Quadratic diameter bounds for dual network flow polyhedra
aa r X i v : . [ m a t h . O C ] A ug Quadratic diameter bounds for dual network flow polyhedra
Steffen Borgwardt, Elisabeth Finhold, and Raymond Hemmecke S. Borgwardt, Technische Universit¨at M¨unchen, +49-89-28916876, [email protected] E. Finhold, Technische Universit¨at M¨unchen, +49-89-28916891, [email protected] R. Hemmecke, Technische Universit¨at M¨unchen, +49-89-28916864, [email protected]
Abstract.
Both the combinatorial and the circuit diameters of polyhedra are of interest tothe theory of linear programming for their intimate connection to a best-case performanceof linear programming algorithms.We study the diameters of dual network flow polyhedra associated to b -flows on directedgraphs G = ( V, E ) and prove quadratic upper bounds for both of them: the minimum of( | V |− ·| E | and | V | for the combinatorial diameter, and | V |· ( | V |− for the circuit diameter.The latter strengthens the cubic bound implied by a result in [De Loera, Hemmecke, Lee;2014].Previously, bounds on these diameters have only been known for bipartite graphs. Thesituation is much more involved for general graphs. In particular, we construct a familyof dual network flow polyhedra with members that violate the circuit diameter bound forbipartite graphs by an arbitrary additive constant. Further, it provides examples of circuitdiameter | V | − Keywords: combinatorial diameter, circuit diameter, Hirsch Conjecture, edges, circuits, Graverbasis, linear program, integer program
In the context of a best-case performance of the Simplex algorithm, the studies of the combinatorialdiameter of polyhedra are a classical field in the theory of linear programming. In particular, if onecan find an n -dimensional polyhedron with f facets with a diameter that is exponential in f and n , then the existence of a polynomial pivot rule for the Simplex algorithm would be disproved.In 1957, Hirsch stated the famous conjecture [3] claiming that the combinatorial diameter of apolyhedron is at most f − n . For (unbounded) polyhedra there are low-dimensional counterexamples[9]. For polytopes however, the Hirsch conjecture stood for more than 50 years, until Santos gave afirst counterexample [11]. Nonetheless the bound holds for several well-known families of polyhedra,like 0 / f and n is open. See the survey by Kim and Santos for the currentstate-of-the-art [8].For our discussion, we use the following notation. Let v (1) and v (2) be two vertices of a polyhedron P . We call a sequence of vertices v (1) = y (0) , . . . , y ( k ) = v (2) an edge walk of length k if every pairof consecutive vertices is connected by an edge. The (combinatorial) distance of v (1) and v (2) isthe minimum length of an edge walk between v (1) and v (2) . The combinatorial diameter diam( P )of P then is the maximum distance between any two vertices of P .On such edge walks we only go along edges of the polyhedron P , in particular we never leave itsboundary. In contrast to this, circuit walks also use only ’potential’ edge directions, but may walkthrough the interior of the polyhedron: Let a polyhedron P be given by P = (cid:8) z ∈ R n : A z = b , A z ≥ b (cid:9) for matrices A i ∈ Q d i × n and vectors b i ∈ R d i , i = 1 ,
2. The circuits or elementary vectors C ( A , A )of A and A are those vectors g ∈ ker( A ) \ { } , for which A g is support-minimal in the set (cid:8) A x : x ∈ ker (cid:0) A (cid:1) \{ } (cid:9) , where g is normalized to coprime integer components. It can be shownthat the set of circuits consists exactly of all edge directions of P for varying b and b [7]. Circuitsand their integer programming equivalents, Graver bases, play an important role in the theory ofinteger programming. We refer the reader to the book [4] for a thorough introduction to the topic.The circuit analogues to the notions of combinatorial distance and diameter for P are then definedas follows [2]: For two vertices v (1) , v (2) of P , we call a sequence v (1) = y (0) , . . . , y ( k ) = v (2) a circuit walk of length k if for all i = 0 , . . . , k − y ( i ) ∈ P ,2. y ( i +1) − y ( i ) = α i g ( i ) for some g ( i ) ∈ C ( A , A ) and α i >
