EExtended Formulations in CombinatorialOptimization
Volker Kaibel ∗ April 7, 2011
Abstract
The concept of representing a polytope that is associated with somecombinatorial optimization problem as a linear projection of a higher-dimensional polyhedron has recently received increasing attention. Inthis paper (written for the newsletter
Optima of the Mathematical Opti-mization Society), we provide a brief introduction to this topic and sketchsome of the recent developments with respect to both tools for construct-ing such extended formulations as well as lower bounds on their sizes.
Linear Programming based methods and polyhedral theory form the backboneof large parts of Combinatorial Optimization. The basic paradigm here is toidentify the feasible solutions to a given problem with some vectors in such away that the optimization problem becomes the problem of optimizing a linearfunction over the finite set X of these vectors. The optimal value of a linearfunction over X is equal to its optimal value over the convex hull conv( X ) = { (cid:80) x ∈ X λ x x : (cid:80) x ∈ X λ x = 1 , λ ≥ O } of X . According to the Weyl–MinkowskiTheorem [35, 27], every polytope (i.e., the convex hull of a finite set of vec-tors) can be written as the set of solutions to a system of linear equations andinequalities. Thus one ends up with a linear programming problem.As for the maybe most classical example, let us consider the set M ( n ) of allmatchings in the complete graph K n = ( V n , E n ) on n nodes (where a matchingis a subset of edges no two of which share a common end-node). Identifyingevery matching M ⊆ E n with its characteristic vector χ ( M ) ∈ { , } E n (where χ ( M ) e = 1 if and only if e ∈ M ), we obtain the matching polytope P match ( n ) = conv { χ ( M ) : M ∈ M ( n ) } . In one of his seminal papers, Edmonds [14] proved that P match ( n ) equals theset of all x ∈ R E n + that satisfy the inequalities x ( δ ( v )) ≤ v ∈ V n and ∗ Institut f¨ur Mathematische Optimierung, Fakult¨at f¨ur Mathematik, Otto-von-GuerickeUniversit¨at Magdeburg, Universit¨atsplatz 2, 39108 Magdeburg, Germany, [email protected] a r X i v : . [ m a t h . C O ] A p r ( E n ( S )) ≤ (cid:98)| S | / (cid:99) for all subsets S ⊆ V n of odd cardinality 3 ≤ | S | ≤ n (where δ ( v ) is the set of all edges incident to v , E n ( S ) is the set of all edges with bothend-nodes in S , and x ( F ) = (cid:80) e ∈ F x e ). No inequality in this system, whose sizeis exponential in n , is redundant.The situation is quite similar for the permutahedron P perm ( n ), i.e., the con-vex hull of all vectors that arise from permuting the components of (1 , , . . . , n ).Rado [31] proved that P perm ( n ) is described by the equation x ([ n ]) = n ( n + 1) / x ( S ) ≥ | S | ( | S | + 1) / ∅ (cid:54) = S (cid:40) [ n ] (with [ n ] = { , . . . , n } ), none of the 2 n − σ : [ n ] → [ n ] we consider the corresponding permutation ma-trix y ∈ { , } n × n (satisfying y ij = 1 if and only if σ ( i ) = j ) rather thanthe vector ( σ (1) , . . . , σ ( n )), we obtain a much smaller description of the re-sulting polytope, since, according to Birkhoff [8] and von Neumann [34], theconvex hull P birk ( n ) (the Birkhoff-Polytope ) of all n × n -permutation matri-ces is equal to the set of all doubly-stochastic n × n -matrices (i.e., nonnegative n × n -matrices all of whose row- and column sums are equal to one). It is easyto see that the permutahedron P perm ( n ) is a linear projection of the Birkhoff-polytope P birk ( n ) via the map defined by p ( y ) i = (cid:80) nj =1 jy ij . Since, for everylinear objective function vector c ∈ R n , we have max {(cid:104) c, x (cid:105) : x ∈ P perm ( n ) } =max { (cid:80) ni =1 (cid:80) nj =1 jc i y ij : y ∈ P birk ( n ) } , one can use P birk ( n ) (that can be de-scribed by the n nonnegativity-inequalities) instead of P perm ( n ) (whose de-scription requires 2 n − extension of a polytope P ⊆ R n is a polyhedron Q ⊆ R d (i.e., an intersection of finitely many affine hyperplanes and halfspaces) togetherwith a linear projection p : R d → R n satisfying P = p ( Q ). Any descriptionof Q by linear equations and linear inequalities then (together with p ) is an extended formulation of P . The size of the extended formulation is the numberof inequalities in the description. Note that we neither account for the numberof equations (we can get rid of them by eliminating variables) nor for the numberof variables (we