Undirected Multiflow Problems and Related Topics

Undirected Multiflow Problems and Related Topics Some Recent Developments and Results Alexander V. Karzanov Institute for System Studies, Academy of Sciences of USSR, 9, Prospect 60 Let Oktyabrya 117312 Moscow, USSR

Abstract. A multiflow (multicommodity flow), arising as a natural extension of the well-known notion of a network flow, is a popular object studied in linear programming and combinatorial optimization. We discuss some recent results on undirected multiflow problems and ideas behind them, concerning combinatorial and computational aspects, such as: (i) the existence of feasible and optimal solutions with small denominators, (ii) special solvabilility criteria and minimax relations, (iii) efficient solution algorithms, (iv) a relationship between multiflows and packings of cuts and metrics, and some others.

1. Preliminaries Fractionality. We start with some basic notion. Suppose that Jf is a collection (a class) of linear programs P of the form : (i) find x G Q" satisfying Ax < b (the feasibility problem), or (ii) maximize (or minimize) cTx subject to Ax 0 eeE

holds for each metric monV

(6)

ueu

such that

m is primitive and has an extremal graph T with Er ^U.

(7)

Here by a metric on V we mean a nonnegative rational-valued function m on the set of unordered pairs in V satisfying m(xx) = 0 and m(xy) + m(yz) > m(xz) for any x,y,z e V (we use the term "metric" rather than "semimetric"); m is called primitive if mf-\-m" = m, where m! and m" are metrics, implies m' = Xm for some X; an extremal graph of m is a minimal graph F = (Vr,Er) with Vr ^ V such that for any distinct u,v e V there is s£ G Er for which m(st) = m(su) + m(uv) + m(vt). We say that a metric m on V is induced by a graph Q = (VQ, EQ) if there is a mapping 0 for all e G E, and st g E whenever st G U. For a

Undirected Multiflow Problems and Related Topics

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pair TI = {uv,vw} of edges of G let a = a(n) < min{c(uv), c(vw)} be the maximum rational for which D(G',H,c',d) is still solvable, where Gf and c' arise from G and c by decreasing c(uv) and c(vw) and increasing c(uw) by a (if uw £ E we add the edge uw of capacity a); this operation is called splitting off n by a. Let n be chosen so that a is maximum. If a > 1 we split off n by 1, obtaining a "simpler" solvable problem D(G',H,c",d) for which (c",d) is again Eulerian, and the result follows by induction. But if a < 1 then, by Theorem 1, there is X £ V such that A" := A(df,d,X) < 0 (A(-,-,X) is defined in (5)). Since A := A(c,d,X) and zl" are even, zl > 0 and, obviously, c(ö(X)) — c"(apÓ) < 2, we have zf = 0, whence a = 0. This implies f = 0, and hence d = 0; a contradiction. The above proof can be transformed to a strongly polynomial algorithm as follows. Choose a vertex i; G F of the current network (G, i/, c), consider pairs % = {uv,vw}, one by one, and split offn by La(7c)J. If v G 7 \ T , remove i> from G. Repeat the same for a new vertex */, and so on. As a result, one eventually gets G = (V,Ë) and c such that V = T and c(e) > d(e) for e € U. Now a required multiflow in the original network is constructed in a natural way by using the obtained numbers a. One shows that calculation of a(n) can be reduced to solving 0(1) minimum cut problems. This provides a strongly polynomial algorithm. Now let T\ + ... + rp denote the graph that is the union of disjoint graphs r\,...,rp. For H = K2 + K2 + K2 and arbitrary k G Z + one can construct a solvable problem (2) of fractionality at least k [Loi, Lo3]. This and simple observation that if H' is a subgraph of H then cp(D(H!)) < cp(D(H)) imply the following result. Theorem 3. If H contains a matching of 3 edges then (p(D(H)) = 00. The only graphs different from those in Theorems 2 and 3 are: (i) certain subgraphs of K5, (ii) the union of Ki and a 1-star, (iii) K3 + K3. The case (ii) is easily reduced to (i). The following theorem generalizes Theorem 2. Theorem 4 [Ka6]. If H = K5 then cp(De(H)) = 1. The proof of this theorem given in [Ka6] and based on splitting-off techniques is rather complicated. First, one shows that a metric satisfying (7) for H = K5 is either a cut metric or a metric induced by ^2,3, called a 2,3-metric (KM is the complete bipartite graph with parts of p and q vertices). Thus (2) is solvable if and only if (6) holds for all cut metrics and 2,3-metrics on V. Second, unlike the proof above (when a = a(7c) is always an integer), in our case a can take half-integer values; in particular, a = 1/2 is possible (in which case the "obstacle" m violating (6) after splitting off n by 1 is a 2,3-metric). The core of the proof is to show, using combinatorial properties of 2,3-metrics, that if a(n) < 1 for all n then a is 0 everywhere. This proof also can be turned into a strongly polynomial algorithm (however, using the ellipsoid method). To get such an algorithm, one shows that determining a(n) is reduced to solving 0(1) problems P: given d G Z^ and a metric o on T that is either a cut metric or a 2,3-metric, find a metric m on V such that 777 coincides with Q on T and ^(d(e)m(e)\e G E) is minimum. The size of the constraint matrix for P is a polynomial in \V\, hence P is solvable in a strongly polynomial time, by [Ta].

