Integrality Gap of the Hypergraphic Relaxation of Steiner Trees: a short proof of a 1.55 upper bound

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1 Integrality Gap o the Hypergraphic Relaxation o Steiner Trees: a short proo o a.55 upper ound Deeparna Chakraarty University o Pennsylvania Jochen önemann University o Waterloo David Pritchard École Polytechnique Fédérale de Lausanne Astract Recently, Byrka, Grandoni, Rothvoß and Sanità gave a.9-approximation or the Steiner tree prolem, using a hypergraph-ased linear programming relaxation. They also upper-ounded its integrality gap y.55. We descrie a shorter proo o the same integrality gap ound, y applying some o their techniques to a randomized loss-contracting algorithm. Introduction In the Steiner tree prolem, we are given an undirected graph G = (V,E) with costs c on edges and its vertex set partitioned into terminals (denoted R V ) and Steiner vertices (V \R). A Steiner tree is a tree spanning all o R plus any suset o V \R, and the prolem is to ind a minimum-cost such tree. The Steiner tree prolem is APX-hard, thus the est we can hope or is a constant-actor approximation algorithm. The est known ratio is otained y Byrka, Grandoni, Rothvoß and Sanità []: their randomized iterated rounding algorithm gives approximation ratio ln(4) + ǫ.9. The prior est was a + ln + ǫ.55 ratio, via the deterministic loss-contracting algorithm o Roins and Zelikovsky [6]. The algorithm o [] diers rom previous work in that it uses a linear programming (LP) relaxation; the LP is ased on hypergraphs, and it has several dierent-looking ut equivalent [, 5] nice ormulations. A second result o [] concerns the LP s integrality gap, which is deined as the worst-case ratio (max over all instances) o the optimal Steiner tree cost to the LP s optimal value. Byrka et al. show the integrality gap is at most.55, and their proo uilds on the analysis o [6]. In this note we give a shorter proo o the same ound using a simple LP-rounding algorithm. We now descrie one ormulation or the hypergraphic LP. Given a set R o terminals, a ull component on is a tree whose lea set is and whose internal nodes are Steiner vertices. Without loss o generality, Steiner trees have no Steiner nodes o degree, and under this condition they decompose in a unique edge-disjoint way into ull components; Figures (i) and (ii) show an example. Moreover, one can show that a set o ull components on sets {,..., r } orms a Steiner tree i and only i the hypergraph (V, {,..., r }) is a hyper-spanning tree. Here, a hyper-spanning tree means there is a

2 a c d e (i) a d e (ii) c d a e (iii) c Figure : In (i) we show a Steiner tree; circles are terminals and squares are Steiner nodes. In (ii) we show its decomposition into ull components, and their losses in old. In (iii) we show the ull components ater loss contraction. unique path (alternating vertex-hyperedge sequence o incidences) connecting every pair o vertices. Let F() denote a minimum-cost ull component or terminal set R, and let C e its cost. The hypergraphic LP is as ollows: min C x : (S) S R : : S x ( S ) S x ( ) = R : x 0 The integral solutions o (S) correspond to the ull component sets o Steiner trees. As an aside, the r-restricted ull component method (e.g. [4]) allows us to assume there are a polynomial numer o ull components while aecting the optimal Steiner tree cost y a + ǫ actor. Then, it is possile to solve (S) in polynomial time [, 8]. Here is our goal: Theorem. [] The integrality gap o the hypergraphic LP (S) is at most + (ln )/.55. Randomized Loss-Contracting Algorithm In this section we descrie the algorithm. We introduce some terminology irst. The loss o ull component F(), denoted y Loss(), is a minimum-cost suset o F() s edges that connects the Steiner vertices to the terminals. For example, Figure (ii) shows the loss o the two ull components in old. We let loss() denote the total cost o all edges in Loss(). The loss-contracted ull component o, denoted y LC(), is otained rom F() y contracting its loss edges (see Figure (iii) or an example). For clarity we make two oservations. First, or each the edges o LC() correspond to the edges o F()\Loss(). Second, or terminals u,v, a uv edge may appear in LC( ) andlc( ) or distinct ull components and ; ut we think o them as distinct parallel edges. Our randomized rounding algorithm, RLC, is shown elow. We choose M to have value at least x such that t = M ln is integral. MST( ) denotes a minimum spanning tree and mst its cost.