can ensure that there are not more variables than inequalitiesby projecting Q to the orthogonal complement of its lineality space , where thelatter is the space of all directions of lines contained in Q ). If T ∈ R n × d is thematrix with p ( y ) = T y , then, for every c ∈ R n , we have max {(cid:104) c, x (cid:105) : x ∈ P } =max {(cid:104) T t c, y (cid:105) : y ∈ Q } .In the example described above, P birk ( n ) thus provides an extended formu-lation of P perm ( n ) of size n . It is not known whether one can do somethingsimilar for the matching polytopes P match ( n ) (we will be back to this questionin Section 4.2). However there are many other examples of nice and small ex-tended formulations for polytopes associated with combinatorial optimizationproblems. The aim of this article (that has appeared in [1]) is to show a fewof them and to shed some light on the geometric, combinatorial and algebraicbackground of this concept that recently has received increased attention. Thepresentation is not meant to be a survey (for this purpose, we refer to Vander-beck and Wolsey [33] as well as to Cornu´ejols, Conforti, and Zambelli [12]) but2ather an appetizer for investigating alternative possibilities to express combi-natorial optimization problems by means of linear programs.While we will not be concerned with practical aspects here, extended formu-lations have also proven to be useful in computations. You can find more on thisin Laurence Wolsey’s discussion column in [1]. Fundamental work with respectto understanding the concept of extended formulations and its limits has beendone by Mihalis Yannakakis in his 1991-paper Expressing Combinatorial Opti-mization Problems by Linear Programs [36] (see Sect. 3.3 and 4). He discussessome of his thoughts on the subject in another discussion column in [1].
The spanning tree polytope P spt ( n ) associated with the complete graph K n =( V n , E n ) on n nodes is the convex hull of all characteristic vectors of spanningtrees, i.e., of all subsets of edges that form connected and cycle-free subgraphs.In another seminal paper, Edmonds [15] proved that P spt ( n ) is the set of all x ∈ R E n + that satisfy the equation x ( E n ) = n − x ( E n ( S )) ≤| S | − S ⊆ V n with 2 ≤ | S | < n . Again, none of the exponentially manyinequalities is redundant.However, by introducing additional variables z v,w,u for all ordered triples( v, w, u ) of pairwise different nodes meant to encode whether the edge { v, w } is contained in the tree and u is in the component of w when removing { v, w } from the tree, it turns out that the system consisting of the equations x { v,w } − z v,w,u − z w,v,u = 0 and x { v,w } + (cid:80) u ∈ [ n ] \{ v,w } z v,u,w = 1 (for all pairwise dif-ferent v, w, u ∈ V n ) along with the nonnegativity constraints and the equation x ( E n ) = n − spt ( n ) of size O( n ) (withorthogonal projection to the space of x -variables). This formulation is due toMartin [25] (see also [36, 12]). You will find an alternative one in LaurenceWolsey’s discussion column below. If P i ⊆ R n is a polytope for each i ∈ [ q ], then clearly P = conv( P ∪ · · · ∪ P q ) isa polytope as well, but, in general, it is difficult to derive a description by linearequations and inequalities in R n from such descriptions of the polytopes P i .However constructing an extended formulation for P in this situation is verysimple. Indeed suppose that each P i is described by a system A i x ≤ b i of f i lin-ear inequalities (where, in order to simplify notation, we assume that equationsare written, e.g., as pairs of inequalities). Then the system A i z i ≤ λ i b i for all i ∈ [ q ], (cid:80) qi =1 λ i = 1, λ ≥ O with variables z i ∈ R n for all i ∈ [ q ] and λ ∈ R q is an extended formulation for P of size f + · · · + f q + q , where the projectionis given by ( z , . . . , z q , λ ) (cid:55)→ z + · · · + z q . This has been proved first by Balas(see, e.g., [4]), even for polyhedra that are not necessarily polytopes (where in3his general case P needs to be defined as the topological closure of the convexhull of the union). When a combinatorial optimization problem can be solved by a dynamic pro-gramming algorithm, one often can derive an extended formulation for the as-sociated polytope whose size is roughly bounded by the running time of thealgorithm.A simple example is the 0/1-Knapsack problem, where we are given a non-negative integral weight vector w ∈ N n , a weight bound