Uncapacitated Flow-based Extended Formulations
aa r X i v : . [ m a t h . O C ] J un Uncapacitated Flow-based Extended Formulations
Samuel Fiorini ∗ Kanstantsin Pashkovich † August 16, 2018
Abstract
An extended formulation of a polytope is a linear description of this polytope usingextra variables besides the variables in which the polytope is defined. The interest ofextended formulations is due to the fact that many interesting polytopes have extendedformulations with a lot fewer inequalities than any linear description in the original space.This motivates the development of methods for, on the one hand, constructing extendedformulations and, on the other hand, proving lower bounds on the sizes of extendedformulations.Network flows are a central paradigm in discrete optimization, and are widely usedto design extended formulations. We prove exponential lower bounds on the sizes of un-capacitated flow-based extended formulations of several polytopes, such as the (bipartiteand non-bipartite) perfect matching polytope and TSP polytope. We also give new ex-amples of flow-based extended formulations, e.g., for 0/1-polytopes defined from regularlanguages. Finally, we state a few open problems. An extended formulation (shorthand: EF ) of a polytope P ⊆ R d is a system of linear constraints E x + F y g , E = x + F = y = g = (1)with ( x, y ) ∈ R d + k such that x ∈ R d belongs to P if and only if there exists y ∈ R k such that( x, y ) satisfies (1). An extended formulation of P is simply a linear description of P in anextended space. Geometrically, P is described as the projection of the polyhedron Q ⊆ R d + k defined by (1). More generally, we call a polyhedron Q ⊆ R e an extension (or lift ) of P if thereexists an affine map π : R e → R d such that π ( Q ) = P .Consider a linear description Ax b of P in its original space. If f : R d → R is anyfunction, thensup { f ( x ) | Ax b } = sup { f ( x ) | E x + F y g , E = x + F = y = g = } . (2)Thus every optimization problem on P can be reformulated as an optimization problem overany extension of P . This is why extended formulations are interesting for optimization: in(2), the number of constraints in the right-hand side can be much smaller than the number ofconstraints in the left-hand side. ∗ Partially supported by
Fonds National de la Recherche Scientifique (F.R.S.-FNRS) and the
Actions deRecherche Concert´ees (ARC) fund of the French community of Belgium. † Supported by the Progetto di Eccellenza 2008-2009 of the Fondazione Cassa Risparmio di Padova e Rovigo. We remark that although we allow for now Q to be unbounded, we will soon show that one can restrict tothe case where Q is bounded, that is, a polytope.
1e define the size of an extended formulations as its number of inequalities, and the sizeof an extension as its number of facets; these turn out to be the right measures of size. Notethat the size of an extended formulation is at least the size of the associated extension becauseevery facet of a polyhedron is part of every linear description of this polyhedron (in the space inwhich it is defined), and every extension corresponds to an extended formulation with exactlyits size.The field of extended formulations is attracting more and more attention. In particular,size lower-bounding techniques are becoming increasingly powerful and diverse, see, e.g., [32,21, 6, 17, 18, 5, 8, 7]. The reader will find in the surveys [12, 22, 31] a good description of thefield as it was a few years ago.In this paper, we study some restricted forms of extended formulations (extensions) whichwe call flow-based extended formulations (extensions) , see Section 3 for a definition. Informally,a flow-based extension of a polytope P is another polytope Q that can be realized as the convexhull of all flows in some network. This