Results 1  10
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12
An O(log k) approximate mincut maxflow theorem and approximation algorithm
 SIAM J. Comput
, 1998
"... Abstract. It is shown that the minimum cut ratio is within a factor of O(log k) of the maximum concurrent flow for kcommodity flow instances with arbitrary capacities and demands. This improves upon the previously bestknown bound of O(log 2 k) and is existentially tight, up to a constant factor. A ..."
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Cited by 125 (7 self)
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Abstract. It is shown that the minimum cut ratio is within a factor of O(log k) of the maximum concurrent flow for kcommodity flow instances with arbitrary capacities and demands. This improves upon the previously bestknown bound of O(log 2 k) and is existentially tight, up to a constant factor. An algorithm for finding a cut with ratio within a factor of O(log k) of the maximum concurrent flow, and thus of the optimal mincut ratio, is presented.
On the Solitaire Cone and Its Relationship to MultiCommodity Flows
 PREPRINT CAMS 142 ECOLE DES HAUTES ETUDES EN SCIENCES SOCIALES
, 2001
"... The classical game of Peg Solitaire has uncertain origins, but was certainly popular by the time of Louis XIV, and was described by Leibniz in 1710. The modern mathematical study of the game dates to the 1960s, when the solitaire cone was first described by Boardman and Conway. Valid inequalities ov ..."
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Cited by 7 (3 self)
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The classical game of Peg Solitaire has uncertain origins, but was certainly popular by the time of Louis XIV, and was described by Leibniz in 1710. The modern mathematical study of the game dates to the 1960s, when the solitaire cone was first described by Boardman and Conway. Valid inequalities over this cone, known as pagoda functions, were used to show the infeasibility of various peg games. In this paper we study the extremal structure of solitaire cones for a variety of boards, and relate their structure to the well studied metric cone. In particular we give: 1. an equivalence between the multicommodity flow problem with associated dual metric cone and a generalized peg game with associated solitaire cone; 2. a related NPcompleteness result; 3. a method of generating large classes of facets; 4. a complete characterization of 01 facets; 5. exponential upper and lower bounds (in the dimension) on the number of facets; 6. results on the number of facets, incidence and adjacency relationships and diameter for small rectangular, toric and triangular boards; 7. a complete characterization of the adjacency of extreme rays, diameter, number of 2faces and edge connectivity for rectangular toric boards.
On the relationship between convex bodies related to correlation experiments with dichotomic observables
 Journal of Physics A: Mathematical and General
, 2006
"... In this paper we explore further the connections between convex bodies related to quantum correlation experiments with dichotomic variables and related bodies studied in combinatorial optimization, especially cut polyhedra. Such a relationship was established in Avis, Imai, Ito and Sasaki (2005 J. P ..."
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Cited by 7 (4 self)
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In this paper we explore further the connections between convex bodies related to quantum correlation experiments with dichotomic variables and related bodies studied in combinatorial optimization, especially cut polyhedra. Such a relationship was established in Avis, Imai, Ito and Sasaki (2005 J. Phys. A: Math. Gen. 38 10971–87) with respect to Bell inequalities. We show that several well known bodies related to cut polyhedra are equivalent to bodies such as those defined by Tsirelson (1993 Hadronic J. S. 8 329–45) to represent hidden deterministic behaviors, quantum behaviors, and nosignalling behaviors. Among other things, our results allow a unique representation of these bodies, give a necessary condition for vertices of the nosignalling polytope, and give a method for bounding the quantum violation of Bell inequalities by means of a body that contains the set of quantum behaviors. Optimization over this latter body may be performed efficiently by semidefinite programming. In the second part of the paper we apply these results to the study of classical correlation functions. We provide a complete list of tight inequalities for the two party case with (m, n) dichotomic observables when m = 4, n = 4 and when min{m, n} ≤ 3, and give a new general family of correlation inequalities. PACS classification numbers: 03.65.Ud, 02.40.Ft, 02.10.Ud
Solitaire Cones
 Discrete Applied Mathematics
, 1996
"... The classical game of Peg Solitaire has uncertain origins, but was certainly popular by the time of Louis XIV, and was described by Leibniz in 1710. The modern mathematical study of the game dates to the 1960s, when the solitaire cone was first described by Boardman and Conway. Valid inequalities o ..."
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Cited by 5 (0 self)
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The classical game of Peg Solitaire has uncertain origins, but was certainly popular by the time of Louis XIV, and was described by Leibniz in 1710. The modern mathematical study of the game dates to the 1960s, when the solitaire cone was first described by Boardman and Conway. Valid inequalities over this cone, known as pagoda functions, were used to show the infeasibility of various peg games. In this paper we study the extremal structure of solitaire cones for a variety of boards, and relate their structure to the well studied metric cone. In particular we give: 1. an equivalence between the multicommodity flow problem with associated dual metric cone and a generalized peg game with associated solitaire cone; 2. a related NPcompleteness result; 3. a method of generating large classes of facets; 4. a complete characterization of 01 facets; 5. exponential upper and lower bounds (in the dimension) on the number of facets; 6. results on the number of facets, incidence, adjacency and ...