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Theorems 2-4 yield ç(D(H)) for all H except H = K3 + £ 3 . A simple example shows that cp(De(K3 + K3)) is at least 2, and a conjecture is that it is exactly 2. Theorems 2 and 4 have some consequence in lattice theory. A set S — {a\,..., au} ^ Q n is said to form a Hilbert basis if the intersection of the lattice fliZ + ... + ßfcZ with the cone in Q" generated by S coincides with {X\ai + ...+ Xkak\Xi,..., Xk G Z+} (cf. [GP]). Let S(G, H) be the set of the following vectors in Q £ x Qu: (i) (xL,£st), L is an s — t chain in G, st G U; (ii) (%c,0), C is a circuit in G; (iii) (2ee,0), e G E; here #L (xc) is the incidence vector of EL (Ec) in Q E , and ee (8si) is the e-th (st-th) unit basis vector in Q E (Qu). Then S(G,H) forms a Hilbert basis when H is a subgraph of K5, or a 2-star, or the union of K3 and a 1-star.

3. Maximization Problem The set of if's for which M(H) has bounded fractionality turns out to be larger than that for D(H). The complete list of such i f s is unknown, but the values 1 for each st G U. We say that c is inner Eulerian if c( 1; this enables us to apply induction. The proof can be turned into a "pure combinatorial" strongly polynomial algorithm.


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The expression (8) shows that (p(M*(H)) < 2 for if as in Theorem 5. The exact values of cp(M*(H)) were pointed out in [Ka7] for all H. In particular, the following result was stated there. Specify the set of if's with the property : (9)

for any three pairwise intersecting anticliques A,B, C in H, AnB = BnC = CnA.

Theorem 6. cp(M*(H)) G {1,2,4} if H satisfies (9) and (p(M*(H)) = oo otherwise. The first part can be reformulated in polyhedral terms as follows. Let P(G,H) be the polyhedron {/ G Q £ |/ > 0, dist/(s, t) > 1 for any st G U}. If H is as in (9) then P(G,H) is 1/4-integral, that is, each face in it contains a 1/4-integral point. Another consequence of Theorem 6 is that if H is not as in (9) then cp(M(H)) = oo because (p(M(Hf)) > (p(M*(H')) for all H'. The latter follows from a general statement [Ka7] (extending a result on totally dual integral systems [Fui, EG]): let A be a nonnegative m x 77-matrix, b be an integral 777-vector, and let the program D(c) := max{yTfc|y > 0, yTA ^ c} have a l//c-integral optimal solution for every nonnegative integral 77-vector c; then the polyhedron {x G Q"|x > 0,Ax > b} is l//c-integral. It is unknown whether 3 (that is, sé(H) consists of p pairwise disjoint sets) then cp(C(H)) = 2. Theorem 7 is a consequence of a pseudo-polynomial algorithm (an algorithm of complexity 0(c(E)ß(|F|)), Q(7i) is a polynomial in 77) which finds a half-integral optimal primal solution. This algorithm extends the minimum-cost augmenting path method in [FF] based on ideas of the primal-dual method in linear programming. Recently the author found a strongly polynomial algorithm using a general method in [Ta]. On the other hand, it was shown in [Ka5] that if H is not a complete p-partite (p > 2) then cp(C(H)) = 00.