W ∈ N , and a profitvector c ∈ R n , and the task is to solve max {(cid:104) c, x (cid:105) : x ∈ F ( w, W ) } with F ( w, W ) = { x ∈ { , } n : (cid:104) w, x (cid:105) ≤ W } . A classical dynamic programmingalgorithm works by setting up an acyclic directed graph with nodes s = (0 , t , and ( i, ω ) for all i ∈ [ n ], ω ∈ { , , . . . , W } and arcs from ( i, ω ) to ( i (cid:48) , ω (cid:48) )if and only if i < i (cid:48) and ω (cid:48) = ω + w i (cid:48) , where such an arc would be assignedlength c i (cid:48) , as well as arcs from all nodes to t (of length zero). Then solving the0/1-Knapsack problem is equivalent to finding a longest s - t -path in this acyclicdirected network, which can be carried out in linear time in the number α ofarcs.The polyhedron Q ⊆ R α + of all s - t -flows of value one in that network equalsthe convex hull of all characteristic vectors of s - t -paths (due to the total unimod-ularity of the node-arc incidence matrix), thus it is easily seen to be mapped tothe associated Knapsack-polytope P knap ( w, W ) = conv( F ( w, W )) via the pro-jection given by y (cid:55)→ x , where x i is the sum of all components of y indexed byarcs pointing to nodes of type ( i, (cid:63) ). As Q is described by nonnegativity con-straints, the flow-conservation equations on the nodes different from s and t andthe equation ensuring an outflow of value one from s , these constraints providean extended formulation for P knap ( w, W ) of size α .However quite often dynamic programming algorithms can only be formu-lated as longest-paths problems in acyclic directed hyper graphs with hyperarcsof the type ( S, v ) (with a subset S of nodes) whose usage in the path representsthe fact that the optimal solution to the partial problem represented by node v has been constructed from the optimal solutions to the partial problems repre-sented by the set S . Martin, Rardin, and Campbell [26] showed that, under thecondition that one can assign appropriate reference sets to the nodes, also inthis more general situation nonnegativity constraints and flow-equations sufficeto describe the convex hull of the characteristic vectors of the hyperpaths. Thisgeneralization allows one to derive polynomial size extended formulations formany of the combinatorial optimization problems that can be solved in polyno-mial time by dynamic programming algorithms. A common generalization of the techniques to construct extended formulationsby means of disjunctive programming or dynamic programming is provided by4 ranched polyhedral systems ( BPS ) [21]. In this framework, one starts from anacyclic directed graph that has associated with each of its non-sink nodes v apolyhedron in the space indexed by the out-neighbors of v . From these buildingblocks, one constructs a polyhedron in the space indexed by all nodes. Undercertain conditions one can derive an extended formulation for the constructedpolyhedron from extended formulations of the polyhedra associated with thenodes.Some very nice extended formulations have recently been given by Faenza,Oriolo, and Stauffer [17] for stable set polytopes of claw-free graphs. Herethe crucial step is to glue together descriptions of stable set polytopes of certainbuilding block graphs by means of strip compositions . One of their constructionscan be obtained by applying the BPS-framework, though apparently the mostinteresting one they have cannot.An asymptotically smallest possible extended formulation of size O( n log n )for the permutahedron P perm ( n ) has been found by Goemans [19]. His con-struction relies on the existence of sorting networks of size r = O( n log n ) (Aj-tai, Koml´os, and Szemer´edi [2]), i.e., sequences ( i , j ) , . . . , ( i r , j r ) for whichthe algorithm that in each step s swaps elements a i s and a j s if and only if a i s > a j s sorts every sequence ( a , . . . , a n ) ∈ R into non-decreasing order. Theconstruction principle of Goemans has been generalized to the framework of reflection relations [22], which, for instance, can be used to obtain small ex-tended formulations for all G -permutahedra of finite reflection groups G (see,e.g., Humphreys [20]), including extended formulations of size O(log m ) of reg-ular m -gons, previously constructed by Ben-Tal and Nemirovski [7]. Anotherapplication of reflection relations yields extended formulations of size O( n log n )for Huffman-polytopes , i.e., the convex hulls of the leaves-to-root-distances