Deriving tight Bell inequalities for 2 parties with many 2valued observables from facets of cut polytopes
 PHYS REV. D50
, 2004
"... Relatively few families of Bell inequalities have previously been identified. Some examples are the trivial, CHSH, Imm22, and CGLMP inequalities. This paper presents a large number of new families of tight Bell inequalities for the case of many observables. For example, 44,368,793 inequivalent tight ..."
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Cited by 5 (4 self)
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Relatively few families of Bell inequalities have previously been identified. Some examples are the trivial, CHSH, Imm22, and CGLMP inequalities. This paper presents a large number of new families of tight Bell inequalities for the case of many observables. For example, 44,368,793 inequivalent tight Bell inequalities other than CHSH are obtained for the case of 2 parties each with 10 2valued observables. This is accomplished by first establishing a relationship between the Bell inequalities and the facets of the cut polytope, a well studied object in polyhedral combinatorics. We then prove a theorem allowing us to derive new facets of cut polytopes from facets of smaller polytopes by a process derived from FourierMotzkin elimination, which we call triangular elimination. These new facets in turn give new tight Bell inequalities. We give additional results for projections, liftings, and the complexity of membership testing for the associated Bell polytope.
Compacting cuts: a new linear formulation for minimum cut
 Proceeding of the Symposium on Discrete Algorithms
, 2007
"... For a graph (V, E), existing compact linear formulations for the minimum cut problem require Θ(V E) variables and constraints and can be interpreted as a composition of V −1 polyhedra for minimum st cuts in much the same way as early approaches to finding globally minimum cuts relied on V  ..."
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Cited by 4 (0 self)
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For a graph (V, E), existing compact linear formulations for the minimum cut problem require Θ(V E) variables and constraints and can be interpreted as a composition of V −1 polyhedra for minimum st cuts in much the same way as early approaches to finding globally minimum cuts relied on V  − 1 calls to a minimum st cut algorithm. We present the first formulation to beat this bound, one that uses O(V  2) variables and O(V  3) constraints. An immediate consequence of our result is a compact linear relaxation with O(V  2) constraints and O(V  3) variables for enforcing global connectivity constraints. This relaxation is as strong as standard cutbased relaxations and has applications in solving traveling salesman problems by integer programming as well as finding approximate solutions for survivable network design problems using Jain’s iterative rounding method. Another application is a polynomialtime verifiable certificate of size n for for the NPcomplete problem of l1embeddability of a rational metric on an nset (as opposed to a certificate of size n2 known previously). 1
Fullerenes and Coordination Polyhedra versus HalfCubes Embeddings
, 1997
"... A fullerene F n is a 3regular (or cubic) polyhedral carbon molecule for which the n vertices  the carbons atoms  are arranged in 12 pentagons and ( n 2 \Gamma 10) hexagons. Only a finite number of fullerenes are expected to be, up to scale, isometrically embeddable into a hypercube. Looking fo ..."
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Cited by 2 (0 self)
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A fullerene F n is a 3regular (or cubic) polyhedral carbon molecule for which the n vertices  the carbons atoms  are arranged in 12 pentagons and ( n 2 \Gamma 10) hexagons. Only a finite number of fullerenes are expected to be, up to scale, isometrically embeddable into a hypercube. Looking for the list of such fullerenes, we first check the embeddability of all fullerenes F n for n ! 60 and of all preferable fullerenes C n for n ! 86 and their duals. Then, we consider some infinite families, including fullerenes with icosahedral symmetry, which describe virus capsids, onionlike metallic clusters and geodesic domes. Quasiembeddings and fullerene analogues are considered. We also present some results on chemically relevant polyhedra such as coordination polyhedra and cluster polyhedra. Finally we conjecture that the list of known embeddable fullerenes is complete and present its relevance to the Katsura model for vesicles cells. Contents 1 Introduction and Basic Properties 2 1...
Finite Metrics in Switching Classes
"... Let D be a finite set and g: D × D → R a symmetric function satisfying g(x, x) = 0 and g(x, y) = g(y, x) for all x, y ∈ D. A switch g σ is obtained from g by using a local valuation σ: D → R: g σ (x, y) = σ(x) + g(x, y) + σ(y) for x ̸ = y. It is shown that every symmetric function g has a unique ..."
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Let D be a finite set and g: D × D → R a symmetric function satisfying g(x, x) = 0 and g(x, y) = g(y, x) for all x, y ∈ D. A switch g σ is obtained from g by using a local valuation σ: D → R: g σ (x, y) = σ(x) + g(x, y) + σ(y) for x ̸ = y. It is shown that every symmetric function g has a unique minimal pseudometric switch, and, moreover, there is a switch g σ of g that is isometric to a finite Manhattan metric. Also, for each metric on a finite set D, we associate an extension metric on the set of all nonempty subsets of D, and we show that this extended metric inherits the switching classes on D.