3 e 6 6 e e a c d 6 6 a,,c,d a c d (i) (ii) (iii) Figure : In (i) we show a terminal spanning tree T in red, and a ull component spanning terminal suset = {a,,c,d} in lack; thick edges are its loss. In (ii) we show T/, and Drop T () is shown as dashed edges. In (iii) we show MST(T LC()). Algorithm RLC. : Let T e a minimum spanning tree o the induced graph G[R]. : x Solve (S) : or i t do 4: Sample with replacement a single component i as ollows: with proaility x /M it is the ull component, with proaility x /M it is the empty set 5: T i+ MST(T i LC( i )) 6: end or 7: Output any Steiner tree in ALG := T t+ t i= Loss( i). To prove that ALG actually contains a Steiner tree, we must show all terminals are connected. To see this, note each edge uv o T t+ is either a terminal-terminal edge o G[R] in the input instance, or else uv LC( i ) or some i and thereore a u-v path is created when we add in Loss( i ). Analysis In this section we prove that the tree s expected cost is at most + ln times the optimum value o (S). Each iteration o the main loop o algorithm RLC irst samples a ull component i in step 4, and susequently recomputes a minimum-cost spanning tree in the graph otained rom adding the loss-contracted part o i to T i. The new spanning tree T i+ is no more expensive than T i ; some o its edges are replaced y newly added edges in LC( i ). Bounding the drop in cost will e the centerpiece o our analysis, and this step will in turn e acilitated y the elegant Bridge Lemma o Byrka et al. []. We descrie this lemma irst. We irst deine the drop o a ull component with respect to a terminal spanning tree T (it is just a dierent name or the ridges o []). Let T/ e the graph otained rom T y identiying the terminals spanned y. Then let Drop T () := E(T) \ E(MST(T/)),

4 e the set o edges o T that are not contained in a minimum spanning tree o T/, and drop T () e its cost. We illustrate this in Figure. We state the Bridge Lemma here and present its proo or completeness. Lemma (Bridge Lemma []). Given a terminal spanning tree T and a easile solution x to (S), x drop T () c(t). () Proo. The proo needs the ollowing theorem []: given a graph H = (R, F), the extreme points o the polytope {z R F 0 : z e S ; S R, z e = R } (G) e γ(s) are the indicator variales o spanning trees o H, where γ(s) F is the set o edges with oth endpoints in S. The proo strategy is as ollows. We construct a multigraph H = (R,F) with costs c, and z R F such that: the cost o z equals the let-hand side o (); z (G); and all spanning trees o H have cost at least c(t). Edmonds theorem then immediately implies the lemma. In the rest o the proo we deine H and supply the three parts o this strategy. For each ull component with x > 0, consider the edges in Drop T (). Contracting all edges o E(T)\Drop T (), we see that Drop T () corresponds to edges o a spanning tree o. These edges are copied (with the same cost c) into the set F, and the copies are given weight z e = x. Using the deinition o drop, one can show each e F is a maximum-cost edge in the unique cycle o T {e}. Having now deined F, we see c e z e = x drop T (). e F Note that we introduce edges or each ull component, and that, or any S R, at most S o these have oth ends in S. These two oservations together with the act that x is easile or (S) directly imply that z is easile or (G). To show all spanning trees o H have cost at least c(t), it suices to show T is an MST o T H. In turn, this ollows (e.g. [7, Theorem 50.9]) rom the act that each e F is a maximum-cost edge in the unique cycle o T {e}. We also need two standard acts that we summarize in the ollowing lemma. They rely on the input costs satisying the triangle inequality (i.e. metricity), and that internal nodes o ull components have degree at least, oth o which hold without loss o generality. Lemma. (a) The value mst(g[r]) o the initial terminal spanning tree computed y algorithm RLC is at most twice the optimal value o (S). () For any ull component, loss() C /. Proo. For (a) we use a shortcutting argument along with Edmonds polytope (G) or the graph H = G[R]. In detail, let x e an optimal solution to (S). For each, shortcut a e F 4