vec-tors in rooted binary trees with n labelled leaves. Note that linear descriptionsof these polytopes in the original spaces are very large, rather complicated, andunknown (see Nguyen, Nguyen, and Maurras [28]).The list of combinatorial problems for which small (and nice) extended for-mulations have been found comprises many others, among them perfect match-ing polytopes of planar graphs (Barahona [6]), perfectly matchable subgraphpolytopes of bipartite graphs (Balas and Pulleyblank [5]), stable-set polytopesof distance claw-free graphs (Pulleyblank and Shepherd [30]), packing and parti-tioning orbitopes [16], subtour-elimination polytopes (Yannakakis [36] and, forplanar graphs, Rivin [32], Cheung [10]), and certain mixed-integer programs(see, e.g., Conforti, di Summa, Eisenbrand, and Wolsey [13]). Any intersection of a polyhedron P with the boundary hyperplane of some affinehalfspace containing P is called a face of P . The empty set and P itself are5onsidered to be (non-proper) faces of P as well. For instance, the proper facesof a three-dimensional polytope are its vertices, edges, and the polygons thatmake up the boundary of P . Partially ordered by inclusion, the faces of apolyhedron P form a lattice L ( P ), the face lattice of P . The proper faces thatare maximal with respect to inclusion are the facets of P . Equivalently, thefacets of P are those faces whose dimension is one less than the dimension of P .An irredundant linear description of P has exactly one inequality for each facetof P .If Q ⊆ R d is an extension of the polytope P ⊆ R n with a linear projection p : R d → R n , then mapping each face of P to its preimage in Q under p definesan embedding of L ( P ) into L ( Q ). Figure 3.1 illustrates this embedding for thetrivial extension Q = { y ∈ R V + : (cid:80) x ∈ X y x = 1 } of P = conv( X ) via p ( y ) = (cid:80) x ∈ X y x x for X = { e , − e , . . . , e , − e } (thus P is the cross-polytope in R with 16 facets and Q is the standard-simplex in R with 8 facets). As this figureFigure 1: Embedding of the face lattice of the 4-dimensional cross-polytope intothe face lattice of the 7-dimensional simplex.suggests, constructing a small extended formulation for a polytope P means tohide the facets of P in the fat middle part of the face lattice of an extensionwith few facets. Let P = { x ∈ A : Ax ≤ b } ⊆ R n be a polytope with affine hull A = aff( P ), A ∈ R m × n , and b ∈ R m . The affine map ϕ : A → R m with ϕ ( x ) = b − Ax (the slack map of P w.r.t. Ax ≤ b ) is injective. We denote its inverse (the inverseslack map ) on its image, the affine subspace ˜ A = ϕ ( A ) ⊆ R m , by ˜ ϕ : ˜ A → A .6he polytope ˜ P = ˜ A ∩ R m + , the slack-representation of P w.r.t. Ax ≤ b , isisomorphic to P with ϕ ( P ) = ˜ P and ˜ ϕ ( ˜ P ) = P .If Z ⊆ R m + is a finite set of nonnegative vectors whose convex conic hull ccone( Z ) = { (cid:80) z ∈ Z λ z z : λ ≥ O } ⊆ R m + contains ˜ P = ˜ A ∩ R m + , then we have˜ P = ˜ A ∩ ccone( Z ), and thus, the system (cid:80) z ∈ Z λ z z ∈ ˜ A and λ z ≥ z ∈ Z ) provides an extended formulation of P of size | Z | via the projection λ (cid:55)→ ˜ ϕ ( (cid:80) z ∈ Z λ z z ). Let us call such an extension a slack extension and theset Z a slack generating set of P (both w.r.t. Ax ≤ b ).Now suppose conversely that we have any extended formulation of P of size q defining an extension Q that is pointed (i.e., the polyhedron Q does not containa line). As for polytopes above (which in particular are pointed polyhedra),we can consider a slack representation ˜ Q ⊆ R q of Q and the correspondinginverse slack map ˜ ψ . Then we have ϕ ( p ( ˜ ψ ( ˜ Q ))) = ˜ P , where p is the projectionmap of the extension. If the system Ax ≤ b is binding for P , i.e., each of itsinequalities is satisfied at equation by some point from P , then one can show(by using strong LP-duality) that there is a nonnegative matrix T ∈ R m × q + with ϕ ( p ( ˜ ψ (˜ z ))) = T ˜ z for all ˜ z ∈ ˜ Q , thus ˜ P = T ˜ Q . Hence the columns of T form aslack generating set of P (w.r.t. Ax ≤ b ), yielding a slack extension of size q . Asevery non-pointed extension of a polytope can be turned into a pointed one ofthe same size by projection to the orthogonal complement of the lineality space,we obtain the following result, where the extension complexity of a polytope P is the smallest size of any extended formulation of P . Theorem 1 ([18])
