## Domination analysis of combinatorial optimization algorithms and problems (2005)

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Venue: | In Graph Theory, Combinatorics and Algorithms: Interdisciplinary Applications (M.C. Golumbic and I. Ben-Arroyo |

Citations: | 12 - 7 self |

### BibTeX

@INPROCEEDINGS{Gutin05dominationanalysis,

author = {Gregory Gutin and Anders Yeo},

title = {Domination analysis of combinatorial optimization algorithms and problems},

booktitle = {In Graph Theory, Combinatorics and Algorithms: Interdisciplinary Applications (M.C. Golumbic and I. Ben-Arroyo},

year = {2005},

publisher = {Springer-Verlag}

}

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### Abstract

We provide an overview of an emerging area of domination analysis (DA) of combinatorial optimization algorithms and problems. We consider DA theory and its relevance to computational practice. 1

### Citations

502 |
Matroid Theory
- Oxley
- 1992
(Show Context)
Citation Context ...) = 2, the domination number of greedy is 1 for this instance of the (E ′ , F ′ )-optimization problem. Note that if we add (I3) below to (I1),(I2), then we obtain one of the definitions of a matroid =-=[33]-=-: (I3) If U and V are in F and |U| > |V |, then there exists x ∈ U − V such that V ∪ {x} ∈ F. It is well-known that domination number of greedy for every matroid (E, F) is |F|: greedy always finds an ... |

288 |
The traveling salesman problem: A case study
- Johnson, McGeoch
- 1997
(Show Context)
Citation Context ... at least (n − 2)!/2 when n is even and (n − 2)! when n is odd. Observe that this result is of restricted interest since to reach a k-opt local optimum one may need exponential time (cf. Section 3 in =-=[27]-=-). However, Punnen, Margot and Kabadi [34] managed to prove the following result. Theorem 2.9 For the STSP the best improvement 2-opt LS produces a tour, which is not worse than at least Ω((n − 2)!) o... |

260 | Theory, Algorithms and Applications
- Bang-Jensen, Gutin, et al.
- 2001
(Show Context)
Citation Context ... the weight wt ′ (u, w), for u, w ∈ V ′ , equals wt(u, x) if w = va, wt(y, w) if u = va, and wt(u, w), otherwise. The above definition has an obvious extension to a set of arcs; for more details, see =-=[6]-=-. For an undirected graph G, V (G) (E(G)) denotes the vertex (edge) set of H. A tour in a graph G is a Hamilton cycle in G. A complete graph Kn is a graph in which every pair x, y of distinct vertices... |

180 | Complexity and Approximation
- Ausiello, Crescenzi, et al.
- 1999
(Show Context)
Citation Context ...is is of definite importance, it cannot cover all possible families of instances of the CO problem at hand and, in particular, it normally does not cover the hardest instances. Approximation Analysis =-=[4]-=- is a frequently used tool for theoretical evaluation of CO heuristics. Let H be a heuristic for a combinatorial minimization problem P and let In be the set of instances of P of size n. In approximat... |

121 |
Location of bank accounts to optimize float: An analytic study of exact and approximate algorithms
- Cornuejols, Fisher, et al.
- 1977
(Show Context)
Citation Context ...r TSP. Thus, it seems that both max TSP and TSP should be in the same class of CO problems. The above asymmetry was already viewed as a drawback of performance ratio already in the 1970’s, see, e.g., =-=[11, 30, 40]-=-. Notice that from the DA point view max TSP and TSP are equivalent problems. Zemel [40] was the first to characterize measures of quality of approximate solutions (of binary integer programming probl... |

119 | A Survey of Very Large-Scale Neighborhood
- Ahuja, Ergun, et al.
- 2002
(Show Context)
Citation Context ...ucting heuristics for CO problems. Recently, several researchers started inves3sd 12 9 10 7 a 5 8 Figure 1: A complete weighted digraph tigation of LS with Very Large Scale Neighbourhoods (see, e.g., =-=[1, 12, 26]-=-). The hypothesis behind this approach is that the larger the neighbourhood the better quality solution are expected to be found [1]. However, some computational experiments do not support this hypoth... |