0, and3. y ( i ) + α g ( i ) is infeasible for all α > α i .The circuit distance dist C ( v (1) , v (2) ) from v (1) to v (2) then is the minimum length of a circuit walkfrom v (1) to v (2) . The circuit diameter diam C ( P ) of P is the maximum circuit distance betweenany two vertices of P .Clearly, the circuit diameter of a polyhedron is at most as large as the combinatorial diameter of thepolyhedron, as a walk along the 1-skeleton/edges of the polyhedron is a special circuit walk. Onceagain, if there exists a polyhedron with exponential circuit diameter, there can be no polynomialpivot rule for the Simplex algorithm. This is one of several reasons to study it in the context oflinear programming; see [2]. In fact, the circuit diameter gives a lower bound for any augmentationalgorithm along circuit directions [4].In fact, it is open whether there is a polyhedron with a circuit diameter that exceeds f − n , as in theHirsch conjecture (see Conjecture 1 in [2]). The polyhedra giving counterexamples to the Hirsch2onjecture do not violate this bound for the circuit diameter. This raises the natural question howthese two diameters are related to one another. In this paper, we study the diameters for thefamily of dual network flow polyhedra , for which we prove quadratic upper bounds on both thecombinatorial diameter and the circuit diameter.Let G = ( V, E ) be a directed connected graph and let A ∈ { − , , } | V |×| E | be its node-arcincidence matrix, where a ie = − a je = 1 if arc e has node i as its tail and node j as its head.Let b ∈ R | V | . A b -flow on G is given by any solution to A x = b , x ≥ , that is, in each node i ∈ V the resulting flow (incoming minus outgoing flow) is given by b i . For some cost function c : E → R + , the min-cost b -flow problem and its dual are given bymin (cid:8) c T x : A x = b , x ≥ (cid:9) and max (cid:8) u T b : A T u ≤ c (cid:9) . In the following we are interested in the dual network flow polyhedron associated to some graph G and vector c ∈ R | E | . These polyhedra can be written as P G, c = n u ∈ R | V | : − u a + u b ≤ c ab ∀ ab ∈ E, u = 0 o . As is standard, we set u = 0 to make P G, c pointed (to actually have vertices). Then linearprogramming over P G, c is a viable approach for solving the corresponding min-cost b -flow problemand is another reason for the interest in the diameters of this family of polyhedra.In [1] and [2] the diameters of dual transportation polyhedra were studied. They are associatedto undirected bipartite graphs and can be interpreted as dual network flow polyhedra on directedbipartite graphs on node sets V = V ˙ ∪ V , where all edges point from V to V . Hence thesediameter results transfer to special cases of our more general setting:Balinski [1] proved that the combinatorial diameter of a dual transportation polyhedron associatedwith a complete bipartite graph on M × N nodes is bounded above by ( M − N −
1) and thatthis bound is sharp for all
M, N . Observe that this bound is quadratic in the number of nodes andlinear in the number of edges. The circuit diameter of a dual transportation polyhedron definedon an arbitrary bipartite graph on M × N nodes is bounded above by M + N − | V | − M + N − | V | − M + N .For general graphs, we cannot expect similar bounds. The following example gives a graph forwhich the upper bound | V | − Example 1.
The dual network flow polyhedron P G, c associated with the following graph on 4 nodeshas circuit diameter at least | V | = 4. (See Section 4 for a proof.) The edges are labeled with thecorresponding values of c . 3 v v v
00 0 2 (cid:3)
We extend this graph to a family of graphs with associated polyhedra of circuit diameter greaterthan | V | + k − k . To do so, we introduce what we call a glueing construction : If weglue k graphs together at a single, arbitrary node, we obtain a larger graph. The circuit diameter,respectively combinatorial diameter, of this larger graph then is the sum of the circuit diameters,respectively combinatorial diameters, of the polyhedra associated to the smaller graphs; see Lemma3 in Section 4.Applying this construction to k copies of Example 1 above, we get a family of graphs on 3 k + 1nodes with associated dual network flow polyhedra that admit a circuit diameter of at least 4 k = | V | + k −
1. Hence we violate the circuit diameter bound for bipartite graphs by an arbitraryadditive constant. This further yields a family of polyhedra whose circuit diameter approaches | V | : Lemma 1.