definition is inspired by the prominent role played bynetwork flows in discrete optimization: many algorithms and structural results crucially relyon network flows [1, 27]. Quite a lot of known extended formulations are based on networkflows, such as those obtained from dynamic programming algorithms [24].Here, we focus on uncapacitated networks. Our main contribution is to prove size lowerbounds of the form 2 Ω( n ) for uncapacitated flow-based extended formulations of several poly-topes, such as the perfect matching polytope of (bipartite and non-bipartite) complete graphsand the traveling salesman polytope of the complete graph. Our results are summarized inTable 1. Below, the notations O ∗ ( · ), Ω ∗ ( · ) and Θ ∗ ( · ) have the same meaning as the usualnotations O ( · ), Ω( · ) and Θ( · ), except that polynomial factors are ignored.Polytope Size bounds for general EFs Size bounds for flow-based EFsP perfect matching ( K n,n ) Θ( n ) [3] Θ ∗ ( n )P perfect matching ( K n ) Ω( n ), O ∗ (2 n ) [21, 15] Ω ∗ ( n2 ), O ( . )P traveling salesman ( K n ) 2 Ω( √ n ) [18], O ∗ (2 n ) [20] Ω ∗ ( n4 ), O ∗ (2 n ) [20]Table 1: Table of results. New results are indicated in boldface. The bounds for flow-basedEFs assume that the network is uncapacitated.Before giving an outline of the paper, we briefly discuss our motivations. Lower bounds onrestricted types of extended formulations have been studied by quite many authors, startingwith the work of Yannakakis [32] on symmetric extended formulations. There has been workon hierarchies such as the Sherali-Adams [29] and Lov´asz-Schrijver hierarchies [23], see, e.g.,[9, 26, 16, 11, 19, 2]; further work on symmetric extended formulations [21, 25, 6] and also workon extended formulations from low variance protocols [15].We think that the restriction of being flow-based is as natural as the restrictions studiedin the aforementioned papers. Combinatorial optimization offers a variety of modeling toolsbeyond flows, which are the most basic and important modeling tool: e.g., matchings, polyma-troids and polymatroid intersections [27]. It seems a worthy research goal to characterize theexpressivity of these modeling tools, and give theoretical explanations of the fact that someproblems can be efficiently expressed by some modeling tools and not by others. This paper isa first step in that direction.Of particular interest are separations between modeling tools. It is striking that all our lowerbounds rely on a separation between uncapacitated and capacitated flows: while the perfectmatching polytope of the complete bipartite graph K n,n has a O ( n )-size capacitated flow-basedextended formulation, we show a Ω ∗ (2 n ) lower bound on the size of every uncapacitated flow-based extended formulations of that polytope. Via reductions, we derive from this the otherlower bounds reported in Table 1. 2e conclude this discussion by focussing on the traveling salesman polytope. Held andKarp [20] gave a O ∗ (2 n )-complexity dynamic programming algorithm for the traveling sales-man problem based on subsets. In our terminology, this yields a O ∗ (2 n )-size uncapacitatedflow-based extended formulation for the traveling salesman polytope. In a survey paper onexact algorithms for combinatorial optimization problems, Woeginger [30] stated as an openproblem the question of determining if the traveling salesman problem has an exact algorithmof complexity (2 − ε ) n for some ε >
