Random Shortest Paths: Non-Euclidean Instances for Metric Optimization Problems
Karl Bringmann, Christian Engels, Bodo Manthey, B.V. Raghavendra Rao
aa r X i v : . [ c s . D S ] M a y Random Shortest Paths: Non-Euclidean Instances forMetric Optimization Problems ∗ Karl Bringmann † , Christian Engels , Bodo Manthey , andB. V. Raghavendra Rao Max Planck Institute for Informatics, Saarbr¨ucken, Germany, [email protected] Saarland University, Saarbr¨ucken, Germany, [email protected] University of Twente, Enschede, Netherlands, [email protected] Indian Institute of Technology Madras, Chennai, India, [email protected]
May 23, 2014
Probabilistic analysis for metric optimization problems has mostly been con-ducted on random Euclidean instances, but little is known about metric instancesdrawn from distributions other than the Euclidean. This motivates our study ofrandom metric instances for optimization problems obtained as follows: Every edgeof a complete graph gets a weight drawn independently at random. The distancebetween two nodes is then the length of a shortest path (with respect to the weightsdrawn) that connects these nodes.We prove structural properties of the random shortest path metrics generated inthis way. Our main structural contribution is the construction of a good clustering.Then we apply these findings to analyze the approximation ratios of heuristics formatching, the traveling salesman problem (TSP), and the k -median problem, aswell as the running-time of the 2-opt heuristic for the TSP. The bounds that weobtain are considerably better than the respective worst-case bounds. This suggeststhat random shortest path metrics are easy instances, similar to random Euclideaninstances, albeit for completely different structural reasons. For large-scale optimization problems, finding optimal solutions within reasonable time is of-ten impossible, because many such problems, like the traveling salesman problem (TSP), areNP-hard. Nevertheless, we often observe that simple heuristics succeed surprisingly quickly infinding close-to-optimal solutions. Many such heuristics perform well in practice but have a ∗ To appear in
Algorithmica . An extended abstract of this work has appeared in the
Proceedings of the 38thInt. Symp. on Mathematical Foundations of Computer Science (MFCS 2013) . † Karl Bringmann is a recipient of the
Google Europe Fellowship in Randomized Algorithms , and this researchis supported in part by this Google Fellowship. random shortest path metrics .This model is also known as first-passage percolation , and has been introduced by Broadbentand Hammersley as a model for passage of fluid in a porous medium [10, 11]. More recently, ithas also been used to model shortest paths in networks such as the Internet [16]. The appealingfeature of random shortest path metrics is their simplicity, which enables us to use them forthe analysis of heuristics.
There has been significant study of random shortest path metrics or first-passage percolation.The expected length of an edge is known to be ln n/n [13, 24]. Asymptotically the samebound holds also for the longest edge almost surely [21, 24]. These results hold not only forthe exponential distribution, but for every distribution with distribution function F satisfying F ( x ) = x + o ( x ) for small values of x [24]. (See also Section 6.) This model has been usedto analyze algorithms for computing shortest paths [20, 21, 34]. Kulkarni and Adlakha havedeveloped algorithmic methods to compute distribution and moments of several optimizationproblems [30–32]. Beyond shortest path algorithms, random shortest path metrics have beenapplied only rarely to analyze algorithms. Dyer and Frieze [15], answering a question raisedby Karp and Steele [27, Section 3.4], analyzed the patching heuristic for the asymmetric TSP(ATSP) in this model. They showed that it comes within a factor of 1 + o (1) of the optimalsolution with high probability. Hassin and Zemel [21] applied their findings to the 1-centerproblem.From a more structural point of view, first-passage percolation has been analyzed in the areaof complex networks, where the hop-count (the number of edges on a shortest path) and thelength of shortest path trees have been analyzed [23]. These properties have also been studiedon random graphs with random edge weights in various settings [7–9, 22, 29]. Addario-Berryet al. [1] have shown that the number of edges in the longest of the shortest paths is O (log n )with high probability, and hence the shortest