb. Th. (B), 1975), Johnson (J. of Alg., 1983), and Kornhauser, Miller and

### CS5314 Randomized Algorithms. Lecture 16: Balls, Bins, Random Graphs (Random Graphs, Hamiltonian Cycles)

CS5314 Randomized Algorithms Lecture 16: Balls, Bins, Random Graphs (Random Graphs, Hamiltonian Cycles) 1 Objectives Introduce Random Graph Model used to define a probability space for all graphs with

### Connectivity and cuts

Math 104, Graph Theory February 19, 2013 Measure of connectivity How connected are each of these graphs? > increasing connectivity > I G 1 is a tree, so it is a connected graph w/minimum # of edges. Every

### Spectral graph theory

Spectral graph theory Uri Feige January 2010 1 Background With every graph (or digraph) one can associate several different matrices. We have already seen the vertex-edge incidence matrix, the Laplacian

### Ph.D. Thesis. Judit Nagy-György. Supervisor: Péter Hajnal Associate Professor

Online algorithms for combinatorial problems Ph.D. Thesis by Judit Nagy-György Supervisor: Péter Hajnal Associate Professor Doctoral School in Mathematics and Computer Science University of Szeged Bolyai

### School Timetabling in Theory and Practice

School Timetabling in Theory and Practice Irving van Heuven van Staereling VU University, Amsterdam Faculty of Sciences December 24, 2012 Preface At almost every secondary school and university, some

### Solutions to Exercises 8

Discrete Mathematics Lent 2009 MA210 Solutions to Exercises 8 (1) Suppose that G is a graph in which every vertex has degree at least k, where k 1, and in which every cycle contains at least 4 vertices.

### 1 Basic Definitions and Concepts in Graph Theory

CME 305: Discrete Mathematics and Algorithms 1 Basic Definitions and Concepts in Graph Theory A graph G(V, E) is a set V of vertices and a set E of edges. In an undirected graph, an edge is an unordered

### Weighted Sum Coloring in Batch Scheduling of Conflicting Jobs

Weighted Sum Coloring in Batch Scheduling of Conflicting Jobs Leah Epstein Magnús M. Halldórsson Asaf Levin Hadas Shachnai Abstract Motivated by applications in batch scheduling of jobs in manufacturing

### THEORY OF SIMPLEX METHOD

Chapter THEORY OF SIMPLEX METHOD Mathematical Programming Problems A mathematical programming problem is an optimization problem of finding the values of the unknown variables x, x,, x n that maximize

### Ecient approximation algorithm for minimizing makespan. on uniformly related machines. Chandra Chekuri. November 25, 1997.

Ecient approximation algorithm for minimizing makespan on uniformly related machines Chandra Chekuri November 25, 1997 Abstract We obtain a new ecient approximation algorithm for scheduling precedence

### Scheduling Parallel Jobs with Monotone Speedup 1

Scheduling Parallel Jobs with Monotone Speedup 1 Alexander Grigoriev, Marc Uetz Maastricht University, Quantitative Economics, P.O.Box 616, 6200 MD Maastricht, The Netherlands, {a.grigoriev@ke.unimaas.nl,

### The Congestion of n-cube Layout on a Rectangular Grid

The Congestion of n-cube Layout on a Rectangular Grid S.L. Bezrukov J.D. Chavez L.H. Harper M. Röttger U.-P. Schroeder Abstract We consider the problem of embedding the n-dimensional cube into a rectangular

### Cycles and clique-minors in expanders

Cycles and clique-minors in expanders Benny Sudakov UCLA and Princeton University Expanders Definition: The vertex boundary of a subset X of a graph G: X = { all vertices in G\X with at least one neighbor

### Research Article Batch Scheduling on Two-Machine Flowshop with Machine-Dependent Setup Times

Hindawi Publishing Corporation Advances in Operations Research Volume 2009, Article ID 153910, 10 pages doi:10.1155/2009/153910 Research Article Batch Scheduling on Two-Machine Flowshop with Machine-Dependent

### Graph Isomorphism Completeness for Perfect Graphs and Subclasses of Perfect Graphs

Graph Isomorphism Completeness for Perfect Graphs and Subclasses of Perfect Graphs C. Boucher D. Loker May 2006 Abstract A problem is said to be GI-complete if it is provably as hard as graph isomorphism;

### On a tree graph dened by a set of cycles

Discrete Mathematics 271 (2003) 303 310 www.elsevier.com/locate/disc Note On a tree graph dened by a set of cycles Xueliang Li a,vctor Neumann-Lara b, Eduardo Rivera-Campo c;1 a Center for Combinatorics,

### More Mathematical Induction. October 27, 2016

More Mathematical Induction October 7, 016 In these slides... Review of ordinary induction. Remark about exponential and polynomial growth. Example a second proof that P(A) = A. Strong induction. Least

### Coloring Eulerian triangulations of the projective plane

Coloring Eulerian triangulations of the projective plane Bojan Mohar 1 Department of Mathematics, University of Ljubljana, 1111 Ljubljana, Slovenia bojan.mohar@uni-lj.si Abstract A simple characterization