The extension complexity of a polytope P is equal to theminimum size of all slack extensions of P . As every slack extension of a polytope is bounded (and since all bounded poly-hedra are polytopes), Theorem 1 implies that the extension complexity of apolytope is attained by an extension that is a polytope itself. Furthermore,in Theorem 1 one may take the minimum over the slack extensions w.r.t. anyfixed binding system of inequalities describing P . In particular, all these minimaconcide. Now let P = conv( X ) = { x ∈ aff( P ) : Ax ≤ b } ⊆ R n be a polytope withsome finite set X ⊆ R n and A ∈ R m × n , b ∈ R m . The slack matrix of P w.r.t. X and Ax ≤ b is Φ ∈ R [ m ] × X + with Φ i,x = b − (cid:104) A i,(cid:63) , x (cid:105) . Thus the slackrepresentation ˜ P ⊆ R m of P (w.r.t. Ax ≤ b ) is the convex hull of the columnsof Φ. Consequently, if the columns of a nonnegative matrix T ⊆ R [ m ] × [ f ]+ forma slack generating set of P , then there is a nonnegative matrix S ∈ R [ f ] × X + withΦ = T S . Conversely, for every factorization Φ = T (cid:48) S (cid:48) of the slack matrix intononnegative matrices T (cid:48) ∈ R [ m ] × [ f (cid:48) ]+ and S (cid:48) ∈ R [ f (cid:48) ] × X + , the columns of T (cid:48) forma slack generating set for P .Therefore constructing an extended formulation of size f for P amounts tofinding a factorization of the slack matrix Φ = T S into nonnegative matrices T f columns and S with f rows. In particular, we have derived the followingresult that essentially is due to Yannakakis [36] (see also [18]). Here, the non-negative rank of a matrix is the minumum number f such that the matrix canbe written as a product of two nonnegative matrices, where the first one has f columns and the second one has f rows. Theorem 2
The extension complexity of a polytope P is equal to the nonnega-tive rank of its slack matrix (w.r.t. any set X and binding system Ax ≤ b with P = conv( X ) = { x ∈ aff( P ) : Ax ≤ b } ). Clearly, the nonnegative rank of a matrix is bounded from below by its usualrank as known from Linear Algebra. There is also quite some interest in the nonnegative rank of (not necessarily slack) matrices in general (see, e.g., Cohenand Rothblum [11]).
Every extension Q of a polytope P has at least as many faces as P , as the facelattice of P can be embedded into the face lattice of Q (see Sect. 3.1). Since eachface is the intersection of some facets, one finds that the extension complexityof a polyhedron with β faces is at least log β (the binary logarithm of β ). Thisobservation has first been made by Goemans [19] in order to argue that theextension complexity of the permutahedron P perm ( n ) is at least Ω( n log n ).Suppose that Φ = T S is a factorization of a slack matrix Φ of the polytope P into nonnegative matrices T and S with columns t , . . . , t f and rows s , . . . , s f ,respectively. Then we can write Φ = (cid:80) fi =1 t i s i as the sum of f nonnegativematrices of rank one. Calling the set of all non-zero positions of a matrix its support , we thus find that the nonnegative factorization Φ = T S provides away to cover the support of Φ by f rectangles , i.e., sets of the form I × J ,where I and J are subsets of the row- and column-indices of Φ, respectively.Hence, due to Theorem 2, the minimum number of rectangles by which one cancover the support of Φ yields a lower bound (the rectangle covering bound ) onthe extension complexity of P (Yannakakis [36]). Actually, the rectangle cov-ering bound dominates the bound discussed in the previous paragraph [18]. AsYannakakis [36] observed furthermore, the logarithm of the rectangle coveringbound of a polytope P is equal to the nondeterministic communication com-plexity (see, e.g., the book of Kushilevitz and Nisan [24]) of the predicate onthe pairs ( v, f ) of vertices v and facets f of P that is true if and only if v (cid:54)∈ f .One can equivalently describe the rectangle covering bound as the minimumnumber of complete bipartite subgraphs needed to cover the vertex-facet-non-incidence graph of the polytope P . A fooling set is a subset F of the edgesof this graph such that no two of the edges in F are contained in a completebipartite subgraph. Thus every fooling set F proves that the rectangle coveringbound, and hence, the extension complexity of P , is at least | F | . For instance,8or the n -dimensional cube it is not too difficult to come up with a fooling setof size 2 n , proving that for a cube one cannot do better by allowing extendedformulations