0. The question was answered affirmatively by Bjork-lund [4], at least if one tolerates randomized algorithms with small failure probability andrestricts to instances where the coefficients are bounded. Our Ω ∗ (2 n ) lower bound for unca-pacitated flow-based extended formulations for the traveling salesman polytope also applies todynamic programming algorithms for the traveling salesman problem, which sheds some lighton Woeginger’s question.The rest of the paper is organized as follows. We begin with preliminaries in Section 2:after introducing some notations, we define convex polytopes in general as well as the particu-lar convex polytopes studied here. Then, in Section 3, we formally define flow-based extendedformulations, discuss an example and establish basic properties of flow-based extended formu-lations, focussing on the uncapacitated case. Finally, in Section 4, we prove size bounds foruncapacitated flow-based extended formulations described in Table 1. Let I be a finite ground set. The incidence vector of a subset J ⊆ I is the vector χ J ∈ R I defined as χ Ji = (cid:26) i ∈ J i / ∈ J for i ∈ I . For x ∈ R I , we let x ( J ) := P i ∈ J x i .First, let G = ( V, E ) be an undirected graph. For a subset of vertices U ⊆ V , we denote as δ ( U ) the set of edges of G with exactly one endpoint in U . So, δ ( U ) = { uv ∈ E : u ∈ U, v / ∈ U } . Now, let N = ( V, A ) be a directed graph. For U ⊆ V , we denote by δ + ( U ) the set of arcsof N with tail in U and head in V \ U , and by δ − ( U ) the set of arcs of N with head in U andtail in V \ U , i.e. δ + ( U ) = { ( u, v ) ∈ A : u ∈ U, v / ∈ U } , and δ − ( U ) = { ( v, u ) ∈ A : u ∈ U, v / ∈ U } . As usual, for v ∈ V , we use the shortcuts δ ( v ), δ + ( v ) and δ − ( v ) for δ ( { v } ), δ + ( { v } ) and δ − ( { v } )respectively. A (convex) polytope is a set P ⊆ R d that is the convex hull of a finite set of points in R d .Equivalently, P ⊆ R d is a polytope if and only if P is bounded and the intersection of a finitecollection of closed halfspaces. This is equivalent to saying that P is bounded and the set ofsolutions of a finite system of linear inequalities (or equalities, each of which can be representedby a pair of inequalities). A (convex) polyhedron is similar to a polytope, except that it maybe unbounded. Formally, a polyhedron Q ⊆ R d is any set that can be represented as theMinkowski sum of a polytope and a polyhedral cone or, equivalently, as the intersection of afinite collection of closed halfspaces. For more background on polytopes and polyhedra, see thestandard reference [33]. 3 .2 Perfect Matching Polytope A perfect matching of an undirected graph G = ( V, E ) is set of edges M ⊆ E such that everyvertex of G is incident to exactly one edge in M . The perfect matching polytope of the graph G is the convex hull of the incidence vectors of the perfect matchings of G, i.e.,P perfect matching ( G ) = conv { χ M ∈ R E : M perfect matching of G } . Edmonds [14] showed that the perfect matching polytope of G is described by the followingsystem of linear constraints (see also [28], page 438): x ( δ ( U )) > U ⊆ V with | U | odd , (3) x ( δ ( v )) = 1 for v ∈ V ,x e > e ∈ E .
In the case where the graph G is bipartite, that is, when the vertex set V can be partitionedinto two sets A and B such that every edge in E has an endpoint in A and the other in B , theodd cut inequalities (3) may be dropped [3]. Thus the perfect matching polytope of a bipartitegraph G is described as follows: x ( δ ( v )) = 1 for v ∈ V ,x e > e ∈ E . A Hamiltonian cycle of G = ( V, E ) is a connected subgraph of G such that every vertex of G is incident to exactly two edges in C . The traveling salesman polytope of the graph G is theconvex hull of the incidence vectors of the hamiltonian cycles of G, i.e.,P traveling salesman ( G ) = conv { χ E ( C ) ∈ R E : C Hamiltonian cycle of G } . In the formula above, E ( C ) denotes the edge set of Hamiltonian cycle C .No linear description of the traveling salesman polytope of the complete graph K n is known.Moreover no “reasonable” linear description of this polytope should be expected unless N P =co-
N P (see Corollary 5.16a [28]).