99 | The Traveling Salesman Problem and Its Variations
- Gutin, Punnen
- 2002
(Show Context)
Citation Context ...rea of domination analysis (DA) of combinatorial optimization algorithms and problems. We consider DA theory and its relevance to computational practice. 1 Introduction In the recently published book =-=[19]-=-, Chapter 6 is partially devoted to domination analysis (DA) of the Traveling Salesman Problem (TSP) and its heuristics. The aim of this chapter is to provide an overview of the whole area of DA. In p... |

88 |
The Probabilistic Method, 2nd Edition
- Alon, Spencer
- 2000
(Show Context)
Citation Context ... of satisfied clauses in a random truth assignment is E = � m i=1 pi. For simplicity, in the sequel true (false) will be replaced by the binaries 1 (0). By a construction described in Section 15.2 of =-=[3]-=-, there exists a binary matrix A = (aij) with n columns and r = O(n ⌊k/2⌋ ) rows such that the following holds: Let B be an arbitrary submatrix of A, consisting of k of its columns (chosen arbitrarily... |

76 |
Graphs
- Berge
- 1985
(Show Context)
Citation Context .... A decomposition of A(K ∗ n) into tours, is a collection of tours in K ∗ n, such that every arc in K ∗ n belongs to exactly one of the tours. The following lemma was proved for odd n by Kirkman (see =-=[9]-=-, p. 187), and the even case result was established in [39]. Lemma 2.1 For every n ≥ 2, n �= 4, n �= 6, there exists a decomposition of A(K ∗ n) into tours. An automorphism, f, of V (K ∗ n) is a bijec... |

70 |
On a lemma of Littlewood and
- Erdös
- 1945
(Show Context)
Citation Context ...lowing: Proposition 3.2 The number of vectors (ɛ1, . . . , ɛn) ∈ Bn for which | �n i=1 ɛiai| < � 2n−k . ak+1 is at most � k ⌊k/2⌋ To prove this proposition, we will use the following lemma: Lemma 3.3 =-=[13]-=- Let a1 ≥ a2 ≥ · · · ≥ ak and let (a, b) be an arbitrary open interval such that b − a ≤ 2ak. Then the number of vectors (δ1, . . . , δk) ∈ Bk such that �k i=1 δiai ∈ (a, b) is at most � k � ⌊k/2⌋ . F... |

54 |
The Approximation of Maximum Subgraph Problems
- Lund, Yannakakis
- 1993
(Show Context)
Citation Context ... i.e., every induced subgraph of a graph with property Π has property Π, and non-trivial, i.e., it is satisfied for infinitely many graphs 22sand false for infinitely many graphs. Lund and Yannakakis =-=[32]-=- proved that MISP is not approximable within n ɛ for some ɛ > 0 unless P=NP, if Π is hereditary, non-trivial and is false for some clique or independent set (e.g., planar, bipartite, triangle-free). T... |

40 |
The traveling salesman problem: New solvable cases and linkages with the development of approximation algorithms
- Glover, Punnen
- 1997
(Show Context)
Citation Context ... by 2. However, Punnen, Margot and Kabadi [34] proved that the domination number of DMST is 1. 1.2 Introduction to domination analysis Domination Analysis was formally introduced by Glover and Punnen =-=[16]-=- in 1997. Interestingly, important results on domination analysis for the TSP can be traced back to the 1970s, see Rublineckii [36] and Sarvanov [37]. Let P be a CO problem and let H be a heuristic fo... |

33 | Clique is hard to approximate within
- H̊astad
- 1999
(Show Context)
Citation Context ...ave considered in the previous subsections have been DOM-easy. We will show that MCl and MVC are DOM-hard unless P=NP. Theorem 3.10 [20] MCl is DOM-hard unless P=NP. Proof: We use a result by H˚astad =-=[29]-=-, which states that, provided that P�=NP, MCl is not approximable within a factor n1/2−ɛ for any ɛ > 0, where n is the number of vertices in a graph. Let G be a graph with n vertices, and let q be the... |

30 | Experimental Analysis of Heuristics for the ATSP
- Johnson, Gutin, et al.
- 2002
(Show Context)
Citation Context ...lems. In other words, the greedy algorithm, in the worst case, produces the unique worst possible solution. This is in line with latest computational experiments with the greedy algorithm, see, e.g., =-=[28]-=-, where the authors came to the conclusion that the greedy algorithm ’might be said to selfdestruct’ and that it should not be used even as ’a general-purpose starting tour generator’. The Asymmetric ... |