for the representation. For more details on bounds of this type werefer to [18].Unfortunately, all in all the currently known techniques for deriving lowerbounds on extension complexities are rather limited and yield mostly quiteunsatisfying bounds. Asking, for instance, about the extension complexity of the matching polytopeP match ( n ) defined in the beginning, one finds that not much is known. It mightbe anything between quadratic and exponential in n . However, in the main partof his striking paper [36], Yannakakis established an exponential lower boundon the sizes of symmetric extended formulations of P match ( n ). Here, symmetric means that the extension polyhedron remains unchanged when renumbering thenodes of the complete graph, or more formally that, for each permutation π ofthe edges of the complete graph that is induced by a permutation of its nodes,there is a permutation κ π of the variables of the extended formulation that mapsthe extension polyhedron to itself such that, for every vector y in the extendedspace, applying π to the projection of y yields the same vector as projectingthe vector obtained from y by applying κ π . Indeed, many extended formula-tions are symmetric in a similar way, for instance the extended formulation ofthe permutahedron by the Birkhoff-polytope mentioned in the Introduction aswell as the extended formulation for the spanning tree polytope discussed inSection 2.1.In order to state Yannakakis’ result more precisely, denote by M (cid:96) ( n ) the setof all matchings of cardinality (cid:96) in the complete graph with n nodes, and byP match (cid:96) ( n ) = conv { χ ( M ) : M ∈ M (cid:96) ( n ) } the associated polytope. In particular,P match n/ ( n ) is the perfect-matching-polytope (for even n ). Theorem 3 (Yannakakis [36])
For even n , the size of every symmetric ex-tended formulation of P match n/ ( n ) is at least Ω( (cid:0) n (cid:98) ( n − / (cid:99) (cid:1) ) . Since P match (cid:98) n/ (cid:99) ( n ) is (isomorphic to) a face of P match ( n ), one easily derives theabove mentioned exponential lower bound on the sizes of symmetric extendedformulations for P match ( n ) from Theorem 3.At the core of his beautiful proof of Theorem 3, Yannakakis shows that, foreven n , there is no symmetric extended formulation in equation form (i.e., withequations and nonnegativity constraints only) of P match n/ ( n ) of size at most (cid:0) nk (cid:1) with k = (cid:98) ( n − / (cid:99) . From such a hypothetical extended formulation EF ,he first constructs an extended formulation EF in equation form on variables y A for all matchings A with | A | ≤ k such that the 0/1-vector valued map s (cid:63) on the vertices of P match n/ ( n ) defined by s (cid:63) ( χ ( M )) A = 1 if and only if A ⊆ M is a section of EF , i.e., s (cid:63) ( x ) maps every vertex x to a preimage under theprojection of EF that is contained in the extension polyhedron. Then it turns9ut that an extended formulation like EF cannot exist. In fact, for an arbitrarypartitioning of the node set into two parts V and V with | V | = 2 k + 1, onecan construct a nonnegative point y (cid:63) in the affine hull of the image of s (cid:63) (thus y (cid:63) is contained in the extension polyhedron of EF that is defined by equationsand nonnegativity constraints only) with y (cid:63) { e } = 0 for all edges e connecting V and V , which implies that the projection of the point y (cid:63) violates the inequality x ( δ ( V )) ≥ match n/ ( n ) (since | V | = 2 k + 1 is odd). The crucialingredient for constructing EF from EF is a theorem of Bocherts’ [9] statingthat every subgroup G of permutations of m elements that is primitive with | G | > m ! / (cid:98) ( m + 1) / (cid:99) ! contains all even permutations. Yannakakis constructs asection s for EF for that he can show—by exploiting Bochert’s theorem—thatthere is a nonnegative matrix C with s ( χ ( M )) = C · s (cid:63) ( χ ( M )) for all M ∈M n/ ( n ), which makes it rather straight forward to construct EF from EF .With respect to the fact that his proof yields an exponential lower boundonly for symmetric extended formulations, Yannakakis [36] remarked “we do notthink that asymmetry helps much” in constructing small extended formulationsof the (perfect) matching polytopes and stated as an open problem to “provethat the matching (. . . ) polytopes cannot be expressed by polynomial size LP’swithout the symmetry assumption”. As indicated above, today we still do notknow whether this is possible. However, at least it turned out recently thatrequiring symmetry can make a big difference for the smallest possible size ofan extended formulation. Theorem 4 ([23])