Let N = ( V, A ) be a network with source node s ∈ V , sink node t ∈ V \ { s } and arc capacities c a ∈ R + ∪ {∞} for a ∈ A . An s – t flow of value k is a vector φ ∈ R A satisfying φ ( δ + ( v )) − φ ( δ − ( v )) = 0 ∀ v ∈ V \ { s, t } , (4) φ ( δ + ( s )) − φ ( δ − ( s )) = k, (5) φ a > ∀ a ∈ A, (6) φ a c a ∀ a ∈ A. (7)For a fixed k ∈ R , the set of all s – t flows of value k in network N defines a polyhedron Q = Q ( V, A, s, t, k, c ) that we call flow polyhedron .In this paper, we will assume most of the time that the network is uncapacitated , that is, c a = ∞ for all a ∈ A . This amounts to ignoring the upper bound inequalities (7).4 Flow-based Extended Formulations
Consider again a network N = ( V, A ) with source node s ∈ V , sink node t ∈ V \ { s } , arccapacities c a ∈ R + ∪ {∞} for a ∈ A and flow value k ∈ R + . We say that the flow polyhedron Q = Q ( V, A, s, t, k, c ) is a flow-based extension of a given polytope P in R d if there exists alinear projection π : R A → R d such that π ( Q ) = P . A flow-based extension is said to be uncapacitated if the associated network is uncapacitated.From now on, we will always assume that the projection π is linear. This causes essentiallyno loss of generality because an affine projection can be made linear at the cost of adding onenew arc ( s ′ , s ) to the network and moving the source to the node s ′ . We denote by M ∈ R d × A the matrix of projection π , that is, the matrix M ∈ R d × A such that π ( φ ) = M φ for all φ ∈ R A .Moreover, we denote by F ∈ R ( V \{ s,t } ) × A the coefficient matrix of the flow balance equations.In other words, F φ = 0 is the matrix form of (4). Then, the flow-based extension Q can bedescribed algebraically as: x = M φ, F φ = 0 , φ ( δ + ( s )) − φ ( δ − ( s )) = k, φ c, (8)We call system (8) a flow-based extended formulation of P .Notice that in the uncapacitated case, the size (that is, number of inequalities) of a flow-based extended formulation is exactly the number of arcs in the corresponding network.Notice also that in the uncapacitated case, we can assume that k = 1 without loss ofgenerality. This is because changing k to 1 simply amounts to replacing Q by (1 /k ) Q . Indeed,if π : R A → R d projects Q to P , then π ′ : R A → R d : φ π ′ ( φ ) := π ( kφ ) projects (1 /k ) Q to P . (In case k = 0, Q is just a point. We will ignore this case in what follows.)We will prove below that in the uncapacitated case, we can furthermore assume that N is acyclic, provided ∅ ( P ⊆ R d + . In this case, Q is a polytope and its vertices are thecharacteristic vectors χ σ of all directed s – t paths σ in network N (this follows from the well-known fact that the system (4)–(6) defining Q is totally unimodular). We call such an extensionan s – t path extension , any corresponding extended formulation an s – t path extended formulation and define the s – t path extension complexity xc s – t path ( P ) of a polytope P as the minimumnumber of arcs of a network whose s – t path polytope is an extension of P . We will show thatthis is also the minimum size of an uncapacitated flow-based extended formulation of P . In order to convince the reader that s – t path extensions are quite powerful, we now discuss anillustrating example that generalizes Carr and Konjevod’s flow-based extended formulation ofthe convex hull of even 0/1-vectors in R n [10].Consider a deterministic finite automaton M over the alphabet { , } , that is, a 4-tuple( Q, δ, q , F ) where Q is now a (nonempty) finite set of states , δ : Q × { , } → Q is the transition function , q ∈ Q is the initial state and F ⊆ Q is the set of accept states . For a giveninput word x = x x · · · x n in { , } ∗ , the automaton M performs a computation starting atthe initial state q and in which the state q i ( i ∈ [ n ]) is determined by the previous state q i − and the i th letter x i of word x through the equation q i = δ ( q i − , x i ). The automaton is said to accept x if the final state q n is an accept state, that is, q n belongs to F .The automaton M defines a language L = L ( M ) over { , } consisting of all words x ∈{ , } ∗ accepted by M . Such a language is said to be regular . Now pick a positive integer n , and consider a word x = x x · · · x n of length n in L . Treating each letter of word x asbelonging to a different coordinate, we see that x defines a 0 / x , x , . . . , x n ) ⊺ in R n .5y taking the convex hull of all 0 / n in L , weobtain a 0 / P n ( L ) in R n .As we show now, one can easily construct compact flow-based extended formulations forsuch 0 / Proposition 1.
