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## Symmetry in mathematical programming

Venue: | MIXED INTEGER NONLINEAR PROGRAMMING. VOLUME IMA |

Citations: | 3 - 2 self |

### Citations

542 | Primal-dual interiorpoint methods for semidefinite programming: Convergence rates, stability and numerical results
- Alizadeh, Haeberly, et al.
- 1998
(Show Context)
Citation Context ...finite constraint cannot be written as a mathematical expression of the operators listed above, SDP is important because it can be solved in polynomial time by a special-purpose interior point method =-=[1]-=-, and because many tight relaxations of polynomial programming problems can be cast as SDPs. Symmetries have been used in MP for analysis purposes or in order to speed up solution methods. The general... |

504 |
Theory of Groups
- Hall
- 1961
(Show Context)
Citation Context ...an orbit of ⋃ the natural action of G on the integers (i.e. the natural action of G on dom(π), which fixes every other integer), then it is easy to show that π∈G G[B] is a transitive constituent of G =-=[21]-=-. In general, G[B] may not be a subgroup of G: take G = 〈(1,2)(3,4),(1,3),(4,2)〉 and B = {1,2}, then G[B] = 〈(1,2)〉 ̸≤ G. Let B,D ⊆ {1,...,n} with B ∩D = ∅; if π ∈ Sn fixes both B,D setwise, it is eas... |

328 | Practical graph isomorphism - McKay - 1981 |

193 | Symmetry-breaking predicates for search problems
- Crawford, Ginsberg, et al.
- 1996
(Show Context)
Citation Context ...nogoods as hyperedges, for m ≤ k. For a k-ary CSP (one whose constraints have maximum arity k), the group of solution symmetries is equal to the automorphism group of its k-nogood hypergraph [11]. In =-=[13]-=- (possibly the first work in which a reduction from formulationtype symmetries to GI was proposed), SAT symmetries are automatically detected by reducing the problem to a bipartite graph, and identifi... |

121 |
Automorphism groups, isomorphism, reconstruction
- Babai
- 1995
(Show Context)
Citation Context ...ing all Vk’s. Thus, the problem of computing GP has been reduced to computing the (generators of the) automorphism group of a certain vertex-coloured DAG. This is in turn equivalent to the GI problem =-=[3]-=-. GI is in NP, but it is not known whether it is in P or NPcomplete. A notion of GI-completeness has therefore been introduced for those graph classes for which solving the GI problem is as hard as so... |

101 | Complete search in continuous global optimization and constraint satisfaction,” in Acta Numerica
- Neumaier
- 2004
(Show Context)
Citation Context ...ds a DAG DP = (VP,AP) (formed by the union of all the DAGs of functions in P followed by the contraction of leaf vertices with same variable index label) which represents the mathematical structure P =-=[48, 57]-=-. 4. Automatic computation of the formulation group. The method proposed in this section also appears (with more details) in [35]. As mentioned in the literature review, similar techniques are availab... |

96 | Symmetry groups, semidefinite programs, and sums of squares
- Gatermann, Parrilo
(Show Context)
Citation Context ...fundamental domain is a subset F ⊂ X such that GF = X. 2.3. Symmetry in Semidefinite Programming. There are several works describing the exploitation of symmetry in semidefinite programming (see e.g. =-=[26, 19, 27]-=-). Much of the material in this section is taken from the commendable tutorial [67]. Consider the following SDP: ⎫ minX C • X ⎬ ∀k ≤ m Ak • X ≤ bi ⎭ X ≽ 0, (2.2) where X is an n×n symmetric matrix abd... |

89 |
Some polyhedra related to combinatorial problems
- Gomory
- 1969
(Show Context)
Citation Context ...roup computations and symmetry-breaking techniques to be used in BB-type solution algorithms. We consider MILPs of the form min{cx | Ax ≤ b ∧ ∀i ∈ Z xi ∈ Z}. Category (a) was established by R. Gomory =-=[20]-=-: given a basis B of the constraint matrix A, it exploits the (abelian) group G = Zn /〈col(B)〉, where Zn is the additive group of integer n-sequences and 〈col(B)〉 is the additive group generated by th... |