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5. Demand Problem for Planar Graphs Speaking of a planar graph G, we mean that G is explicitly embedded in the plane without intersecting edges. There are known several cases of the demand problem (2) when it is solvable provided that the cut condition (5) holds. The most interesting of them are the following: (CI) the graph (V, EuU)is planar [Se5] ; (C2) there is a set Jf of two faces in G such that each edge of H connects vertices in the boundary of some face in J f [Ok] ; (C3) U = {s\ti,...,Sktk} and there are two inner faces / and J in G so that si,...,Sk occur in clockwise order in I and t\,...,tk do so in J [Sc2]. Moreover, in (C1)-(C3), if (c,d) is Eulerian and (5) holds then (2) has an integral solution. Now we consider the case similar to (C2) for |Jf | > 3. A simple example with G = K2,3 shows that (5) is, in general, not sufficient for solvability of (2). However, the above result is extended, in a sense, as follows. Theorem 8 [KalO]. Let |Jf | = 3. (i) (2) is solvable if and only if the metrical condition (6) holds for all m such that m is a cut metric or a 2,3-metric on V. (ii) If (c, d) is Eulerian and (2) is solvable then (2) has an integral solution. It was shown in [KalO] that if | Jf | = 4 (or more) then (ii) is, in general, not true, and there are infinitely many "types" of metrics m necessary for cheking solvability of (2) for all corresponding G and H. The statement (i) follows from a result on packing of cuts and 2,3-metrics (Theorem 10(ii)(b) below). To prove (ii) we use (i) and the splitting-off method as in the proof of Theorem 4. There are certain difficulties when applying this method, because in order to keep planarity we should take only those pairs % of edges of G which are contained in the boundary of a face of G. The core of the proof is to show that if a(n) < 1 for all such rc's then there are three edges of capacity 1 in G such that the graph G' obtained by removing these edges consists of three components, each containing just one face in Jf. Now (ii) is proved by using Okamura's theorem for (C2).

6. Packings of Cuts and Metrics There is a kind of duality that connects solvability conditions for the demand problem (2) with a certain packing problem on metrics. It can be expressed in a general form as follows. Proposition 9. Given G = (V,E), H — (T,U) and a set M of metrics on V, the following statements are equivalent: (i) for any c and d, (2) is solvable if and only if (6) holds for all me M; (ii) for any l G Z j , there exist m\,...,mk G M and X\,...,Xk G Q+ so that: X\m\(e) + ...Xkmk(e) < 1(e) for all e G E ;

(10)

and X\mi(st) + . . . + Xk,mk(st) = distj(st) for all st G U.

(11)


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This is easily proved by applying Farkas' lemma or the cone polarity. Proposition 9 enables us to derive results on packing of metrics directly from corresponding solvability theorems for (2) (like Theorems 1, 4, 8), and vice versa. Note that this relationship gives only theorems on the existence of rational X. There are stronger, integral, version for some of these theorems; as a rule, their proofs are based on special, sometimes complicated, combinatorial approaches. Now we present some results in this area. We say that a vector / G Z^ is bipartite-like if the /-length of every circuit in G is even. Theorem 10. Let I be bipartite-like. (i) (10) and (11) hold for some cut metrics 777/'s and integral X\'s in the following cases: (a) H is K4 or C5 or a 2-star [Ka4] (cf. [Sel] for H = K2 + K2); (b) G and H are as in (CI) 777 Sect. 5 [Se5]; (c) G and H are as in (C2) 777 Sect. 5 [Sci] (see [Ka8] for a strongly polynomial algorithm). (ii) (10) and (11) hold for m\,...,mk, where 777/ is a cut metric or a 2,3-metric, and integers X\,...,Xk in the following cases: (a) H is K$ or the union of K3 and a 1-star [Ka9]; (b) G and H are as in Theorem 8 [KalO]. There is a connection of the problem (10)—(11) and the problem (P): given a metric 777, decide whether 777 is contained in the conic hull of metrics from a certain collection M. Such a connection was demonstrated in [Ka4] in terms of an extremal graph of 777 for M consisting of the set of cut-metrics. It was also shown there that for this M the problem (P) (or, equivalent, the problem "whether m is embeddable isometrically in the space L 1 " [De]) is iVP-hard.

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