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We investigate the problem of omputing the types of the relationships between Internet Autonomous Systems. We refer to the model introdu ed in [6, 17℄ that bases the dis overy of su h relationships on the analysis of the AS paths extra ted from the BGP routing tables. We hara terize the time omplexity of the above problem, showing both NPompleteness results and eÆ ient algorithms for solving spe i ases. Motivated by the hardness of the general problem, we propose heuristi s based on a novel paradigm and show their e e tiveness against publi ly available data sets. The experiments put in eviden e that our heuristi s performs signi antly better than state of the art heuristi s. 2 1 Introdu tion An Autonomous System (AS) is a portion of Internet under a single administrative authority. Currently, there are more than 10; 000 ASes and their number is rapidly growing. They intera t to oordinate the IP traÆ delivery, ex hanging routing information with a proto ol alled Border Gateway Proto ol (BGP) [16℄. Several authors (see, e.g. [3, 12℄) have pointed out that the relationships between ASes an be roughly lassi ed into ategories that have both a ommer ial and a te hni al avor. A pair of ASes su h that one sells/o ers Internet onne tivity to the other is said to have a providerustomer relationship. If two ASes simply provide onne tivity between their respe tive ustomers are said to have a peer-to-peer relationship. Finally, if two ASes o er ea h other Internet onne tivity are said to be siblings. Of ourse, this lassi ation does not apture all the shades of the possible ommer ial agreements and te hni al details that govern the traÆ ex hanges between ASes but should be onsidered as an important attempt toward understanding the Internet stru ture. Sin e many appli ations would bene t from the knowledge about the Internet stru ture, the resear h on the subje t has re ently produ ed many ontributions. More spe i ally, there is a wide resear h area fo using on the dis overy of the topology underlying the Internet stru ture, either at the AS and at the router level (see, for example, [10, 11, 18℄). Other resear hers on entrate more dire tly on the above mentioned relationships and on the hierar hy that they indu e on the set of ASes. Govindan and Reddy [10℄ study the interplay between the degree of the ASes and their rank in the hierar hy, where the degree of an AS is the number of ASes that have some kind of relationship with it. Gao [6℄ studies, for the rst time, the following problem. ASes are the verti es of a graph (AS graph) where two ASes are adja ent if they ex hange routing information; the edges of su h a graph should be labeled in order to re e t the type of relationship they have. In order to infer the relationships between ASes, Gao uses the information on the degree of ASes together with the AS paths extra ted from the BGP routing tables. An AS path is the sequen e of the ASes traversed by a onne tivity o er (BGP announ ement). In [6℄ a heuristi is presented together with experimental results. An analysis on the properties of the labeled graphs obtained with su h heuristi s is provided in [9℄. Subramanian et al. [17℄ formally de ne, as a minimization problem, a slightly simpli ed version of the problem addressed in [6℄ and onje ture its NPompleteness. They also propose a heuristi based on the observation of the Internet from multiple vantage points, whi h does not rely on the degree of the ASes. Further, they validate the results obtained by the heuristi against a ri h olle tion of data sets. This paper ontributes to the line of resear h opened in [6, 17℄. Namely, its main results are the following. We solve a problem expli itly stated in [17℄. Namely, we hara terize the omplexity of determining the relationships between ASes while minimizing the number of \anomalies". In parti ular: { We show that su h a problem is NPomplete in the general ase; { We produ e a linear time algorithm for determining the AS relationships in the parti ular ase in whi h the problem admits a solution without anomalies; and 3 { We use su h a linear time algorithm to show that for large portions of the Internet (e.g., data obtained from single points of view) it is often possible to determine the relationships between ASes with no anomalies. We introdu e heuristi s, based on a novel approa h, for determining the relationships between ASes with a small number of anomalies. We experimentally show that the proposed approa h leads to heuristi s that performs signi antly better than the utting edge heuristi s of [17℄. The paper is stru tured as follows. Se tion 2 des ribes the addressed problem. Se tions 3 and 4 show an algorithm for testing if the problem admits a solution with no anomalies, and show how to nd a solution if it exists. In Se tion 5 we prove the NPompleteness of the problem in the general ase. Se tion 6 shows new heuristi s and ompare the results with the state of the art. Finally, Se tion 7 ontains on lusions and open problems. 