76 |
Branching and bounds tightening techniques for non-convex MINLP. Optimization Methods and Software
- Belotti, Lee, et al.
- 2009
(Show Context)
Citation Context ...bal optimum more quickly. After extensive computational experimentations with all the symmetric instances of most public instance libraries (MIPLib, GlobalLib and MINLPLib) solved by means of Couenne =-=[6]-=- and BARON [56] (both implementing a spatial Branch-andBound (sBB) algorithm) and also RECIPE [37] (based on the Variable Neighbourhood Search (VNS) metaheuristic [22]), my own very personal opinion i... |

75 | Symmetry definitions for constraint satisfaction problems
- Cohen, Jeavons, et al.
- 2005
(Show Context)
Citation Context ...,xj = b) indicate that the two assignments xi = a and xj = b are incompatible either because of a8 LEO LIBERTI constraint in the CSP or because i = j and a ̸= b. Constraint symmetries are defined in =-=[11]-=- as the automorphisms of the microstructure complement. A k-ary nogood is a k-partial solution (i.e. an assignment of values to k variables) which cannot be extended to a full solution of the given CS... |

72 |
Bounds for unrestricted codes, by linear programming
- Delsarte
- 1972
(Show Context)
Citation Context ...]. The problem for D = 5 is still open: a lower bound taken from lattice theory is 40, and an upper bound derived with Bachoc and Vallentin’s extension [4] of Delsarte’s Linear Programming (LP) bound =-=[14]-=- is 45. We formulate the decision version of the KNP as a nonconvex NLP: ⎫ maxx,α α ⎪⎬ ∀i ≤ N ‖xi‖2 = 4 ∀i < j ≤ N ‖xi − xj‖2 ≥ 4α ∀i ≤ N xi ∈ [−2,2] D α ∈ [0,1]. ⎪⎭ (6.1) For any given N,D > 1, if a ... |

53 | New upper bounds for kissing numbers from semidefinite programming
- Bachoc, Vallentin
(Show Context)
Citation Context ...m for D = 4 was settled recently with N = 24 [47]. The problem for D = 5 is still open: a lower bound taken from lattice theory is 40, and an upper bound derived with Bachoc and Vallentin’s extension =-=[4]-=- of Delsarte’s Linear Programming (LP) bound [14] is 45. We formulate the decision version of the KNP as a nonconvex NLP: ⎫ maxx,α α ⎪⎬ ∀i ≤ N ‖xi‖2 = 4 ∀i < j ≤ N ‖xi − xj‖2 ≥ 4α ∀i ≤ N xi ∈ [−2,2] D... |

51 |
A symbolic reformulation/spatial branch-and-bound algorithm for the global optimisation of nonconvex MINLPs
- Smith, Pantelides
- 1999
(Show Context)
Citation Context ...ld perhaps be attributed to folklore (it is mentioned e.g. in [28], Sect. 2.3). DAG representable functions are routinely used in Global Optimization (GO) to automatically build relaxations of MINLPs =-=[61, 31, 6]-=- or tighten the variable bounds [57]. In the context of symmetry in MP, precise definitions for DAG representable functions and the ≡ oracle implementation are given in [35, 12]. The formulation group... |

50 |
Permutation Group Algorithms
- Seress
- 2003
(Show Context)
Citation Context ... the positive impact of the proposed approach. 1.1. Notation. We follow the notation style common in classical algebra, see e.g. [10, 2], with some modifications drawn from computational group theory =-=[59]-=-. Most of the groups considered in this paper act on vectors in Rn by permuting the components. Permutations act on sets of vectors by acting on each vector in the set. We denote the identity permutat... |

47 | A.: Interval analysis on directed acyclic graphs for global optimization
- Schichl, Neumaier
(Show Context)
Citation Context ... mentioned e.g. in [28], Sect. 2.3). DAG representable functions are routinely used in Global Optimization (GO) to automatically build relaxations of MINLPs [61, 31, 6] or tighten the variable bounds =-=[57]-=-. In the context of symmetry in MP, precise definitions for DAG representable functions and the ≡ oracle implementation are given in [35, 12]. The formulation group of P can now be defined as: GP = {π... |