path trees have depth O (log n ).2 .2 Our Results As far as we are aware, simple heuristics such as greedy heuristics have not been studied inthis model yet. Understanding the performance of such algorithms is particularly importantas they are easy to implement and used in many applications.We provide a probabilistic analysis of simple heuristics for optimization under random short-est path metrics. First, we provide structural properties of random shortest path metrics(Section 3). Our most important structural contribution is proving the existence of a goodclustering (Lemma 3.9). Then we use these structural insights to analyze simple algorithmsfor minimum weight matching and the TSP to obtain better expected approximation ratioscompared to the worst-case bounds. In particular, we show that the greedy algorithm forminimum-weight perfect matching (Theorem 4.2), the nearest-neighbor heuristic for the TSP(Theorem 4.4), and every insertion heuristic for the TSP (Theorem 4.6) achieve constant ex-pected approximation ratios. We also analyze the 2-opt heuristic for the TSP and show that theexpected number of 2-exchanges required before the termination of the algorithm is boundedby O ( n log n ) (Theorem 4.7). Investigating further the structural properties of random short-est path metrics, we then consider the k -median problem (Section 5), and show that the mosttrivial procedure of choosing k arbitrary vertices as k -median yields a 1 + o (1) approximationin expectation, provided k = O ( n − ε ) for some ε > We consider undirected complete graphs G = ( V, E ) without loops. First, we draw edge weights w ( e ) independently at random according to the exponential distribution with parameter 1.Second, let the distances d : V × V → [0 , ∞ ) be given as follows: the distance d ( u, v ) between u and v is the minimum total weight of a path connecting u and v . In particular, we have d ( v, v ) = 0 for all v ∈ V , d ( u, v ) = d ( v, u ) because G is undirected, and the triangle inequality: d ( u, v ) ≤ d ( u, x ) + d ( x, v ) for all u, x, v ∈ V . We call the complete graph with distances d obtained from random weights w a random shortest path metric .We use the following notation: Let ∆ max = max u,v d ( u, v ) denote the diameter of the randomshortest path metric. Let B ∆ ( v ) = { u ∈ V | d ( u, v ) ≤ ∆ } be the ball of radius ∆ around v ,i.e., the set of all nodes whose distance to v is at most ∆.We denote the minimal ∆ such that there are at least k nodes within a distance of ∆ of v by τ k ( v ). Formally, we define τ k ( v ) = min { ∆ | | B ∆ ( v ) | ≥ k } .By Exp( λ ), we denote the exponential distribution with parameter λ . If a random variable X is distributed according to a probability distribution P , we write X ∼ P . In particular, X ∼ P mi =1 Exp( λ i ) means that X is the sum of m independent exponentially distributedrandom variables with parameters λ , . . . , λ m .By exp, we denote the exponential function. For n ∈ N , let [ n ] = { , . . . , n } and let H n = P ni =1 /i be the n -th harmonic number. Exponential distributions are technically the easiest to handle because they are memoryless. We will discussother distributions in Section 6. Structural Properties of Shortest Path Metrics
To understand random shortest path metrics, it is convenient to fix a starting vertex v and seehow the lengths from v to the other vertices develop. In this way, we analyze the distributionof τ k ( v ).The values τ k ( v ) are generated by a simple birth process as follows. (The same process hasbeen analyzed by Davis and Prieditis [13], Janson [24], and also in subsequent papers.) For k = 1, we have τ k ( v ) = 0.For k ≥
1, we are looking for the closest vertex to any vertex in B τ k ( v ) ( v ) in order to obtain τ k +1 ( v ). This conditions all edges ( u, x ) with u ∈ B τ k ( v ) ( v ) and x / ∈ B τ k ( v ) ( v ) to be of lengthat least τ k ( v ) − d ( v, u ). The set B τ k ( v ) ( v ) contains k vertices. Thus, there are k · ( n − k ) edgesto the rest of the graph. Consequently, the difference δ k = τ k +1 ( v ) − τ k ( v ) is distributed asthe minimum of k ( n − k ) exponential random variables (with parameter 1), or, equivalently, asExp( k · ( n − k )). We obtain that τ k +1 ( v ) ∼ P ki =1 Exp (cid:0) i · ( n − i ) (cid:1) . Note that these exponentialdistributions as well as the random variables δ , . . . , δ n are independent.Exploiting linearity of expectation and that the expected value of Exp( λ ) is 1 /λ we obtainthe following lemma. Lemma 3.1.