For any n ≥ , there is a graph G = ( V, E ) on | V | = n nodes and a vector c ∈ R | E | such that diam C ( P G, c ) ≥ | V | − | V | on the previous diameter bounds. Hencewe get quadratic upper bounds on both the combinatorial and the circuit diameter. Theorem 1 (Combinatorial diameter).
The combinatorial diameter of dual network flow poly-hedra P G, c is bounded above by min { ( | V | − · | E | , | V | } . Theorem 2 (Circuit diameter).
The circuit diameter of dual network flow polyhedra P G, c isbounded above by | V |· ( | V |− . Throughout this paper, we will exploit the special structure of dual network flow polyhedra P G, c byrelating the vertices and edges of such polyhedra to subgraphs of the defining graph G . For u ∈ P G, c ,we denote by G ( u ) the graph with nodes V and with edges ab ∈ E for which − u a + u b ≤ c ab istight. If the polyhedron P G, c is non-degenerate, these graphs have no cycles.The vertices of P G, c are determined by the sets of inequalities − u a + u b ≤ c ab that are tight.It can be shown that that u ∈ P G, c is a vertex if and only if G ( u ) is a spanning subgraph of G . In particular, every such spanning subgraph contains a spanning tree of G with | V | − − u a + u b ≤ c ab that are tight at the vertex. Thisspanning tree uniquely determines the vertex u , since we assume u = 0.The circuit directions of P G, c can be described as follows: Let R, S ⊆ V be connected nonemptynode sets with R ˙ ∪ S = V (which implies R ∩ S = ∅ ). W.l.o.g., we may assume 0 ∈ R . Then thevector g ∈ R M + N with g i = ( , if i ∈ R, , if i ∈ S, (1)is an edge direction of P G, c for some right-hand side c . In fact, it can be shown that these are allpotential edge directions and hence they constitute the set of circuits C G associated to the matrixdefining P G, c .Let y ∈ P G, c . We apply a circuit step given by R ˙ ∪ S = V or the corresponding g by setting y ′ := y ± ǫ g , where ǫ is the smallest non-negative number such that an inequality − u a + u b ≤ c ab with a ∈ R and b ∈ S (respectively b ∈ R and a ∈ S ) becomes tight. This means that we increase(respectively decrease) all components y s with s ∈ S until an edge from R to S (respectively from S to R ) is inserted.Two vertices u (1) , u (2) of P G, c are connected by an edge if and only if the subgraph of G with edgeset E (cid:0) G ( u (1) ) (cid:1) ∩ E (cid:0) G ( u (2) ) (cid:1) consists of exactly two connected components. Then the node sets R and S of these components describe the edge direction via Equation (1).We continue with some advanced tools and results that we will need in Section 3. The idea of contracting edges simplifies the proofs of Theorems 1 and 2: Assume that we have a vertex (feasible5oint) y of a polyhedron P G, c from which we want to construct an edge walk (circuit walk) to somevertex w , and assume that E ( G ( y )) and E ( G ( w )) have an edge ab in common. Then we wish tokeep this edge on the remaining edge walk (circuit walk). Therefore, the difference between u a and u b has to remain constant, which means that in every edge step (circuit step) given by V = R ˙ ∪ S , a and b are assigned both to R or both to S . To simplify this idea, we interpret a and b as onenode in the following sense: We contract the edge ab and continue our edge walk (circuit walk) ina smaller polyhedron defined on a graph with one node less and adjusted edge set.Geometrically this corresponds