5 tour o F() to otain a spanning tree o with c-cost at most twice C (y the triangle inequality) and add these edges to F with z-value x. Like eore, since x is easile or (S), z is easile or (G), and so there is a spanning tree o G[R] whose c-cost is at most e F c ez e C x. The result () is standard (e.g. [4, Lemma 4.]) ut we give a sketch. In the ull component, take a Steiner node x with at most one Steiner neighour. Thus x has terminal neighours. Include the cheapest edge rom x to a terminal neighour in the loss; then treat x as a terminal and iterate rom the eginning. We end when there are no Steiner nodes, at which point we have spent at most hal o C to construct the loss. We are ready to prove the main theorem. Proo o Theorem. Let x e an optimal solution to (S) computed in step, deine lp to e its ojective value, and loss = x loss() its ractional loss. Our goal will e to derive upper ounds on the expected cost o tree T i maintained y the algorithm at the eginning o iteration i. Ater selecting i, one possile candidate spanning tree o T i LC( i ) is given y the edges o T i \Drop Ti ( i ) LC( i ), and thus c(t i+ ) c(t i ) drop Ti ( i ) + c(lc( i )). () Let us ound the expected value o T i+, given any ixed T i. Due to the distriution rom which i is drawn, and using () with linearity o expectation, conditioning on any T i we have E[c(T i+ ) T i ] c(t i ) x drop M Ti () + x (C loss()). M Applying the ridge lemma on the terminal spanning tree T i, and using the deinitions o lp and loss, we have E[c(T i+ ) T i ] ( M )c(t i) + (lp loss )/M We can now remove the conditioning and use induction to get E[c(T t+ )] ( M )t c(t ) + (lp loss )( ( M )t ) lp ( + ( M )t ) loss ( ( M )t ), where the second inequality comes rom Lemma (a). The cost o the inal Steiner tree is at most c(alg) c(t t+ ) + t i= loss( i). Moreover, E[c(ALG)] = E[c(T t+ )] + t loss /M lp ( + ( M )t ) +loss (( M )t + t M ( ) lp + ( ) ) t t + M M lp (/ + / exp( t/m) + t/m). 5

6 Here the second inequality uses loss lp /, a weighted average o Lemma (), as well as ( M )t t/m; the third inequality uses ( M )t exp( t/m). The last line explains our choice o t = M ln since λ = ln minimizes + e λ + λ ln, with value +. Thus the algorithm outputs a Steiner tree o expected cost at most ( + ln )lp, which implies the claimed upper ound o + ln on the integrality gap. We now discuss a variant o the result just proven. A Steiner tree instance is quasiipartite i there are no Steiner-Steiner edges. For quasiipartite instances, Roins and Zelikovsky tightened the analysis o their algorithm to show it has approximation ratio α, where α.8 satisies α = + exp( α)). Here, we ll show an integrality gap ound o α (the longer proo o [] via the Roins-Zelikovsky algorithm can e similarly adapted). We can reine Lemma (a) (like in [6]) to show that in quasi-ipartite instances, mst(g[r]) (lp loss ), which inserted into the previous argument gives E[c(ALG)] ( M )t (lp loss ) + (lp loss )( ( M )t ) +loss t/m exp( t/m)(lp loss ) +lp + (t/m )loss = lp ( + exp( t/m)) +loss (t/m exp( t/m)) and setting t = αm gives E[c(ALG)] α lp, as needed. We note that in quasi-ipartite instances the hypergraphic relaxation is equivalent [] to the so-called idirected cut relaxation thus we get an α integrality gap ound there as well. We close with two suggestions or uture work. First, + ln arose in the analysis o two very dierent algorithms (RLC and Roins-Zelikovsky); a simple explanation or this act would e very interesting. Second, the RLC algorithm works or any large enough value o M and in the limit as M, it can e seen that RLC is equivalent to the algorithm which picks each ull component independently with proaility x. The key to see this equivalence is that or any collection L o ull components, RLC picks a set o ull components disjoint rom L with proaility lim M ( L x /M) M ln = P L x, the same as the independent sampling algorithm. It would e nice to analyze this version o the algorithm directly. Reerences [] J. Byrka, F. Grandoni, T. Rothvoß, and L. Sanità. An improved LP-ased approximation or Steiner tree. In Proc. 4nd STOC, pages 58 59, 00. [] Deeparna Chakraarty, Jochen önemann, and David Pritchard. Hypergraphic LP relaxations or Steiner trees. In Proc. 4th IPCO, pages 8 96, 00. Full version at arxiv: [] J. Edmonds. Matroids and the greedy algorithm. Math. Programming, :7 6, 97. [4] C. Gröpl, S. Hougardy, T. Nierho, and H. J. Prömel. Approximation algorithms or the Steiner tree prolem in graphs. In X. Cheng and D.Z. Du, editors, Steiner trees in industries, pages luwer Academic Pulishers, Norvell, Massachusetts, 00. 6

7 [5] Toias Polzin and Siavash Vahdati Daneshmand. On Steiner trees and minimum spanning trees in hypergraphs. Oper. Res. Lett., (): 0, 00. [6] G. Roins and A. Zelikovsky. Tighter ounds or graph Steiner tree approximation. SIAM J. Discrete Math., 9(): 4, 005. Preliminary version appeared in Proc. th SODA, pages , 000. [7] A. Schrijver. Cominatorial optimization. Springer, New York, 00. [8] D.M. Warme. A new exact algorithm or rectilinear Steiner trees. In P.M. Pardalos and D.-Z. Du, editors, Network Design: Connectivity and Facilities Location, pages American Mathematical Society, 997. (Result therein attriuted to M. Queyranne.). 7