46 |
Improving discrete model representations via symmetry considerations
- Sherali, Smith
(Show Context)
Citation Context ...d papers. Many types of combinatorial problems exhibit a certain amount of symmetry. Symmetries are usually broken by means of specific branching techniques (e.g. [40]), appropriate global cuts (e.g. =-=[60]-=-) or special formulations [30, 9] based on the problem structure. The main limitation of the methods in this category is that they are difficult to generalize and/or to be made automatic. Category (c)... |

45 |
Variable neighbourhood search: Principles and applications
- Hansen, Mladenovic
- 2001
(Show Context)
Citation Context ...LPLib) solved by means of Couenne [6] and BARON [56] (both implementing a spatial Branch-andBound (sBB) algorithm) and also RECIPE [37] (based on the Variable Neighbourhood Search (VNS) metaheuristic =-=[22]-=-), my own very personal opinion is that removing symmetries is good when solving with sBB and bad when solving with VNS. The sBB algorithm is a tree search based on bisecting the variable bounds at ea... |

41 | Pruning by isomorphism in branch-and-cut
- Margot
(Show Context)
Citation Context ... methods in this category is that they are difficult to generalize and/or to be made automatic. Category (c) contains two main research streams. The first was established by Margot in the early 2000s =-=[38, 39]-=-, and is applicable to Binary Linear Programs (BLPs) in the form: min cx Ax ≤ b x ∈ {0,1} n . Margot [38, 42] defines the relaxation group G LP (P) of a BLP P as: G LP (P) = {π ∈ Sn | cπ = c ∧ ∃σ ∈ Sn... |

39 |
Automatic detection of variable and value symmetries
- Puget
(Show Context)
Citation Context ...the constraint group of the corresponding CSP instance, and might be considered the CP equivalent of Thm. 4.1 and Thm. 4.2 appearing below (although the proof techniques are completely different). In =-=[51]-=-, a systematic reduction of many types of constraints to an equivalent graph form is proposed; an improved representation and extensive computational results are given in [45]. The problem of determin... |

37 |
2002 The N-vortex problem
- Newton
(Show Context)
Citation Context ...cohol, getting into a brawl about whether twelve or thirteen spheres might kiss a central one if the billiard table was tridimensional. This theory disregards the alleged scholarly note (mentioned in =-=[62]-=-) about the problem arising from an astronomical question. When D = 2, the maximum feasible N is of course 6 (hexagonal lattice). When D = 3, the maximum feasible N was conjectured by Newton to be 12 ... |

36 | nauty User’s Guide, Version 2.4
- McKay
- 2009
(Show Context)
Citation Context ...Optimization Software Engine [36] AMPL-hooked solver is then called (with ROSE’s Rsymmgroup reformulator) to produce a file representation of the problem expression DAG. This is then fed into nauty’s =-=[44, 43]-=- dreadnaut shell to efficiently compute the generators of Aut(DP). A system of shell scripts and Unix tools parses the nauty output to form a valid GAP [18] input, used to print the actual group descr... |

35 | Exploiting orbits in symmetric ILP
- Margot
(Show Context)
Citation Context ... methods in this category is that they are difficult to generalize and/or to be made automatic. Category (c) contains two main research streams. The first was established by Margot in the early 2000s =-=[38, 39]-=-, and is applicable to Binary Linear Programs (BLPs) in the form: min cx Ax ≤ b x ∈ {0,1} n . Margot [38, 42] defines the relaxation group G LP (P) of a BLP P as: G LP (P) = {π ∈ Sn | cπ = c ∧ ∃σ ∈ Sn... |

33 |
der Waerden, Das Problem der dreizehn
- Schütte, van
- 1953
(Show Context)
Citation Context ... = 2, the maximum feasible N is of course 6 (hexagonal lattice). When D = 3, the maximum feasible N was conjectured by Newton to be 12 and by Gregory to be 13 (Newton was proven right 180 years later =-=[58]-=-). The problem for D = 4 was settled recently with N = 24 [47]. The problem for D = 5 is still open: a lower bound taken from lattice theory is 40, and an upper bound derived with Bachoc and Vallentin... |