Let L denote a regular language over { , } and M = ( Q, δ, q , F ) any deter-ministic finite automaton recognizing the language L . For each positive integer n , there existsan s – t path extended formulation of P n ( L ) with size at most | Q | n .Proof. We define a network N from automaton M . Besides source node s and sink node t ,network N has n − q, q, n −
1) for each state q ∈ Q . To simplify notations,we also denote s by ( q , V of N . For i ∈ [ n − q, i −
1) to each of the nodes ( δ ( q, , i ) and ( δ ( q, , i ) by an arc. Moreover, for each transition q ′ = δ ( q, σ ) with q ′ ∈ F we add an arc from node ( q, n −
1) to sink node t . This defines thearc set A of N . See Figure 1 for an example. In a formula, we have (with a slight abuse ofnotation because the network can have parallel arcs) V = { ( q , | {z } = s } ∪ { ( q, i ) | q ∈ Q, i ∈ [ n − } ∪ { t } ,A = { (( q, i − , ( δ ( q, σ ) , i )) | ( q, i − ∈ N, i ∈ [ n − , σ ∈ { , }}∪ { (( q, n − , t ) | ∃ σ ∈ { , } : δ ( q, σ ) ∈ F } . Each arc a ∈ A corresponds to a transition q ′ = δ ( q, σ ), and is said to carry the label σ ∈ { , } .Thus the label carried by an arc is the symbol that caused the transition.10 0 0 01 11 1 1 11 1 10011 s t N = ( V, A ), we send k = 1 units of flow from s to t , setting all capacities c a to ∞ . The column of the projection matrix corresponding to arc a ∈ A from node ( q, i −
1) isthe 0 / , . . . , , σ, , . . . , ⊺ with σ in position i and 0 everywhere else, where σ ∈ { , } is the label carried by arc a . We leave it to the reader to perform the straightforward checkthat this defines an s - t path extended formulation of P n ( L ).The size of this extended formulation is the number of arcs in the network, that is,2 + 2 | Q | ( n − | Q | n. A linear projection π : R A → R d is called nonnegative if its projection matrix is (entry-wise)nonnegative. Lemma 2.
For every uncapacitated flow-based extension Q ⊆ R A , π : R A → R d of a polytope P ⊆ R d + , there is a nonnegative linear projection π ′ : R A → R d such that π ′ ( Q ) = P . roof. As above, let M denote the matrix of π . It suffices to show that for every row M i of thematrix M there exists a row vector Λ i ∈ ( R V \{ s,t } ) ∗ such that M i + Λ i F >
0, since due to (8)the system
F φ = 0 holds for all φ ∈ Q and thus ( M + Λ F ) φ = M φ + Λ
F φ = M φ .Suppose, for the sake of contradiction, that no such Λ i exists for some i . Then by Farkas’lemma, there exists a vector ψ ∈ R A such that F ψ = 0 , ψ > M i ψ < . Thus ψ is an s – t flow in N . Because the network is uncapacitated, we can assume that thevalue of ψ is precisely k , by scaling ψ if necessary, hence ψ ∈ Q . Now, the inequality M i ψ < i th coordinate of the projection π ( ψ ) = M ψ is negative, which gives the desiredcontradiction.
The network associated to every minimum size uncapacitated flow-based extension Q ⊆ R A of a nonempty polytope P ⊆ R d + is acyclic.Proof. By Lemma 2 the projection π : φ M φ may be assumed nonnegative. Consider adirected cycle C in network N and the corresponding columns of M . Take a point φ ∈ Q andconsider the projection π ( φ + Kχ C ) where K ∈ R + . By linearity, π ( φ + Kχ C ) = π ( φ )+ Kπ ( χ C ).If π ( χ C ) is a non-zero vector and K is chosen large enough, π ( φ ) + Kπ ( χ C ) would be outsideof polytope P , a contradiction to the fact that φ + Kχ C satisfies (8) and thus lies in Q .Hence π ( χ C ) is a zero vector. Due to nonegativity of π , for every arc a ∈ A containedin at least one directed cycle, the corresponding column of M is zero, that is, π ( χ { a } ) = 0.Therefore, if N contains a directed cycle, we can contract every strongly connected componentof N to a node and obtain a smaller flow-based extension of P , a contradiction. Note that if s and t are in the same strongly connected component of