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The Combinatorial Structure of Small Cut and Metric Polytopes

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The Combinatorial Structure of Small Cut and Metric Polytopes The Combinatorial Structure of Small Cut and Metric Polytop es Antoine DEZA Michel DEZA Lab oratoire dInformatique URA du CNRS Departement de Mathematiques et dInformatique Ecole Normale Superieure LIENS January The Combinatorial Structure of Small Cut and Metric Polytop es Antoine DEZA Michel DEZA January Abstract We study the combinatorial structure of the cut and metric p olytop es on n no des for n Those two p olytop es have a complicated geometrical structure but using their large symmetry group we can completely describ e their face lattices We present for any n some orbits of faces and give new result on the tightness of the wrapping of the cut p olytop e by the metric p olytop e disproving a conjecture of on their lattices Key words metric p olytop e complete bipartite subgraphs p olytop e face lattice Intro duction We rst recall the denition of the metric polytope m the cut polytope c and their n n relatives the metric cone and the cut cone Then we present some applications to well known optimization problems and some combinatorial and geometric prop erties of those p olyhedra The general references are for p olytop es and for graphs For a complete study of the applications and the combinatorial optimization asp ects of those p olyhedra we refer resp ectively to the surveys and For all sets fi j k g f ng we consider the following inequalities x x x ij ik j k x x x ij ik j k The inequalities dene the metric cone and the metric p olytop e m is obtained by n n b ounding the latter by the inequalities The facets dened by the inequalities 3 which can b e seen as triangle inequalities for distance x on f ng are called ij n homogeneous triangle facets and are denoted by Tr The facets dened by the ijk 3 inequalities are called nonhomogeneous triangle facets and are denoted by Tr ij k Given a subset S of V f ng the cut determined by S consists of the pairs n i j of elements of V such that exactly one of i j is in S S denotes b oth the cut n n 2 and its incidence vector in IR that is S if exactly one of i j is in S and ij otherwise for i j n By abuse of language we use the term cut for b oth the cut n 2 itself and its incidence vector so S are considered as co ordinates of a p oint in IR ij The cut p olytop e of the complete graph c which is also called the complete bipartite n n1 subgraphs polytope is the convex hull of all cuts and the cut cone is the conic hull n1 of all nonzero cuts Those p olyhedra were considered by many authors see for instance and references therein One of the motivations for the study of these p olyhedra comes from their applications in combinatorial optimiza tion the most imp ortant b eing the maxcut and multicommo dity ow problems Given a graph G V E and nonnegative weights w e E assigned to its edges n e P w is as large the maxcut problem consists in nding a cut S whose weight e e (S ) as p ossible It is a wellknown NP complete problem By setting w if e is not an e edge of G we can consider without loss of generality the complete graph on V Then the n maxcut problem can b e stated as a linear programming problem over the cut p olytop e c as follows n T max w x x c n T Since the metric p olytop e is a relaxation of the cut p olytop e optimizing w x over c n instead of m provides an upp er b ound for the maxcut problem n With E the set of edges of the complete graph on V an instance of the multicom n modity ow problem is given by two nonnegative vectors indexed by E a capacity ce and a requirement r e for each e E Let U fe E r e g If T denotes the subset of V spanned by the edges in U then we say that the graph G T U denotes n the suppor t of r For each edge e s t in the supp ort of r we seek a ow of r e units b etween s and t in the complete graph The sum of all ows along any edge e E must not exceed ce If such a ow exists we call c r feasible A necessary and sucient condition for feasibility is given by the Japanese theorem a pair c r is feasible if and T only if c r x is valid over the metric cone For example Tr can b e seen as ijk an elementary solvable ow problem with cij r ik r j k and ce r e T otherwise so the inequalities corresp ond to c r x for x in the metric cone Therefore the metric cone is the dual cone to the cone of feasible multicommo dity ow problems Combinatorial and geometric prop erties of the cut and metric p olytop es n n1 The p olytop e c is a dimensional p olyhedron with vertices and m is n n 2 n n 2 a p olytop e of same dimension with facets inscrib ed in the cub e We have 3 1 1 1 c m with