30 |
Writing global optimization software
- Liberti
- 2006
(Show Context)
Citation Context ...ld perhaps be attributed to folklore (it is mentioned e.g. in [28], Sect. 2.3). DAG representable functions are routinely used in Global Optimization (GO) to automatically build relaxations of MINLPs =-=[61, 31, 6]-=- or tighten the variable bounds [57]. In the context of symmetry in MP, precise definitions for DAG representable functions and the ≡ oracle implementation are given in [35, 12]. The formulation group... |

30 | Symmetry in integer linear programming
- Margot
- 2010
(Show Context)
Citation Context ...roblem. Once (some) problem symmetries are known, either the MP is reformulated so that some symmetric optima become infeasible and then solved via standard solution methods (static symmetry breaking =-=[42]-=-), or a known solution method is modified so that it recognizes and exploits symmetry dynamically as it goes along. Symmetries in MP can be broadly classified in two types: solution symmetries, i.e. t... |

29 | Exploiting group symmetry in semidefinite programming relaxations of the quadratic assignment problem
- Klerk, Sotirov
(Show Context)
Citation Context ...fundamental domain is a subset F ⊂ X such that GF = X. 2.3. Symmetry in Semidefinite Programming. There are several works describing the exploitation of symmetry in semidefinite programming (see e.g. =-=[26, 19, 27]-=-). Much of the material in this section is taken from the commendable tutorial [67]. Consider the following SDP: ⎫ minX C • X ⎬ ∀k ≤ m Ak • X ≤ bi ⎭ X ≽ 0, (2.2) where X is an n×n symmetric matrix abd... |

26 |
A combinatorial equivalence of matrices
- Tucker
- 1960
(Show Context)
Citation Context ...ric polyhedra [53]. Such results, however, are mostly about the classification of symmetric polyhedra and are rarely used in algorithmics. The inherent symmetry of the simplex algorithm is studied in =-=[63, 64, 65]-=-. Given two m × n matrices A,B, let SA = {x ∈ R m+n | (I|A)x = 0} (where (I|A) is the m × (m + n) matrix formed by the m × m identity followed by the columns of A) and SB = {x ∈ R m+n | (I|B)x = 0}; A... |

25 | Packing and partitioning orbitopes
- Kaibel, Pfetsch
- 2008
(Show Context)
Citation Context ...uces a feasible division of the search space; orbital branching restricts this disjunction to xh = 1 ∨ ∑ i∈O xi where h is an arbitrary index in O. The second was established by Kaibel et al. in 2007 =-=[25, 15]-=-, with the introduction of the packing and partitioning orbitopes, i.e. convex hulls ⎫ ⎬ ⎭6 LEO LIBERTI of certain 0-1 matrices that represent possible solutions to sets of packing and partitioning c... |

24 |
Problems Polynomially Equivalent to Graph Isomorphism
- Booth, Colbourn
- 1977
(Show Context)
Citation Context ...mplete. A notion of GI-completeness has therefore been introduced for those graph classes for which solving the GI problem is as hard as solving it on general graphs [66]. Rooted DAGs are GI-complete =-=[8]-=- but there is an algorithm for solving the GI problem on trees which is linear in the number of vertices in the tree ([55], Ch. 8.5.2). This should give an insight as to the type of difficulty inheren... |

21 | Reformulations in mathematical programming: Definitions and systematics
- Liberti
(Show Context)
Citation Context ... Symmetry Breaking Constraints. Once the formulation group is detected, we can adjoin constraints to (1.1) in order to make some of the symmetric optima infeasible. According to the classification in =-=[34]-=-, this is a reformulation of the narrowing type. Definition 5.1. Given a problem P, a narrowing Q of P is a formulation (1.1) such that (a) there is a function η : F(Q) → F(P) for which12 LEO LIBERTI... |

21 | F.: Reformulations in Mathematical Programming: A Computational Approach
- Liberti, Cafieri, et al.
- 2009
(Show Context)
Citation Context ...tem (called symmgroup) that automatically detects the formulation group of a problem (1.1). Our system first calls AMPL [16] to parse the instance; the ROSE Reformulation/Optimization Software Engine =-=[36]-=- AMPL-hooked solver is then called (with ROSE’s Rsymmgroup reformulator) to produce a file representation of the problem expression DAG. This is then fed into nauty’s [44, 43] dreadnaut shell to effic... |