to intersecting the dual network flow polyhedron with the hyper-plane (cid:8) u ∈ R | V | : − u a + u b = c ab (cid:9) . This defines a face of the polyhedron, which is a dual networkflow polyhedron in its own right. We then continue the edge walk (circuit walk) on this face. Moreformally, let ab be the common edge in G = ( V, E ). The new polyhedron P G ′ ,c ′ is defined by anew graph G ′ = ( V ′ , E ′ ) and a new vector c ′ (for a simple notation we use c ij = ∞ if ij / ∈ E ) asfollows: V ′ = V \{ b } E ′ = { ij : ij ∈ E and i, j = a, b }∪ { aj : aj ∈ E or bj ∈ E } ∪ { ia : ia ∈ E or ib ∈ E } c ′ ij = c ij for i, j = a, ij ∈ E ′ min { c aj , c bj + c ab } for i = a, aj ∈ E ′ min { c ia + c ab , c ib } for j = a, ia ∈ E ′ For the definition of c ′ , observe that if ab exists in G ( u ) (i.e. − u a + u b = c ab ), and aj, bj ∈ E forsome j , then − u a + u j ≤ c aj will become tight before − u b + u j ≤ c bj when decreasing both u a and u b if and only if c aj ≤ c bj + c ab . Hence, when keeping ab , the latter case will never occur and onlythe first inequality is relevant. On the other hand c aj > c bj + c ab implies that only − u b + u j ≤ c bj can become tight, such that we only need to consider this inequality in the following. In this casewe further have to adjust the value for c (observe u b = u a + c ab ). The other case is analogous.Hence, every edge walk (circuit walk) in P G ′ ,c ′ admits an edge walk (circuit walk) in P G,c thatkeeps the edge ab , such that we can continue the walk in the smaller polyhedron.Next we show that the existence of a feasible point whose graph contains a certain edge ab impliesthe non-existence of a feasible point whose graph contains a different directed path from a to b . Lemma 2.
Let P G, c be a dual network flow polyhedron. Let v v k ∈ E such that in G there isanother directed path P from v to v k , i.e. there are nodes v , v , . . . , v k ∈ V , k ≥ , such that v i v i +1 ∈ E for all i = 0 , . . . , k − .Assume there is a feasible point w ∈ P G,c with v v k ∈ E ( G ( w )) and let u ∈ P G,c with
P ⊂ G ( u ) .Then also v v k ∈ E ( G ( u )) . Thus, if P G, c is non-degenerate, there can be no such u . roof. The feasible point w ∈ P G,c satisfies c v v k = − w v + w v k = k − X i =0 (cid:0) − w v i + w v i +1 (cid:1) ≤ k − X i =0 c v i v i +1 . u ∈ P G,c satisfies − u v i + u v i +1 = c v i v i +1 for i = 0 , . . . , k − − u v + u v k ≤ c v v k . We then see k − X i =0 c v i v i +1 = k − X i =0 (cid:0) − u v i + u v i +1 (cid:1) = − u v + u v k ≤ c v v k ≤ k − X i =0 c v i v i +1 . Hence, all inequalities must be satisfied with equality and we get − u v + u v k = c v v k ; that is, v v k ∈ E ( G ( u )). (cid:3) Observe that Lemma 2 can easily be generalized to a slightly stronger statement: Assume thatthere is a feasible point whose graph contains a directed path from some node v to some node v k . Then every point of the dual network flow polyhedron, whose graph contains another directed v − v k -path, must contain the first path as well. This can only happen in the degenerate case. We begin with the proof of Theorem 1. Note that for proving upper bounds on the combinatorialdiameter of polyhedra it is enough to consider non-degenerate polyhedra, as by perturbation anypolyhedron can be turned into a non-degenerate polyhedron whose diameter is at least as large asthe one of the original polyhedron.
Proof of Theorem 1.