2 Problem Des ription A pre x is a blo k of destination IP addresses. An Internet Autonomous System (AS) applies lo al poli ies to sele t the best route for ea h pre x and to de ide whether to export this route to neighboring ASes. Several authors have pointed out that ASes typi ally have providerustomer or peerto-peer relationships (see, e.g. [3, 12, 7, 17℄). A ustomer exports to a provider its routes and the routes learned from its own ustomers, but does not export routes learned from other providers or peers. A provider exports to a ustomer its routes, the routes learned from the other ustomers, its providers, and its peers. Peers export to ea h other their own routes and the routes learned from their ustomers but do not export the routes learned from their providers and other peers. Consider the AS paths that are asso iated with the BGP announ ements of the routes. If all the ASes adopted export poli ies a ording to the above model, then the AS paths would have a pe uliar stru ture [6, 17℄. Namely, (1) no AS path an ontain more than one pair of ASes having a peer-to-peer relationship; and (2) on e a providerustomer or a peer-to-peer pair of ASes is met in the AS path, no ustomer-provider an be found in the remaining part of it. Further, the above mentioned pe uliarities of the AS paths have been formally stated in a theorem of [6℄, that has been also reasted in [17℄. A graph-theoreti formulation of the same theorem will be given in what follows. 2.1 Type-of-Relationship problem The relationships between ASes in the Internet may be represented as a graph G whose edges are either dire ted or undire ted. Ea h vertex is an AS, a dire ted edge from vertex u to vertex v indi ates that u is a ustomer of v (providerustomer relationship), and an undire ted edge between vertex w and vertex z indi ates that w and z are peers (peerto-peer relationship). A BGP AS path orresponds to a path on G. Suppose path p is omposed by the sequen e of verti es v1; : : : ; vn, then p is valid if it is of one of the following two types. 4 Type 1: p is omposed by a (possibly empty) sequen e of forward edges followed by a (possibly empty) sequen e of ba kward edges; more formally, there exists a vertex vi of p su h that for j 2 1; : : : ; i 1 edge (vj; vj+1) is dire ted from vj to vj+1 and for j 2 i; : : : ; n 1 edge (vj; vj+1) is dire ted from vj+1 to vj. Type 2: p is omposed by a (possibly empty) sequen e of forward edges, followed by an undire ted edge, followed by a (possibly empty) sequen e of ba kward edges; more formally, there exists a vertex vi of p su h that for j 2 1; : : : ; i 1 edge (vj; vj+1) is dire ted from vj to vj+1, edge (vi; vi+1) is undire ted, and for j 2 i + 1; : : : ; n 1 edge (vj; vj+1) is dire ted from vj+1 to vj. An invalid path is a path that is not valid. At this point the above mentioned theorem [17℄ an be restated as follows: if every AS obeys the ustomer, peer, and provider export poli ies, then every advertised path is either of Type 1 or of Type 2. However, the Internet is more omplex. To give a few examples: ASes operated by the same ompany an have a sibling relationship, where ea h AS exports all its routes to the other; two ASes may agree a ba kup relationship between them, to over ome possible failures; or ASes may have peering relationships through intermediate ASes. However, nding out whi h is the portion of Internet that obeys the ustomer, peer, and provider export poli ies an be onsidered as the rst step toward a omplete omprehension of the relationships between ASes. Su h motivations have pushed the authors of [17℄ toward identifying the following problem. Type-of-Relationship (ToR) Problem [17℄: Given an undire ted graph G and a set of paths P , give an orientation to some of the edges of G to minimize the number of invalid paths in P . Figure 1 shows an instan e of the ToR problem for whi h an orientation without invalid paths annot be found. In parti ular, ea h orientation of edge (AS701, AS5056) yields at least one invalid path. Suppose, in fa t, that edge (AS701, AS5056) was dire ted from AS701 to AS5056. Path AS5056, AS701, AS4926, AS6461, AS2914, AS174, AS14318 (drawn solid in the gure) would be valid only if edge (AS4926, AS6461) was dire ted from AS6461 to AS4926. Similarly, path AS5056, AS701, AS6461, AS4926, AS4270, AS4387 (drawn dotted in the gure) would be valid only if edge (AS4926, AS6461) was dire ted from AS4926 to AS6461. Hen e, we have a ontradi tion, sin e edge (AS4926, AS6461) should have opposite orientations. Now, suppose that edge (AS701, AS5056) was undire ted. The same arguments apply, leading to the same ontradi tion. Finally, suppose that edge (AS701, AS5056) was dire ted from AS5056 to AS701. It is easy to see that in this ase we have a ontradi tion on the orientation of edge (AS1, AS1239). Figure 2 shows an instan e of the ToR problem that admits an orientation without invalid paths. Figures 3 and 4 show a possible orientation. 2.2 Simplifying the problem The Type-of-Relationship Problem is a minimization problem. In order to studying it, following a standard te hnique [8℄, we onsider its orresponding de ision version as follows. 5