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RNC-Approximation Algorithms for the Steiner Problem

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RNC-Approximation Algorithms for the Steiner Problem RNCApproximation Algorithms for the Steiner Problem Hans Jurgen Promel Institut fur Informatik Humb oldt Universitat zu Berlin Berlin Germany proemelinformatikhuberlinde Angelika Steger Institut fur Informatik TU Munc hen Munc hen Germany stegerinformatiktumuenchende Abstract In this pap er we present an RN C algorithm for nding a min imum spanning tree in a weighted uniform hyp ergraph assuming the edge weights are given in unary and a fully p olynomial time randomized approxima tion scheme if the edge weights are given in binary From this result we then derive RN C approximation algorithms for the Steiner problem in networks with approximation ratio for all Intro duction In recent years the Steiner tree problem in graphs attracted considerable attention as well from the theoretical p oint of view as from its applicabili ty eg in VLSIlayout It is rather easy to see and has b een known for a long time that a minimum Steiner tree spanning a given set of terminals in a graph or network can b e approximated in p olyno mial time up to a factor of cf eg Choukhmane or Kou Markowsky Berman After a long p erio d without any progress Zelikovsky Berman and Ramaiyer Zelikovsky and Karpinski and Zelikovsky improved the approximation factor step by step from to In this pap er we present RN C approximation algorithm for the Steiner problem with approximation ratio for all The running time of these algo rithms is p olynomial in and n Our algorithms also give rise to conceptually much easier and faster though randomized sequential approximation algorithms than the KarpinskiZelikovsky approach which almost match their approximation factor The core of our algorithm is a new RN C algorithm for nding a minimum spanning tree in uniform hyp ergraphs The problem of nding a minimum spanning tree in a given graph is well studied and known to b e sequentially solvable almost in linear time Moreover this problem can also b e solved eciently in parallel On the other hand the problem of nding a minimum spanning tree in a k uniform hyp ergraph is known to b e NP hard whenever k The status of the case k however is 1 not yet completely decided For the unweighted case Lovasz provided a very com plicated O n algorithm which was later improved to a O n algorithm by Gab ow and Stallmann Here n denotes the numb er of vertices of the hyp ergraph Also Lovasz presented a conceptionally simple randomized algorithm for the unweighted case by reducing it to a sequence of computations of determinants of appropriate matri ces For the weighted case Camerini Galbiati and Maoli generalized the approach of Lovasz to obtain a randomized pseudop olynomial time algorithm In this pap er we present an RN C algorithm for nding a minimum spanning tree in a weighted uniform hyp ergraph with edgeweights in unary This algorithm simplies to an RN C algorithm for deciding the existence of a spanning tree in an unweighted uniform hyp ergraph and implies a fully p olynomial approximation scheme if the edge weights are given in binary To achieve this result we combine ideas from Mulmuley Vazirani Vazirani which they used to obtain an RN C algorithm for nding a p erfect matching in a given graph with ideas from Lovasz resp Camerini Galbiati and Maoli where they presented a random pseudop olynomial time algorithm for the general problem of nding a base of sp ecied value in a weighted represented matroid sub ject to parity conditions Minimum Spanning Trees in Uniform Hyp ergraphs A hypergraph H V F is a generalization of graphs where F is an arbitrary family of subsets of V and not just a family of element subsets An r uniform hyp ergraph is a hyp ergraph all of whose edges have cardinality exactly r Many notions and results of graph theory generalize to hyp ergraphs Here we just need cycles and trees A cycle of length l in H is a sequence x e x e l l x e of vertices and edges such that the x are distinct vertices the e are distinct l l i i edges x e e and x e e for all i l A hyp ergraph H is a tree l i i i i H is connected and contains no cycles A spanning tree of a hyp ergraph H is a subhyp ergraph T of H that is a tree and satises V T V H In contrary to graphs not every connected hyp ergraph contains a spanning tree Minimum Spanning Tree Problem MST Input A weighted hyp ergraph H V F w where w F N Output Find a spanning tree of H of minimum weight Restricted to uniform hyp ergraphs that is to graphs the minimum spanning tree problem is easily solved eciently sequentially as well as in parallel On the other hand a trivial