21 | Symmetry in semidefinite programs
- Vallentin
(Show Context)
Citation Context ...mming. There are several works describing the exploitation of symmetry in semidefinite programming (see e.g. [26, 19, 27]). Much of the material in this section is taken from the commendable tutorial =-=[67]-=-. Consider the following SDP: ⎫ minX C • X ⎬ ∀k ≤ m Ak • X ≤ bi ⎭ X ≽ 0, (2.2) where X is an n×n symmetric matrix abd M1 •M1 = trace(M1 ⊤ M2) is the trace product between matrices M1,M2. Let G SDP be ... |

20 | Constraint orbital branching
- Ostrowski, Linderoth, et al.
(Show Context)
Citation Context ...elected BB tree nodes (Margot extended his work to general integer variables in [41]). Further results along the same lines (named orbital branching) are obtained for covering and packing problems in =-=[49, 50]-=-: if O is an orbit of some subgroup of the relaxation group, at each BB node the disjunction (∨ i∈O xi = 1 ) ∨ ∑ i∈O xi = 0 induces a feasible division of the search space; orbital branching restricts... |

17 | Graph isomorphism completeness for chordal bipartite graphs and strongly chordal graphs
- Uehara, Toda, et al.
- 2005
(Show Context)
Citation Context ...t known whether it is in P or NPcomplete. A notion of GI-completeness has therefore been introduced for those graph classes for which solving the GI problem is as hard as solving it on general graphs =-=[66]-=-. Rooted DAGs are GI-complete [8] but there is an algorithm for solving the GI problem on trees which is linear in the number of vertices in the tree ([55], Ch. 8.5.2). This should give an insight as ... |

16 | Small covering designs by branch-and-cut
- Margot
(Show Context)
Citation Context ...ichest in terms of number of published papers. Many types of combinatorial problems exhibit a certain amount of symmetry. Symmetries are usually broken by means of specific branching techniques (e.g. =-=[40]-=-), appropriate global cuts (e.g. [60]) or special formulations [30, 9] based on the problem structure. The main limitation of the methods in this category is that they are difficult to generalize and/... |

14 | Group symmetry in interior point methods for semidefinite programs
- Kanno, Ohsaki, et al.
(Show Context)
Citation Context ...fundamental domain is a subset F ⊂ X such that GF = X. 2.3. Symmetry in Semidefinite Programming. There are several works describing the exploitation of symmetry in semidefinite programming (see e.g. =-=[26, 19, 27]-=-). Much of the material in this section is taken from the commendable tutorial [67]. Consider the following SDP: ⎫ minX C • X ⎬ ∀k ≤ m Ak • X ≤ bi ⎭ X ≽ 0, (2.2) where X is an n×n symmetric matrix abd... |

14 | G.: A good recipe for solving MINLPs
- Liberti, Mladenović, et al.
- 2009
(Show Context)
Citation Context ...nstances of most public instance libraries (MIPLib, GlobalLib and MINLPLib) solved by means of Couenne [6] and BARON [56] (both implementing a spatial Branch-andBound (sBB) algorithm) and also RECIPE =-=[37]-=- (based on the Variable Neighbourhood Search (VNS) metaheuristic [22]), my own very personal opinion is that removing symmetries is good when solving with sBB and bad when solving with VNS. The sBB al... |

14 |
Automatically exploiting symmetries in constraint programming
- Ramani, Markov
- 2005
(Show Context)
Citation Context ...proposed), SAT symmetries are automatically detected by reducing the problem to a bipartite graph, and identified by solving the corresponding GI instance, similarly to the approach taken in [49]. In =-=[52]-=-, constraints involving the arithmetic operations +, −, × are reduced to Directed Acyclic Graphs (DAG) whose leaf vertices represent variables and intermediate vertices represent operators; vertex col... |

13 |
Elements of Abstract Algebra
- Clark
(Show Context)
Citation Context ...try breaking inequalities, and discuss computational results which show the positive impact of the proposed approach. 1.1. Notation. We follow the notation style common in classical algebra, see e.g. =-=[10, 2]-=-, with some modifications drawn from computational group theory [59]. Most of the groups considered in this paper act on vectors in Rn by permuting the components. Permutations act on sets of vectors ... |