All symmetric extended formulations of P match (cid:98) log n (cid:99) ( n ) havesize at least n Ω(log n ) , while there are polynomial size non-symmetric extendedformulations for P match (cid:98) log n (cid:99) ( n ) (i.e., the extension complexity of P match (cid:98) log n (cid:99) ( n ) is boun-ded from above by a polynomial in n ). Thus, at least when considering matchings of size (cid:98) log n (cid:99) instead of perfect(or arbitrary) matchings, asymmetry indeed helps much.While the proof of the lower bound on the sizes of symmetric extendedformulations stated in Theorem 4 is a modification of Yannakakis’ proof indi-cated above, the construction of the polynomial size non-symmetric extendedformulation of P match (cid:98) log n (cid:99) ( n ) relies on the principle of disjunctive programming(see Section 2.2). For an arbitrary coloring ζ of the n nodes of the completegraph with 2 k colors, we call a matching M (with | M | = k ) ζ -colorful if, ineach of the 2 k color classes, there is exactly one node that is an end-node ofone of the edges from M . Let us denote by P ζ the convex hull of the char-acteristic vectors of ζ -colorful matchings. The crucial observation is that P ζ can be described by O(2 k + n ) inequalities (as opposed to Ω(2 n ) inequalitiesneeded to describe the polytope associated with all matchings, see the Intro-duction). On the other hand, according to a theorem due to Alon, Yuster, andZwick [3], there is a family of q such colorings ζ , . . . , ζ q with q = 2 O( k ) log n such that, for every 2 k -element subset W of the n nodes, in at least one ofthe colorings the nodes from W receive pairwise different colors. Thus we haveP match k ( n ) = conv( P ζ ∪ · · · ∪ P ζ q ), and hence (as described in Section 2.2) we10btain an extended formulation of P match k ( n ) of size 2 O( k ) n log n , which, for k = (cid:98) log n (cid:99) , yields the upper bound in Theorem 4.Yannakakis [36] moreover deduced from Theorem 3 that there are no polyno-mial size symmetric extended formulations for the traveling salesman polytope(the convex hull of the characteristic vectors of all cycles of lengths n in thecomplete graph with n nodes). Similarly to Theorem 4, one can also prove thatthere are no polynomial size symmetric extended formulations for the polytopesassociated with cycles of length (cid:98) log n (cid:99) , while these polytopes nevertheless havepolynomially bounded extension complexity [23].Pashkovich [29] further extended Yannakakis’ techniques in order to provethat every symmetric extended formulation of the permutahedron P perm ( n ) hassize at least Ω( n ), showing that the Birkhoff-polytope essentially provides anoptimal symmetric extension for the permutahedron. Many polytopes associated with combinatorial optimization problems can berepresented in small, simple, and nice ways as projections of higher dimensionalpolyhedra. Moreover, though we have not touched this topic here, sometimessuch extended formulations are also very helpful in deriving descriptions in theoriginal spaces. What we currently lack are on the one hand more techniques toconstruct extended formulations and on the other hand a good understandingof the fundamental limits of such representations. For instance, does everypolynomially solvable combinatorial optimization problem admit an extendedformulation of polynomial size? We even do not know this for the matchingproblem. How about the stable set problem in perfect graphs? The best upperbound on the extension complexity of these polytopes for graphs with n nodesstill is n O(log n ) (Yannakakis [36]).Progress on such questions will eventually shed more light onto the principlepossiblities to express combinatorial problems by means of linear constraints.Moreover, the search for extended formulations yields new modelling ideas someof which may prove to be useful also in practical contexts. In any case, work onextended formulations can lead into fascinating mathematics. Acknowldgements
We are grateful to Sam Burer, Samuel Fiorini, KanstantsinPashkovich, Britta Peis, Laurence Wolsey, and Mihalis Yannakakis for commentson a draft of this article and to Matthias Walter for producing Figure 3.1.
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