N , in which case we are not allowed tocontract this component because we assume s = t , then necessarily P = { } and a minimumsize flow-based extension of P is given by a network with two nodes connected by a single arc.The result follows. Let the equation c x = δ be valid for a nonempty polytope P ⊆ R d . Then forevery node v in the network N = ( V, A ) associated to a minimum-size uncapacitated flow-basedextension Q ⊆ R A of P , there is a unique ǫ ∈ R such that c π ( χ σ ) = ǫ for every s – v path σ .Proof. Let σ , σ be two paths from source s to node v . Due to minimality of the extensionthere is also a path σ from v to t . Since σ ∪ σ and σ ∪ σ define paths from s to t , theprojections π ( χ σ ∪ σ ) and π ( χ σ ∪ σ ) lie in the polytope P , and thus satisfy the equation c x = δ .Therefore, 0 = c π ( χ σ ∪ σ ) − c π ( χ σ ∪ σ ) = c π ( χ σ ) − c π ( χ σ ) . To conclude the proof, we may define ǫ as the value c π ( χ σ ). For every polytope P = ∅ and face F of P , there holds xc s – t path ( P ) > xc s – t path ( F ) .Proof. Let Q be a minimum size s – t path extension of P and let N = ( V, A ) denote thecorresponding network. The polytope π − ( F ) ∩ Q is a face of Q . From the linear descriptionof Q , see (4)–(6), we infer π − ( F ) ∩ Q = { φ ∈ Q | φ a = 0 , a ∈ A ′ } A ′ ⊆ A . Hence, the s – t path polytope Q ′ associated with the network N ′ = ( V, A \ A ′ )together with the projection π defines an s – t path extension of face F . Because the size of theextension Q ′ of F is not larger than the size of the extension Q of P , we have xc s – t path ( F ) xc s – t path ( P ). Now we provide lower bounds on the size of uncapacitated flow-based extensions or, equivalently(by Lemmas 2 and 3), s – t path extensions of the (bipartite and non-bipartite) perfect matchingpolytope and traveling salesman polytope. We start by proving that the s – t path extension com-plexity of the perfect matching polytope of K n,n is Θ ∗ (2 n ). This is striking because this polytopehas Θ( n ) facets, and a size-Θ( n ) capacitated flow-based extension. Perhaps less striking areour exponential lower bounds for the perfect matching polytope and traveling salesman poly-tope of K n . We derive these by combining our lower bound on xc s – t path (P perfect matching ( K n,n ))and Lemma 5. Theorem 6.
Every uncapacitated flow-based extension (or, equivalently, s – t path extension) ofthe perfect matching polytope of the complete bipartite graph K n,n has size Ω (cid:16) n √ n (cid:17) .Proof. Due to Lemma 2, we may assume that the projection π : R A → R d is given by a linearnonnegative map. Consider an s – t path extension Q ⊆ R A with network N = ( V, A ) andnonnegative linear projection π : R A → R d .For each vertex u of K n,n , the equation x ( δ ( u )) = 1 ⇐⇒ X e ∈ δ ( u ) x e = 1is valid for P perfect matching ( K n,n ). From Lemma 4, we conclude that for every node v of N thereis a nonnegative vector ǫ v ∈ R n such that for every s – v path σ in the network N and everyvertex u of the graph K n,n the following holds: X e ∈ δ ( u ) π e ( χ σ ) = ǫ vu . We base our analysis on the support of ǫ v , which we denote supp( ǫ v ).Now consider a node v of network N . For every s – t path σ going through v and such that π ( χ σ ) = χ M for some perfect matching M of K n,n , matching M and cut δ (supp( ǫ v )) do nothave an edge in common.Hence if | supp( ǫ v ) | = n the s – t paths of N going through v define at most n ! n ! perfectmatchings M of K n,n .Moreover, for every arc a = ( v , v ) in N with | supp( ǫ v ) | = n < n and | supp( ǫ v ) | = n > n there are at most n ! n − n ! n ! n ! perfect matchings M such that there is an s – t path σ in N with a ∈ σ and χ M = π ( χ σ ), since in this case σ contains both nodes v and v and every suchmatching M must contain all the edges