23 | Polynomial approximation algorithms for the TSP and the QAP with a factorial domination number - Gutin, Yeo - 2002 |

23 | Traveling Salesman Should Not Be Greedy: Domination Analysis of Greedy Type Heuristics for the TSP
- Gutin, Yeo, et al.
(Show Context)
Citation Context ...rices A = [aij] and B = [bij] of integers. Our aim is to find a permutation π of {1, 2, . . . , n} that minimizes the sum n� i=1 j=1 n� aijbπ(i)π(j). Using group-theoretical approaches, Gutin and Yeo =-=[25]-=- proved only that QAP is DOM-easy when n is a prime power. Conjecture 3.14 QAP is DOM-easy (for every value of n). It was noted in [20] that Theorem 3.10 holds for some cases of the following general ... |

22 | Algorithms with large domination ratio
- Alon, Gutin, et al.
- 2004
(Show Context)
Citation Context ...values of ɛ π(1), ɛ π(2), . . . , ɛ π(k). (This can be trivially done in time O(n p ).) Now for each j > k, if � j−1 i=1 ɛ π(i)a π(i) < 0, then set ɛ π(j) = +1, and otherwise ɛ π(j) = −1. Theorem 3.1 =-=[2]-=- The domination ratio of B is at least 1 − � k 1 − Θ( 1 √ k ). ⌊k/2⌋ � /2 k = To prove this theorem, without loss of generality, we may assume a1 ≥ a2 ≥ . . . ≥ an. Observe that if min | � k i=1 ɛiai|... |

22 |
Measuring the quality of approximate solutions to zero-one programming problems
- Zemel
- 1981
(Show Context)
Citation Context ...r TSP. Thus, it seems that both max TSP and TSP should be in the same class of CO problems. The above asymmetry was already viewed as a drawback of performance ratio already in the 1970’s, see, e.g., =-=[11, 30, 40]-=-. Notice that from the DA point view max TSP and TSP are equivalent problems. Zemel [40] was the first to characterize measures of quality of approximate solutions (of binary integer programming probl... |

21 |
Approximate solution of the traveling salesman problem by a local algorithm with scanning neighborhoods of factorial cardinality in cubic time
- Sarvanov, Doroshko
- 1981
(Show Context)
Citation Context ...t coincide, we carry out the next iteration. Otherwise, we output Ti. One of the first exponential size TSP neighborhoods (called assign in [12]) was considered independently by Sarvanov and Doroshko =-=[38]-=-, and Gutin [17]. We describe this neighborhood and establish a simple upper bound on the domination number of the best improvement LS based on this neighborhood. We will see that the domination numbe... |

19 | Algorithmic Number Theory - Volume 1 - Bach, Shallit - 1996 |

16 |
Exponential Neighborhoods and Domination Analysis for the TSP
- Gutin, Yeo, et al.
- 2002
(Show Context)
Citation Context ...ucting heuristics for CO problems. Recently, several researchers started inves3sd 12 9 10 7 a 5 8 Figure 1: A complete weighted digraph tigation of LS with Very Large Scale Neighbourhoods (see, e.g., =-=[1, 12, 26]-=-). The hypothesis behind this approach is that the larger the neighbourhood the better quality solution are expected to be found [1]. However, some computational experiments do not support this hypoth... |

15 |
A study of exponential neighbourhoods for the traveling salesman problem and the quadratic assignment problem
- Deineko, Woeginger
(Show Context)
Citation Context ...ucting heuristics for CO problems. Recently, several researchers started inves3sd 12 9 10 7 a 5 8 Figure 1: A complete weighted digraph tigation of LS with Very Large Scale Neighbourhoods (see, e.g., =-=[1, 12, 26]-=-). The hypothesis behind this approach is that the larger the neighbourhood the better quality solution are expected to be found [1]. However, some computational experiments do not support this hypoth... |

13 |
TSP heuristics: domination analysis and complexity,” Algorithmica 35
- Punnen, Margot, et al.
- 2003
(Show Context)
Citation Context ...ur H. It is well-known and easy to prove that the weight of H is at most twice the weight of the optimal tour. Thus, the performance ratio for DMST is bounded by 2. However, Punnen, Margot and Kabadi =-=[34]-=- proved that the domination number of DMST is 1. 1.2 Introduction to domination analysis Domination Analysis was formally introduced by Glover and Punnen [16] in 1997. Interestingly, important results... |