### Graphs without proper subgraphs of minimum degree 3 and short cycles

Graphs without proper subgraphs of minimum degree 3 and short cycles Lothar Narins, Alexey Pokrovskiy, Tibor Szabó Department of Mathematics, Freie Universität, Berlin, Germany. August 22, 2014 Abstract

### GROUPS SUBGROUPS. Definition 1: An operation on a set G is a function : G G G.

Definition 1: GROUPS An operation on a set G is a function : G G G. Definition 2: A group is a set G which is equipped with an operation and a special element e G, called the identity, such that (i) the

### On the Unique Games Conjecture

On the Unique Games Conjecture Antonios Angelakis National Technical University of Athens June 16, 2015 Antonios Angelakis (NTUA) Theory of Computation June 16, 2015 1 / 20 Overview 1 Introduction 2 Preliminary

### Near Optimal Solutions

Near Optimal Solutions Many important optimization problems are lacking efficient solutions. NP-Complete problems unlikely to have polynomial time solutions. Good heuristics important for such problems.

### 2. (a) Explain the strassen s matrix multiplication. (b) Write deletion algorithm, of Binary search tree. [8+8]

Code No: R05220502 Set No. 1 1. (a) Describe the performance analysis in detail. (b) Show that f 1 (n)+f 2 (n) = 0(max(g 1 (n), g 2 (n)) where f 1 (n) = 0(g 1 (n)) and f 2 (n) = 0(g 2 (n)). [8+8] 2. (a)

### Definition 11.1. Given a graph G on n vertices, we define the following quantities:

Lecture 11 The Lovász ϑ Function 11.1 Perfect graphs We begin with some background on perfect graphs. graphs. First, we define some quantities on Definition 11.1. Given a graph G on n vertices, we define

### Notes on Matrix Multiplication and the Transitive Closure

ICS 6D Due: Wednesday, February 25, 2015 Instructor: Sandy Irani Notes on Matrix Multiplication and the Transitive Closure An n m matrix over a set S is an array of elements from S with n rows and m columns.

### Matching and Scheduling Algorithms on Large Scale Dynamic Graphs

Matching and Scheduling Algorithms on Large Scale Dynamic Graphs Author: Xander van den Eelaart Supervisors: Prof. dr. Rob H. Bisseling ir. Marcel F. van den Elst Second reader: dr. Tobias Müller Master

### Labeling outerplanar graphs with maximum degree three

Labeling outerplanar graphs with maximum degree three Xiangwen Li 1 and Sanming Zhou 2 1 Department of Mathematics Huazhong Normal University, Wuhan 430079, China 2 Department of Mathematics and Statistics

### Chapter 4. Trees. 4.1 Basics

Chapter 4 Trees 4.1 Basics A tree is a connected graph with no cycles. A forest is a collection of trees. A vertex of degree one, particularly in a tree, is called a leaf. Trees arise in a variety of applications.

### All trees contain a large induced subgraph having all degrees 1 (mod k)

All trees contain a large induced subgraph having all degrees 1 (mod k) David M. Berman, A.J. Radcliffe, A.D. Scott, Hong Wang, and Larry Wargo *Department of Mathematics University of New Orleans New

### Minimize subject to. x S R

Chapter 12 Lagrangian Relaxation This chapter is mostly inspired by Chapter 16 of [1]. In the previous chapters, we have succeeded to find efficient algorithms to solve several important problems such

### Single Machine Batch Scheduling with Release Times

Single Machine Batch Scheduling with Release Times Beat Gfeller Leon Peeters Birgitta Weber Peter Widmayer Institute of Theoretical Computer Science, ETH Zurich Technical Report 514 pril 4, 2006 bstract

### The Real Numbers. Here we show one way to explicitly construct the real numbers R. First we need a definition.

The Real Numbers Here we show one way to explicitly construct the real numbers R. First we need a definition. Definitions/Notation: A sequence of rational numbers is a funtion f : N Q. Rather than write

### Measuring Rationality with the Minimum Cost of Revealed Preference Violations. Mark Dean and Daniel Martin. Online Appendices - Not for Publication

Measuring Rationality with the Minimum Cost of Revealed Preference Violations Mark Dean and Daniel Martin Online Appendices - Not for Publication 1 1 Algorithm for Solving the MASP In this online appendix

### Two General Methods to Reduce Delay and Change of Enumeration Algorithms

ISSN 1346-5597 NII Technical Report Two General Methods to Reduce Delay and Change of Enumeration Algorithms Takeaki Uno NII-2003-004E Apr.2003 Two General Methods to Reduce Delay and Change of Enumeration

### NP-Hardness Results Related to PPAD

NP-Hardness Results Related to PPAD Chuangyin Dang Dept. of Manufacturing Engineering & Engineering Management City University of Hong Kong Kowloon, Hong Kong SAR, China E-Mail: mecdang@cityu.edu.hk Yinyu

### Approximating the entropy of a 2-dimensional shift of finite type

Approximating the entropy of a -dimensional shift of finite type Tirasan Khandhawit c 4 July 006 Abstract. In this paper, we extend the method used to compute entropy of -dimensional subshift and the technique