Let u (1) and u (2) be two vertices of the polyhedron P G, c given by spanningtrees T = G ( u (1) ) and T = G ( u (2) ). We construct an edge walk from u (1) to u (2) as follows:Being at a vertex y of P G, c with spanning tree T = G ( y ), we choose an edge rs ∈ T \ T we wish toinsert. We show how to construct an edge walk of length at most | E | that leads to a vertex x forwhich rs ∈ E ( G ( x )), that is, our specified edge is added to the corresponding spanning tree. Thenwe contract this edge to ensure that we do not delete it again. Starting at y = u (1) and repeatingthis for all | V | − T proves the claimed bound of ( | V | − · | E | .Now, let y be the current vertex in our edge walk and let T = G ( y ) be the corresponding spanningtree. We choose an arbitrary edge rs ∈ T we wish to insert. Given a spanning tree T and the node s we distinguish forward and backward edges in E ( T ): We see s as the root of the tree T . Thenevery edge in E ( T ) lies on a unique path starting at s (independent of the directions of the edges).We call the edges pointing away from s backward edges , the edges pointing towards s forward edges .In T there is a unique path (undirected) connecting r and s . Let e be the last backward edge onthis path. Note that by Lemma 2 such an edge must exist. Let R and S be the node sets of the7onnected components of T − e such that r ∈ R and s ∈ S . Observe that in particular all nodesfrom which we can reach s on a directed path in the spanning tree T are assigned to S (and thesenodes form an arborescence of forward edges with root s ).We wish to include the edge rs in our graph, that is, we wish to make the inequality − u r + u s ≤ c rs tight. W.l.o.g. we assume 0 ∈ R , therefore we add an ǫ to all components y i of y with i ∈ S . (If0 / ∈ R , we would subtract ǫ from all components y i with i ∈ R .) We choose as ǫ the smallestnon-negative number such that any inequality − u a + u b ≤ c ab with a ∈ R and b ∈ S becomestight. Due to non-degeneracy there is only one such inequality. This creates a new feasible point y ′ , which is indeed a neighboring vertex of y by construction. r w v sa bef e ′ R S
So, in this edge step e = vw is deleted and f = ab is inserted. If we inserted f = rs , we contract thisedge and start over again, aiming to insert another edge r ′ s ′ from E ( T ). Otherwise we considerthe path connecting r and s in the new spanning tree T ′ . As before the last backward edge e ′ defines sets R ′ and S ′ and we repeat the same procedure until eventually rs is inserted. It remainsto prove that this indeed happens after at most | E | steps. It is enough to show that the deletededge e = vw is not inserted again: As there is a directed path from v to s in G ( y ), v and all nodeson this path will always be assigned to S (in particular, no edge on this path is deleted). As onlyedges from R to S are inserted, e = vw with v ∈ S cannot be inserted twice. This proves theclaimed upper bound ( | V | − · E ).To see the upper bound | V | , we only have to change the way we count the number of steps thatwe need to insert the edge rs in a current graph on i nodes: Note that it has at most i · ( i − (cid:0) i (cid:1) edges e = vw with v ∈ S and w ∈ R . As we only insert edgesfrom R to S , this tells us an upper bound of (cid:0) i (cid:1) steps until rs inserted. After contracting this edge,we start this process again on a graph with i − | V | X i =2 i − X j =1 j = | V | X i =2 (cid:20) i · ( i − (cid:21) = 12 | V | X i =2 i − | V | X i =2 i = | V | − | V | ≤ | V | . We continue with the proof of Theorem 2. Here we cannot simply assume that the polyhedron isnon-degenerate, as it is not clear whether for every degenerate polyhedron there is a perturbednon-degenerate polyhedron bounding the circuit diameter of the original one from above [2].
Proof of Theorem 2.