reduction from Exact Cover by Sets shows that for unweighted uniform hyp ergraphs even deciding whether there exists a spanning tree is NP complete The status of the minimum spanning tree problem for uniform hyp ergraphs is not yet completely resolved Lovasz showed that the problem is in P for unweighted uniform hyp ergraphs His algorithm however is quite complicated and not very ecient Later Gab ow and Stallmann reduced the complexity from O n to O n In Lovasz presented a conceptionally simple randomized algorithm for the unweighted case by reducing it to a sequence of computations of determinants of appropriate matrices Camerini Galbiati and Maoli generalized this approach 2 of Lovasz to obtain a randomized pseudop olynomial time algorithm for the weighted case In this section we develop a randomized parallel algorithm for the minimum span ning tree problem in uniform hyp ergraphs It is also based on the algebraic approach of Lovasz The parallelization is based on ideas from Mulmuley Vazirani Vazi rani We start with some denitions T Let A a b e a skewsymmetric matrix ie A A of size n n and ij let P b e the set of all partitions of f ng into two element sets For an element p ffi i g fi i gg of P we denote by p the sign of the p ermutation n n n n i i i i n n and by p the pro duct n Y p a i i 2j 1 2j j One easily veries that p p is indep endent of the order of the classes and the order within the classes of p Therefore X pfA p p pP is well dened It is called the pfaan of A A wellknown result from linear algebra cf eg says Lemma If A is a skewsymmetric matrix A of size n n and B an arbitrary n n matrix then T detA pfA and pfB AB detB pfA 2 n Lemma Let m n and a b a b be vectors in R and let x x m m m be m indeterminants Then the n n matrix m X T T A x a b b a i i i i i i is skewsymmetric and satises X pfA x x deta jb ja jb i i i i i i 1 n 1 1 n n i i m 1 n With the notation j we just mean the concatenation of the column vectors a and b i i j j Let H V F b e a uniform hyp ergraph on n vertices For every edge f fi j k g in F we pick one vertex arbitrarily say i and let e fi j g ande fi k g f f Let G V E b e a multigraph on the same vertex set as H and with edge set E fe e j f F g f f In fact neither Lovasz nor Camerini Galbiati and Maoli study the minimum spanning tree problem directly They b oth consider the matroid parity problem which contains the minimum spanning tree problem in uniform hyp ergraphs as a sp ecial case 3 Fact A set ff f g F forms a spanning tree in H if and only if fe e n f f 1 1 e e g forms a spanning tree in G 2 f f n n That is the problem of nding a minimum spanning tree in a uniform hyp ergraph is equivalent to the problem of nding a minimum spanning tree in a multigraph where the edges are pairedie either b oth are in the tree or none For every pair of edges e e we dene two ndimensional vectors a and b as f f f f follows if i e if i e f f a b f i f i otherwise otherwise Note that the a s and b s are essentially the incidence vectors of e and e except f f f f that the n st comp onent has b een cut o The following fact resembles a well known prop erty of the incidence matrix of a graph Fact Let f f be n edges in F Then n if f f form a spanning tree in H n j deta jb j ja jb j f f f f 1 1 n n 2 otherwise Finally let A denote the n n matrix X w f T T a b b a A f f f f f F With the ab ove facts and notation at hand it is now relatively straightforward to design an algorithm for constructing a minimum spanning tree whenever this tree is unique Lemma Assume H is a uniform hypergraph on n vertices and w F N is spanning tree T of minimum weight say a weightfunction such that H has a unique w 0 is the highest power of that divides detA Moreover w Then detA and w f T T if we let A A a b b a for al l f F then f f f f f detA f is even f T if and only if w 0 Proof Combining Lemma and Fact we deduce that X w T detA pfA T T where the sum is over all spanning trees T of H and f g for all such trees T Hence X w i 0 detA c with c and appropriate c c Z i i The rst part of the theorem follows For the second part just observe that A is the f matrix corresp onding to the hyp ergraph H f The reasoning ab ove therefore implies that w w 0 0 c if f T f detA f w 0 c if f T f for appropriate constants c Z 2 f 4 To achieve the uniqueness of a minimum spanning tree in general hyp ergraphs we use randomization Lemma Let H V F be a hypergraph on n vertices For every vertex v V choose uniformly and independently at random an integer r v from n and dene P the weight w f of an edge f F as w f r v Then v f ProbThere exists a unique edge of minimum weight 2 Corollary Let H V F w be a weighted uniform hypergraph on n vertices
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