from the support of π ( χ { a } ).Since the polytope Q is an extension of P perfect matching ( K n,n ), for every perfect matching M in K n,n there is an s – t path σ such that χ σ projects to χ M . But since ǫ s is an all zero vectorand ǫ t is an all one vector, this path σ must go through a node v with | supp( ǫ v ) | = n or containan arc a = ( v , v ) with | supp( ǫ v ) | < n < | supp( ǫ v ) | .8ince the total number of perfect matchings in K n,n equals n !, network N contains at least n !2 n ! n ! = Ω (cid:18) n √ n (cid:19) nodes v with | supp( ǫ v ) | = n or arcs a = ( v , v ) with | supp( ǫ v ) | < n < | supp( ǫ v ) | . The resultfollows.The lower bound in Theorem 6 is tight, up to polynomial factors. Indeed, consider acomplete bipartite graph K n,n with bipartition U = { u , . . . , u n } and W = { w , . . . , w n } . Weconstruct the network N = ( V, A ) with V := 2 W and A := { ( S , S ) ∈ V × V | S ⊆ S and | S | + 1 = | S |} and a linear projection π such that for every arc a = ( S , S ) ∈ Aπ u i ,w j ( χ { a } ) := ( i = | S | , { w j } ∪ S = S . It is not hard to see that every ∅ – W path in this network defines a perfect matching. This factcan be seen algorithmically, as follows. Start with S = ∅ and repeat the following step until S = W : having matched the vertices v , . . . , v | S | with the vertices in S , select a mate w ∈ W \ S for vertex v | S | +1 and replace S by S ∪ { w } . It follows that the projection of the ∅ – W pathpolytope of network N coincides with the perfect matching polytope of K n,n . Since network N has n n − = O ∗ (2 n ) arcs, we conclude that xc(P perfect matching ( K n,n )) = Θ ∗ (2 n ). Theorem 7.
Every uncapacitated flow-based extension (or, equivalently, s – t path extension) ofthe perfect matching polytope of the complete graph K n,n has size Ω (cid:16) n √ n (cid:17) .Proof. Indeed, the polytope P perfect matching ( K n , n ) is a face of the polytope P perfect matching ( K n ),and thus Lemma 5 gives the lower bound.In order to construct an s – t path extension of size close to the lower bound in Theorem 7,we consider a complete graph K n with vertex set U = { u , . . . , u n } and construct the network N = ( V, A ) with V := { S ⊆ U | | S | = 2 k, k n ∀ i k : u i ∈ S } A := { ( S , S ) ∈ V × V | S ⊆ S and | S | + 2 = | S |} and a linear projection π such that for every arc a = ( S , S ) ∈ Aπ u i ,u j ( χ { a } ) = ( { u i , u j } ∪ S = S . It is once again easy to verify that this defines an s – t path extension, this time of the perfectmatching polytope of K n . The idea is that every ∅ – U path in network N defines a perfectmatching of K n and conversely, every perfect matching of K n corresponds to at least one(actually many) ∅ – U path in N . The ∅ – U paths in N actually correspond to perfect matchingswhose edges are ordered in such a way that for each i , vertex u i is covered by one of the first i S, S ∪ { u i , u j } ) in such a path corresponds to the addition ofedge u i u j to the matching.Up to a polynomial factor, the size of the network equals the number of nodes in the network,that is, n X k =0 (cid:18) n − kk (cid:19) . This is due to the fact that the nodes S in the k th level of network N are of the form S = { u , . . . , u k } ∪ T , where T is contained in U \ { u , . . . , u k } and has size k . Since the number ofsummands in the above expression is n + 1, the size of the constructed extension is O ∗ (cid:18) max k n (cid:18) n − kk (cid:19)(cid:19) = O ∗ (cid:18) max Every uncapacitated flow-based extension (or, equivalently, s – t path extension) ofthe traveling salesman polytope of the complete graph K n has size Ω (cid:16) n √ n (cid:17) .Proof. Assume for now that n = 4 k for some k ∈ N , the other cases will be dealt with later.Take a partition of the vertices of K n in U = { u , . . . , u k } and W = { w , . . . , w k } , and considerthe following sets of edges in the graph K n : E := { u i w j | i = j, i, j k } and E := { u i w i | i k } . Define the face F of the polytope P traveling salesman ( K n ) as the set of points in P traveling salesman ( K n )such that x e = 0 for every e ∈ E and x e = 1 for every e ∈ E .Let us show that the face F together with an orthogonal projection on the variables corre-sponding to the edges u i u j for 0 i, j k gives an extension of the perfect matching polytopeP perfect matching ( K k ) (here the complete graph K k is defined on the vertex set U ).First, every Hamiltonian cycle C in the graph K n restricted to the edges contained in U isa perfect matching, whenever χ C belongs to the face F . Indeed, for every vertex u i in U theremust be exactly two edges in C adjacent to it. Since the characteristic vector χ C lies in theface F , one of these edges is the edge u i w i and the other is contained in U .Second, every perfect matching M in the graph K k can be extended to a Hamiltonian cycle C in K n such that χ C lies in F . Indeed, extend M by another perfect matching M ′ of K k toa Hamiltonian cycle in K k . Then the desired hamiltonian cycle C can be defined as the unionof M , E and { w i w j | u i u j ∈ M ′ } . Thus the result follows from Theorem 7 and Lemma 5.If n = 4 k + r , for some k, r ∈ N , 1 r 3, the result is obtained in a similar way by takinga bipartition U = { u , . . . , u k } and W = { w , . . . , w k + r } and defining the face F by equations x e = 0 for every e ∈ E , x e = 1 for every e ∈ E and x w k w k +1 = . . . = x w k + r − w k + r = 1, wherethe edge sets E and E are defined as above.For the traveling salesman polytope there is a s – t path extension of size O ∗ (2 n ) constructedin a similar manner as the s – t path extension of the perfect matching polytope of K n,n . Thisextension corresponds to a well-known dynamic programming algorithm of Held and Karp forthe traveling salesman problem [20]. We define this extension here for completeness.10onsider a complete graph K n with vertex set U = { u , . . . , u n } and construct the network N = ( V, A ) with V := { ( S, v ) | S ⊆ U, v ∈ S, u ∈ S } ∪ { ( U, ∅ ) } A := { (( S , v ) , ( S , v )) ∈ V × V | S ∪ { v } = S and | S | + 1 = | S |}∪ { (( U, v ) , ( U, ∅ )) ∈ V × V | v ∈ U } and a linear projection π such that for every arc a = (( S , v ) , ( S , v )) ∈ A , v ∈ U , v ∈ Uπ u i ,u j ( χ { a } ) := ( { u i , u j } = { v , v } a = (( U, v ) , ( U, ∅ )) ∈ A , v ∈ Uπ u i ,u j ( χ { a } ) := ( { u i , u j } = { u , v } . It is straightforward to see that the network with source ( u , { u } ) and sink ( U, ∅ ) generatesthe desired s – t path extension. We conclude this paper by stating three open problems.(i) Obtain lower bounds for capacitated flow-based extensions. Although this type of ex-tensions is more expressive than uncapacitated flow-based extensions, we suspect thatexponential size lower bounds can be obtained for nonbipartite matchings and Hamilto-nian cycles.(ii) How difficult is this to compute a small uncapacitated flow-based extension for a given0/1-polytope? Are there good general lower bounds?(iii) All the lower bounds obtained here are of the type 2 Ω( √ d ) , where d is the dimension of P .Find an explicit 0 / P such that every uncapacitated flow-based extension hassize 2 Ω( d ) . (Notice that every polytope P has an uncapacitated flow-based extension ofsize at most the number of vertices of P , thus this last lower bound would be essentiallytight.)(iv) Davis-Stober, Doignon, Fiorini, Glineur and Regenwetter [13] give uncapacitated flow-based extensions of size O ∗ (2 n ) for the linear ordering polytope and O ∗ (3 n ) for the intervalorder polytope. 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