Let u (1) and u (2) be two vertices of the polyhedron P G, c . Let T be a spanningtree with E ( T ) ⊆ E ( G ( u (2) )). Then u (2) is the unique point of P G, c whose graph contains all edgesin E ( T ). We construct a circuit walk from u (1) to u (2) as follows:Being at a point y ∈ P G, c of our circuit walk, we choose an edge rs ∈ T \ E ( G ( y )) we wish toinsert. We construct a circuit walk to a point x ∈ P G, c with rs ∈ E ( G ( x )). This walk has lengthat most i −
1, where i is the number of nodes in the current underlying graph. As in the proofof Theorem 1, we then contract it to make sure that we do not delete it when continuing ourcircuit walk. We start with y = u (1) and repeat this procedure for all | V | − E ( T ).As the number of nodes decreases after every contraction this then yields the quadratic bound of | V |− P i =1 i = ( | V | · ( | V | − y be a feasible point in the circuit walk. Let rs ∈ E ( T ) \ E ( G ( y )) be an arbitrary edge wewish to insert, that is, we have to make − u r + u s ≤ c rs tight. To this end, we construct a circuitdirection that increases the component y s . This circuit is given by R ˙ ∪ S = V for node sets R and S that are constructed by the following sequence of rules:1. r is assigned to R .2. s is assigned to S .3. All nodes from V \{ r } from which s can be reached on a directed path using edges in E ( G ( y ))are assigned to S . (These edges form an arborescence with root s .)4. All nodes t ∈ V \ S that are connected to r in the underlying undirected graph are assigned to R .5. All remaining nodes are assigned to S .Observe that from s we cannot reach r on a directed path in E ( G ( y )) by Lemma 2, hence the sets R and S are well-defined. Further, they satisfy all the conditions to define a circuit. Let g be thecorresponding circuit direction defined via Equation (1). W.l.o.g. we assume that 0 ∈ R . The case0 ∈ S works analogously by merely switching the roles of R and S and subtracting ǫ g to decrease y r .We now apply the circuit step given by g , that is, we get the next point in our circuit walk as y ′ := y + ǫ g , where ǫ is the smallest non-negative number such that an inequality − u a + u b ≤ c ab with a ∈ R and b ∈ S becomes tight (observe that there could be more than one such inequality,as we do not assume non-degeneracy of the polyhedron P G, c ). In particular, the gap in between − u r + u s and its upper bound c rs becomes smaller. If rs was indeed inserted we contract the edgeand continue in a smaller polyhedron. 9 sbaR S Otherwise, the inserted edge extends the arborescence by at least the node a . We again applya circuit step by constructing sets R ′ and S ′ for y ′ as before, which inserts rs or extends thearborescence further. Continuing like this after at most i − r are containedin the arborescence (if rs was not already inserted). Then the next step must add rs by Lemma 2. (cid:3) Observe that these diameter bounds also hold for dual network flow polyhedra defined on directedgraphs that are not connected. To make the polyhedron pointed, we set, for each connected com-ponent, the value of one variable to zero (just as we fixed u = 0 for connected graphs with justone connected component). Then the algorithmic approaches described in the proofs of Theorem1 and Theorem 2 can be applied to each connected component individually, yielding even betterbounds on the combinatorial diameter and the circuit diameter. In the above, we derived quadratic upper bounds on the circuit and the combinatorial diameterof dual transportation polyhedra. We now complement our discussion by constructing an infinitefamily of graphs that exhibit that the gap between the number of nodes | V | and the circuit diameterof a polyhedron associated with a certain graph can be arbitrarily large.To this end, we begin with a formal introduction of a glueing construction for graphs: Let G i =( V i , E i ), i = 1 , . . . , k be k connected directed graphs. For every graph choose an arbitrary node v i ∈ V i . We construct a new graph G = ( V, E ) by glueing the graphs together at the v i , joiningthem to one node v . Formally, the node sets and the edge set are given by V := { v } ∪ k [ i =1 (cid:0) V i \ (cid:8) v i (cid:9)(cid:1) E := k [ i =1 (cid:0) (cid:8) ab : ab ∈ E i , a, b = v i (cid:9) ∪ (cid:8) v b : v i b ∈ E i (cid:9) ∪ (cid:8) av : av i ∈ E i (cid:9) (cid:1) .
10e depict the graphs G i by highlighting the nodes v i , while all remaining nodes and edges arerepresented by a cycle: G v v G v v G v v G v v Glueing these 4 graphs together yields a graph G that can be illustrated as follows: v G G G G v Now the diameters of the polyhedra associated to these graphs are directly related.
Lemma 3.