11 | Fundamental domains for integer programs with symmetries
- Friedman
- 2007
(Show Context)
Citation Context ... more enlightening and less technical presentation than that given in [25]). Inspired by the work on orbitopes, E. Friedman proposed a similar but more general approach leading to fundamental domains =-=[17]-=-: given a feasible polytope X ⊆ [0,1] n with integral extreme points and a group G acting as an affine transformation on X (i.e. for all π ∈ G there is a matrix A ∈ GL(n) and an n-vector d such that π... |

10 |
Integer Programming: facets, subadditivity and duality for group and semi-group problems
- Johnson
- 1980
(Show Context)
Citation Context ... and only if xB ∈ Zn , setting ϕ(NxN) = ϕ(b) is a necessary and sufficient condition for xB to be integer feasible. Gomory’s seminal paper gave rise to further research, among which [68, 5]. The book =-=[24]-=- is a good starting point. Category (b) is possibly the richest in terms of number of published papers. Many types of combinatorial problems exhibit a certain amount of symmetry. Symmetries are usuall... |

10 | Reformulations in mathematical programming: Automatic symmetry detection and exploitation
- Liberti
(Show Context)
Citation Context ...e review symmetry-based analyses and methods for Linear Programming, Integer Linear Programming, Mixed-Integer Linear Programming and Semidefinite Programming. We then discuss a method (introduced in =-=[35]-=-) for automatically detecting symmetries of general (nonconvex) Nonlinear and Mixed-Integer Nonlinear Programming problems and a reformulation based on adjoining symmetry breaking constraints to the o... |

9 | Extended formulations for packing and partitioning orbitopes
- Faenza, Kaibel
- 2009
(Show Context)
Citation Context ...uces a feasible division of the search space; orbital branching restricts this disjunction to xh = 1 ∨ ∑ i∈O xi where h is an arbitrary index in O. The second was established by Kaibel et al. in 2007 =-=[25, 15]-=-, with the introduction of the packing and partitioning orbitopes, i.e. convex hulls ⎫ ⎬ ⎭6 LEO LIBERTI of certain 0-1 matrices that represent possible solutions to sets of packing and partitioning c... |

8 | Compact mathematical formulation for graph partitioning
- Boulle
(Show Context)
Citation Context ...atorial problems exhibit a certain amount of symmetry. Symmetries are usually broken by means of specific branching techniques (e.g. [40]), appropriate global cuts (e.g. [60]) or special formulations =-=[30, 9]-=- based on the problem structure. The main limitation of the methods in this category is that they are difficult to generalize and/or to be made automatic. Category (c) contains two main research strea... |

8 | Symmetric ILP: Coloring and small integers
- Margot
- 2005
(Show Context)
Citation Context ...) is used to derive effective BB pruning strategies by means of isomorphism pruning and isomorphism cuts local to some selected BB tree nodes (Margot extended his work to general integer variables in =-=[41]-=-). Further results along the same lines (named orbital branching) are obtained for covering and packing problems in [49, 50]: if O is an orbit of some subgroup of the relaxation group, at each BB node... |

8 | On implementating symmetry detection
- Mears, Banda, et al.
- 2006
(Show Context)
Citation Context ...ompletely different). In [51], a systematic reduction of many types of constraints to an equivalent graph form is proposed; an improved representation and extensive computational results are given in =-=[45]-=-. The problem of determining the constraint group of a model (instead of an instance) is discussed in [46] — we pursue a similar line of reasoning when inferring the structure of the KNP group (indepe... |

7 | L.: Formulation symmetries in circle packing
- Costa, Hansen, et al.
(Show Context)
Citation Context ...laxations of MINLPs [61, 31, 6] or tighten the variable bounds [57]. In the context of symmetry in MP, precise definitions for DAG representable functions and the ≡ oracle implementation are given in =-=[35, 12]-=-. The formulation group of P can now be defined as: GP = {π ∈ Sn|Zπ = Z ∧f(xπ) ≡ f(x) ∧ ∃σ ∈ Sm(σg(xπ) ≡ g(x))}. (3.1)10 LEO LIBERTI Because for any function h, h(xπ) ≡ h(x) implies h(xπ) = h(x) for ... |