equality only for n It is easy to see that the p oint n n n 2 2 2 is the center of gravity of b oth c and m and is also the center of the sphere of radius n n p 1 nn where all the cuts lie Another two geometric characteristics of the r 2 cut p olytop e c are its width and geometric diameter We recall that while the width of n a p olytop e P is equal to the minimum distance b etween a pair of parallel hyp erplanes containing P in the slice b etween them the geometric diameter of P is the maximum distance b etween a pair of supp orting hyp erplanes The width of c is and n p n 1 2 its geometric diameter is for n even and n for n o dd see Any facet 2 2 resp ectively ridge that is a face of co dimension of the metric p olytop e contains a facet resp ectively a ridge of the cut p olytop e and the vertices of the cut p olytop e are vertices of the metric p olytop e in fact the cuts are precisely the integral vertices of the metric p olytop e Actually the metric p olytop e m wraps the cut p olytop e c very tightly n n since in addition to the vertices all edges and faces of c are also faces of m n n In other words c is a segment of order of m and its dual m is a segment of order n n n of c in terms of a p olytop e P is a segment of order s of a p olytop e Q if they have n the same dimension and if every iface of P is a face of Q for i s The p olytop e c is neighb ourly Any two cuts are adjacent b oth on c and on m n n n in other words m is quasiintegral in terms of that is the skeleton of the convex n hull of its integral vertices ie the skeleton of the c is an induced subgraph of the edge n is the diameters of graph of the metric p olytop e itself While the diameter of m n c and m are resp ectively conjectured to b e and For a detailed study of n n the combinatorial and geometric prop erties of c and m we refer to n n The metric p olytop e and the cut p olytop e share the same symmetry group that is the group of isometries preserving a p olytop e This group is isomorphic to the automorphism group of the folded ncube that is I sm I sc Aut2 see We recall p p n that the folded ncub e is the graph whose vertices are the partitions of V f ng n into two subsets two partitions b eing adjacent when their common renement contains a set of size one see More precisely for n I sm I sc is induced by n n p ermutations on V f ng and switching reections by a cut Given a cut S the n switching reection r is dened by y r x where y x if i j S and ij ij (S ) (S ) y x otherwise These symmetries preserve the adjacency relations and the linear ij ij indep endency For the study of their face lattices we frequently use the fact that the faces of m and c are partitioned into orbits of their symmetry group n n We nally mention the following link with metrics there is an evident corre sp ondence b etween the elements of the metric cone and all the semimetrics on n p oints and the elements of the cut cone corresp ond precisely to the semimetrics on n p oints n m that are isometrically emb eddable into some l see it is easy to check that m 1 2 Face lattices of small cut and metric p olytop es Face lattice of the c m for n n n For n we have c m moreover c and c are b oth wellknown p olytop es While n n 3 4 p 1 c is combinatorially and volume v c is the regular tetrahedron of edge length 4 3 3 3 2 equivalent to the dimensional cyclic p olytop e with vertices and its volume is v 4 45 The f vector of c is obviously f c more precisely all prop er faces 4 4 of c are partitioned into the following orbits of the symmetry group I sc Aut2 4 4 4 1 the vertices of c form the orbit O 4 0 1 2 the orbits O and O of edges f S S g are resp ectively formed by the 1 1 edges with jS S j o dd and the ones with jS S j even that is resp ectively represented by f g and f g 1 the orbits of faces are O of size which is represented by f g 2 2 and O of size which is represented by f g 2 1 the orbits of faces are O of size which is represented by f 3 2 3 g O of size which is represented by f g and O of 3 3 size and represented by f g 1 the ridges form the orbit O they are the cofaces that is the convex hull of the 4 1 vertices not b elonging to a face corresp onding to the faces from the orbit O 2 1 the facets form the orbit O they are the cofaces corresp onding to the edges 5 1 from the orbit O 1 Remark i The skeleton of c is the grid which is also the line graph of K 2 44 4 4 ii A set of vertices is not a face of c if and only if it contains one of the fol lowing 4 sets of vertices f g and f g Face lattices of c and m 5 5 The face lattices of c and m were obtained in the following way We rst got 5 5 all the nonsimplex faces by systematically checking all p ossible pairwise wise etc
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