Let P G i , c i , i = 1 , . . . , k be arbitrary dual network flow polyhedra with combinatorial(circuit) diameter equal to d i , respectively at least d i . Let G be the graph obtained by glueing these k graphs together, and define c ∈ R | E | by c lj = c ilj , lj ∈ E i .Then P G, c has combinatorial (circuit) diameter P ki =1 d i , respectively at least P ki =1 d i .Proof. Let a circuit direction of P G, c be given by a partition V = R ˙ ∪ S . Assume w.l.o.g. v ∈ R .Then S ⊆ V i \ (cid:8) v i (cid:9) for some i ∈ { , . . . , k } , as the node set S must be connected in the underlyinggraph and v / ∈ S . 11 G G G G v RS Therefore, every step of an edge walk (circuit walk) modifies only variables corresponding to asingle, particular component G i , such that every edge walk (circuit walk) on P G, c of length d ′ directly translates into k edge walks (circuit walks) on P G , c , . . . , P G k , c k of length d ′ , . . . , d ′ k with P ki =1 d ′ i = d ′ and vice versa. (cid:3) We now turn to an example which shows that there are configurations in which there is no circuitstep that inserts an edge from the target tree. Note that in the undirected bipartite case we arealways able to apply such a step. Therefore, recall Example 1 in which we introduced the polyhedron P G, c defined on the following graph. The labels on the edges correspond to the values of c . v v v v
00 0 2
Observe that the polyhedron P G, c is non-degenerate (there can be no cycle of tight inequalities).The following two spanning trees correspond to vertices u (1) and u (2) of P G, c . The nodes are labeledby the values of the corresponding variables. 12 T = G ( u (1) ) 0
23 43 T = G ( u (2) ) These two vertices are connected via the following edge walk of length 4. Hence their circuit distanceand combinatorial distance are at most 4.
00 00 01 01 01
23 43 −→ −→ −→ −→ We now illustrate all possible first circuit steps from u (1) , leading to points y (1) , . . . , y (6) . Thecorresponding circuits are stated below the graphs and are w.l.o.g. given by subsets S ⊆ V suchthat v / ∈ S (note that S = { v , v } is not applicable). Observe that in all cases the inserted (bold)edge is not in E ( T ). y (1) y (2) y (3) −
00 00 10 00 0 { v } { v } { v } (0 , , ,
2) (0 , , ,
2) (0 , , , ) y (4) y (5) y (6)
01 01 00 11 01 11 { v , v } { v , v } { v , v , v } (0 , − , ,
1) (0 , , ,
1) (0 , − , ,
13t then is elementary to verify that the circuit distance from u (1) to u (2) is indeed | V | = 4. Forthis purpose, it is sufficient to see that one was not able to insert an edge from E ( T ) in the firstcircuit step, and that in the remaining circuit walk we cannot insert two edges from E ( T ) at thesame time. Even if the latter property would not hold for a given c , we could always satisfy it bya slight perturbation:For every single step of a circuit walk, a finite number of linear conditions on the right-hand sides c guarantees that only at most one edge from a target tree is inserted. Thus, after k steps on acircuit walk, we only have to exclude the c in the union of a countable number of hyperplanes tobe able to guarantee this property for all steps of circuit walks of length at most k .So we now have a graph G with circuit distance (at least) | V | = 4. Applying Lemma 3 to k copies G i of G yields a new graph G k on 3 k + 1 nodes with (combinatorial and circuit) diameter at least4 k . This gives us a family of graphs G k for which both diameters exceed the number of nodes by anarbitrary constant k − for k → ∞ . In particular we get the following lower bound statement for the circuit diameter(and hence also for the combinatorial diameter) of dual network flow polyhedra. Lemma 4.
For any n ≥ , there is a graph G = ( V, E ) on | V | = n nodes and a vector c ∈ R | E | such that diam C ( P G, c ) ≥ | V | − . Proof.
For n = 3 k + 1 with k ∈ Z the claim follows by choosing G = G k , as k = | V |− and thecircuit diameter is at least 4 k . If n = 3 k + 2 ( n = 3 k + 3) we simply add one leaf (two leaves) to G k . Then k = | V |− ( k = | V |− ) and the circuit diameter is again at least 4 k . (cid:3) Acknowledgments
The second author gratefully acknowledges the support from the graduate program TopMath ofthe Elite Network of Bavaria and the TopMath Graduate Center of TUM Graduate School atTechnische Universit¨at M¨unchen.
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