7 | New formulations for the kissing number problem
- Kucherenko, Belotti, et al.
(Show Context)
Citation Context ... at least one optimum of (1.1) but which are likely to make at least some symmetric optima infeasible. We then present an original application of the proposed techniques to the Kissing Number Problem =-=[29]-=- in Sect. 6: we use our automatic symmetry detection method to formulate a conjecture on the KNP group structure, which we then prove to be true; we derive some symmetry breaking inequalities, and dis... |

7 | On a binary-encoded ILP coloring formulation
- Lee, Margot
(Show Context)
Citation Context ...atorial problems exhibit a certain amount of symmetry. Symmetries are usually broken by means of specific branching techniques (e.g. [40]), appropriate global cuts (e.g. [60]) or special formulations =-=[30, 9]-=- based on the problem structure. The main limitation of the methods in this category is that they are difficult to generalize and/or to be made automatic. Category (c) contains two main research strea... |

6 |
Detecting orbitopal symmetries
- Berthold, Pfetsch
- 2009
(Show Context)
Citation Context ...C1,...,Cq of the variable indices, a permutation π ∈ GLP (P) is an orbitopal symmetry if there are p,r ≤ q such that π is a bijection Cp → Cr that keeps all other Cs elementwise fixed, for s ̸∈ {p,r} =-=[7]-=-. In [25], a complete description of packing/partitioning orbitopes in terms of linear inequalities is provided ([15] gives a much shorter, more enlightening and less technical presentation than that ... |

6 |
Polytopes and symmetry
- Robertson
- 1984
(Show Context)
Citation Context ...r sections (i.e. symmetry detection methods) in Sect. 2.4. 2.1. Symmetry in Linear Programming. The geometrical objects of LP are polyhedra, and there is a very rich literature on symmetric polyhedra =-=[53]-=-. Such results, however, are mostly about the classification of symmetric polyhedra and are rarely used in algorithmics. The inherent symmetry of the simplex algorithm is studied in [63, 64, 65]. Give... |

5 |
and Groups: an Introduction to Abstract Algebra
- Rings
- 1991
(Show Context)
Citation Context ...try breaking inequalities, and discuss computational results which show the positive impact of the proposed approach. 1.1. Notation. We follow the notation style common in classical algebra, see e.g. =-=[10, 2]-=-, with some modifications drawn from computational group theory [59]. Most of the groups considered in this paper act on vectors in Rn by permuting the components. Permutations act on sets of vectors ... |

5 | Reformulations in mathematical programming: Symmetry
- Liberti
(Show Context)
Citation Context ...ies. We present an application of our findings to the Kissing Number Problem. Acknowledgements. I wish to thank François Margot for many useful discussions and suggestions, and for carefully checking =-=[33]-=- (from which some of the present material is taken) as well as one particularly careful referee. This work was financially supported by grants: ANR 07-JCJC-0151 “ARS”, Digiteo Chair 2009-14D “RMNCCO”,... |

4 | Automatic generation of symmetry-breaking constraints, in
- Liberti
- 2008
(Show Context)
Citation Context ...inently in the mathematical programming literature. A method for finding the MILP relaxation group (2.1), based on solving an auxiliary MILP encoding the condition σAπ = A, was proposed and tested in =-=[32]-=- (to the best of our knowledge, the only approach for symmetry detection that does not reduce the problem to a graph). A more practically efficient method consists in finding the automorphism group of... |

3 |
Constructive group relaxations for integer programs
- Bell
- 1976
(Show Context)
Citation Context ...ince ϕ(BxB) = 0 if and only if xB ∈ Zn , setting ϕ(NxN) = ϕ(b) is a necessary and sufficient condition for xB to be integer feasible. Gomory’s seminal paper gave rise to further research, among which =-=[68, 5]-=-. The book [24] is a good starting point. Category (b) is possibly the richest in terms of number of published papers. Many types of combinatorial problems exhibit a certain amount of symmetry. Symmet... |