11 | Domination analysis of some heuristics for the asymmetric traveling salesman problem - Punnen, Kabadi |

10 | Creating very large scale neighborhoods out of smaller ones by compounding moves: A study on the vehicle routing problem. Working paper
- Ergun, Orlin, et al.
- 2003
(Show Context)
Citation Context ... behind this approach is that the larger the neighbourhood the better quality solution are expected to be found [1]. However, some computational experiments do not support this hypothesis, see, e.g., =-=[15]-=-, where an LS with small neighbourhoods proves to be superior to that with large neighbourhoods. This means that some other parameters are responsible for the relative power of a neighbourhood. Theore... |

10 |
Performance analysis of six approximation algorithms for the one-machine maximum lateness scheduling problem with ready times
- Kise, Iberaki, et al.
- 1979
(Show Context)
Citation Context ...r TSP. Thus, it seems that both max TSP and TSP should be in the same class of CO problems. The above asymmetry was already viewed as a drawback of performance ratio already in the 1970’s, see, e.g., =-=[11, 30, 40]-=-. Notice that from the DA point view max TSP and TSP are equivalent problems. Zemel [40] was the first to characterize measures of quality of approximate solutions (of binary integer programming probl... |

8 | Domination analysis of combinatorial optimization problems
- Gutin, Vainshtein, et al.
(Show Context)
Citation Context ...n 1.5 Using the mapping f of Question 1.4 and Theorem 1.2, prove that the greedy algorithm has domination number 1 for the ATSP. 7s1.3 Additional terminology and notation Following the terminology in =-=[20]-=-, a CO problem P is called DOM-easy if there exists a polynomial time algorithm, A, such that domr(A, n) ≥ 1/p(n), where p(n) is a polynomial in n. In other words, a problem is DOM-easy, if, in polyno... |

8 |
Domination analysis of greedy heuristics for the frequency assignment problem
- Koller, Noble
- 2004
(Show Context)
Citation Context ...ivelevich [2] recently proved, using an involved probabilistic argument, that the algorithm of Theorem 3.5 is, in fact, of domination number Ω(2 n ). 19s3.4 Fixed span frequency assignment problem In =-=[31]-=- the domination number is computed for various heuristics for the Fixed Span Frequency Assignment Problem (fs-FAP), which is defined as follows. We are given a set of vertices {x1, x2, . . . , xn} and... |

7 | Combinatorial dominance guarantees for heuristic algorithms
- Berend, Skiena
- 2002
(Show Context)
Citation Context ...sfied. Let U = {x1, . . . , xn} be the set of variables in the instance of max-k-SAT under consideration. Let {C1, . . . , Cm} be the set of clauses. We assume that k is a constant. Berend and Skiena =-=[10]-=- considered some well-known algorithms for maxk-SAT and the algorithms turned out to have domination number at most n + 1. However an algorithm considered in [20] is of domination number at least Ω(2 ... |

6 |
Transformations of Generalized ATSP into ATSP: experimental and theoretical
- Ben-Arieh, Gutin, et al.
(Show Context)
Citation Context ...ts V1, . . . , Vk. We are required to compute a lightest cycle in G containg exactly one vertex from each Vi, i = 1, . . . , k. In the case when all Vi’s are of the same cardinality, Ben-Arieh et al. =-=[8]-=- proved that the Generalized TSP is DOM-easy. The Quadratic Assignment Problem (QAP) can be formulated as follows. We are given two n × n matrices A = [aij] and B = [bij] of integers. Our aim is to fi... |

6 | Upper Bounds on ATSP Neighborhood Size
- Gutin, Yeo
- 2003
(Show Context)
Citation Context ...ristics It is realistic to assume that any ATSP algorithm spends at least one unit of time on every arc of K ∗ n that it considers. We use this assumption in the rest of this subsection. Theorem 2.10 =-=[24, 22]-=- Let A be an ATSP heuristic of complexity t(n). Then the domination number of A does not exceed max1≤n ′ ≤n(t(n)/n ′ ) n′ . Proof: Let D = (K ∗ n, wt) be an instance of the ATSP and let H be the tour ... |