3 | A novel approach for detecting symmetries in CSP models
- Mears, Banda, et al.
- 2008
(Show Context)
Citation Context ...form is proposed; an improved representation and extensive computational results are given in [45]. The problem of determining the constraint group of a model (instead of an instance) is discussed in =-=[46]-=- — we pursue a similar line of reasoning when inferring the structure of the KNP group (independently of the instance) from a sequence of automatically computed KNP instance groups. 3. Groups of a mat... |

3 |
The kissing number in four dimensions. arXiv:math.MG/0309430v2
- Musin
- 2005
(Show Context)
Citation Context ...). When D = 3, the maximum feasible N was conjectured by Newton to be 12 and by Gregory to be 13 (Newton was proven right 180 years later [58]). The problem for D = 4 was settled recently with N = 24 =-=[47]-=-. The problem for D = 5 is still open: a lower bound taken from lattice theory is 40, and an upper bound derived with Bachoc and Vallentin’s extension [4] of Delsarte’s Linear Programming (LP) bound [... |

3 |
Combinator'al Theory Underlying Linear Programs
- Tucker
- 1962
(Show Context)
Citation Context ...ric polyhedra [53]. Such results, however, are mostly about the classification of symmetric polyhedra and are rarely used in algorithmics. The inherent symmetry of the simplex algorithm is studied in =-=[63, 64, 65]-=-. Given two m × n matrices A,B, let SA = {x ∈ R m+n | (I|A)x = 0} (where (I|A) is the m × (m + n) matrix formed by the m × m identity followed by the columns of A) and SB = {x ∈ R m+n | (I|B)x = 0}; A... |

3 |
Group representation theory in integer programming
- Wolsey
- 1969
(Show Context)
Citation Context ...ince ϕ(BxB) = 0 if and only if xB ∈ Zn , setting ϕ(NxN) = ϕ(b) is a necessary and sufficient condition for xB to be integer feasible. Gomory’s seminal paper gave rise to further research, among which =-=[68, 5]-=-. The book [24] is a good starting point. Category (b) is possibly the richest in terms of number of published papers. Many types of combinatorial problems exhibit a certain amount of symmetry. Symmet... |

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A Combinatorial Algorithm for Linear Programs in the General Mixed
- Rockafellar
- 1964
(Show Context)
Citation Context ...r A :: B (among which a formula for constructing all combinatorially equivalent matrices from submatrices of A). In [64] an application to solving matrix games via the simplex method is presented. In =-=[54]-=-, Tucker’s combinatorial equivalence is used to devise a simplex algorithm variant capable of solving a pair of primal/dual LPs directly without many of the necessary pre-processing steps. 2.2. Symmet... |

2 |
Unsolvability of some optimization problems
- Zhu
(Show Context)
Citation Context ... and whether there is a permutation σ ∈ Sm such that σg(xπ) = g(x). Since Nonlinear Equations (determining if a set of general nonlinear equations has a solution) is an undecidable problem in general =-=[69]-=-, such tests are algorithmically intractable. Instead, we assume the existence of a YES/NO oracle ≡ that answers YES if it can establish that f1 = f2 (i.e. f1,f2 have the same domain and are pointwise... |

1 |
The Art of Computer Programming, Part I: Fundamental Algorithms
- Knuth
- 1968
(Show Context)
Citation Context ...ariable vertices can be contracted to obtain a DAG (right). sive graph exploration. The function DAG representation is well known and should perhaps be attributed to folklore (it is mentioned e.g. in =-=[28]-=-, Sect. 2.3). DAG representable functions are routinely used in Global Optimization (GO) to automatically build relaxations of MINLPs [61, 31, 6] or tighten the variable bounds [57]. In the context of... |

1 |
Solving a matrix game by linear programming
- Tucker
- 1960
(Show Context)
Citation Context ...ric polyhedra [53]. Such results, however, are mostly about the classification of symmetric polyhedra and are rarely used in algorithmics. The inherent symmetry of the simplex algorithm is studied in =-=[63, 64, 65]-=-. Given two m × n matrices A,B, let SA = {x ∈ R m+n | (I|A)x = 0} (where (I|A) is the m × (m + n) matrix formed by the m × m identity followed by the columns of A) and SB = {x ∈ R m+n | (I|B)x = 0}; A... |