5 | When the greedy algorithm fails
- Bang-Jensen, Gutin, et al.
- 2004
(Show Context)
Citation Context ...he Asymmetric TSP, Symmetric TSP and AP. Research Question 4.6 Describe new wide families of the (E, F)-optimization problems for which greedy is of domination number 1. 25sBang-Jensen, Gutin and Yeo =-=[7]-=- considered the (E, F)-optimization problems, in which every base is of the same cardinality and wt assumes only a finite number of integral values. For such problems, the authors of [7] completely ch... |

4 |
A survey and annotated bibliography of multicriteria combinatorial optimization
- Ehrgott, Gandibleux
- 2000
(Show Context)
Citation Context ...ne such parameter may well be the domination ratio of the corresponding LS. Sometimes, Approximation Analysis cannot be naturally used. Indeed, a large class of CO problems are multicriteria problems =-=[14]-=-, which have several objective functions. (For example, consider STSP in which edges are assigned both time and cost, and one is required to minimize both time and cost.) We say that one solution s ′ ... |

4 |
Estimates of the accuracy of procedures in the travelling salesman problem
- Rublineckii
- 1973
(Show Context)
Citation Context ...sis Domination Analysis was formally introduced by Glover and Punnen [16] in 1997. Interestingly, important results on domination analysis for the TSP can be traced back to the 1970s, see Rublineckii =-=[36]-=- and Sarvanov [37]. Let P be a CO problem and let H be a heuristic for P. The domination number domn(H, I) of H for a particular instance I of P is the number of feasible solutions of I that are not b... |

4 |
On the minimization of a linear from on a set of all nelements cycles
- Sarvanov
- 1976
(Show Context)
Citation Context ...lysis was formally introduced by Glover and Punnen [16] in 1997. Interestingly, important results on domination analysis for the TSP can be traced back to the 1970s, see Rublineckii [36] and Sarvanov =-=[37]-=-. Let P be a CO problem and let H be a heuristic for P. The domination number domn(H, I) of H for a particular instance I of P is the number of feasible solutions of I that are not better than the sol... |

4 |
A hamiltonian decomposition of K 2m , m 8
- Tillson
- 1980
(Show Context)
Citation Context ...of tours in K ∗ n, such that every arc in K ∗ n belongs to exactly one of the tours. The following lemma was proved for odd n by Kirkman (see [9], p. 187), and the even case result was established in =-=[39]-=-. Lemma 2.1 For every n ≥ 2, n �= 4, n �= 6, there exists a decomposition of A(K ∗ n) into tours. An automorphism, f, of V (K ∗ n) is a bijection from V (K ∗ n) to itself. Note that if C is a tour in ... |

2 | Domination analysis for minimum multiprocessor scheduling
- Gutin, Jensen, et al.
(Show Context)
Citation Context ...ell. For an integer p ≥ 2, a p-partition of a set A is a collection A1, A2, . . . , Ap of subsets of A such that ∪ p i=1 Ai = A and Ai ∩ Aj = ∅ for each 1 ≤ i �= j ≤ p. Theorem 3.1 was generalized in =-=[18]-=-, where the following minimum pprocessor scheduling problem was considered. We are given an integer p ≥ 2 16sand a sequence w1, w2, . . . , wn of positive integers, and we are required to find a p-par... |

1 |
On an approach to solving the TSP
- Gutin
- 1984
(Show Context)
Citation Context ...arry out the next iteration. Otherwise, we output Ti. One of the first exponential size TSP neighborhoods (called assign in [12]) was considered independently by Sarvanov and Doroshko [38], and Gutin =-=[17]-=-. We describe this neighborhood and establish a simple upper bound on the domination number of the best improvement LS based on this neighborhood. We will see that the domination number of the best im... |

1 | Introduction to domination analysis
- Gutin, Yeo
(Show Context)
Citation Context ...ristics It is realistic to assume that any ATSP algorithm spends at least one unit of time on every arc of K ∗ n that it considers. We use this assumption in the rest of this subsection. Theorem 2.10 =-=[24, 22]-=- Let A be an ATSP heuristic of complexity t(n). Then the domination number of A does not exceed max1≤n ′ ≤n(t(n)/n ′ ) n′ . Proof: Let D = (K ∗ n, wt) be an instance of the ATSP and let H be the tour ... |