For any k ∈ [ n ] and any v ∈ V , we have E (cid:0) τ k ( v ) (cid:1) = n · (cid:0) H k − + H n − − H n − k (cid:1) and τ k ( v ) ∼ P k − i =1 Exp (cid:0) i · ( n − i ) (cid:1) .Proof. The proof is by induction on k . For k = 1, we have τ k ( v ) = 0 and H k − + H n − − H n − k = H + H n − − H n − = 0. Now assume that the lemma holds for k for some k ≥
1. In theparagraph preceding this lemma we have seen that τ k +1 ( v ) − τ k ( v ) ∼ Exp( k ( n − k )). Thus, E ( τ k +1 ( v ) − τ k ( v )) = k ( n − k ) . Plugging in the induction hypothesis yields E (cid:0) τ k +1 ( v ) (cid:1) = E (cid:0) τ k ( v ) (cid:1) + 1 k · ( n − k ) = 1 n · (cid:18) H k − + H n − − H n − k + 1 k + 1 n − k (cid:19) = 1 n · (cid:0) H k + H n − − H n − ( k +1) (cid:1) . From this result, we can easily deduce two known results: averaging over k yields that theexpected distance of an edge is H n − n − ≈ ln n/n [13, 24]. The longest distance from v to anyother node is τ n ( v ), which is 2 H n − /n ≈ n/n in expectation [24]. For completeness, let usmention that the diameter ∆ max is approximately 3 ln n/n [24]. However, this does not followimmediately from Lemma 3.1. τ k ( v ) Let us now have a closer look at cumulative distribution function of τ k ( v ) for fixed v ∈ V and k ∈ [ n ]. To do this, the following lemma is very useful. Lemma 3.2.
Let X ∼ P ni =1 Exp( ci ) . Then P ( X ≤ α ) = (1 − e − cα ) n .Proof. The random variable X has the same distribution as max ni =1 Y i , where Y i ∼ Exp( c ).We have X ≤ α if and only if Y i ≤ α for all i ∈ { , . . . , n } .4n the following, let F k denote the cumulative distribution function of τ k ( v ) for some fixedvertex v ∈ V , i.e., F k ( x ) = P ( τ k ( v ) ≤ x ). Lemma 3.3.
For every ∆ ≥ , v ∈ V , and k ∈ [ n ] , we have (cid:0) − exp( − ( n − k )∆) (cid:1) k − ≤ F k (∆) ≤ (cid:0) − exp( − n ∆) (cid:1) k − . Proof.
Lemma 3.1 states that τ k ( v ) ∼ P k − i =1 Exp( i ( n − i )). We approximate the parametersby ci for c ∈ { n − k, n } . The distribution with c = n is stochastically dominated by the truedistribution, which is in turn dominated by the distribution obtained for c = n − k . We applyLemma 3.2 with c = n and c = n − k . Lemma 3.4.
Fix ∆ ≥ and a vertex v ∈ V . Then (cid:0) − exp( − ( n − k )∆) (cid:1) k − ≤ P (cid:0) | B ∆ ( v ) | ≥ k (cid:1) ≤ (cid:0) − exp( − n ∆) (cid:1) k − . Proof.
We have | B ∆ ( v ) | ≥ k if and only if τ k ( v ) ≤ ∆. The lemma follows from Lemma 3.3.We can improve Lemma 3.3 slightly in order to obtain even closer upper and lower bounds.For n, k ≥
2, combining Lemmas 3.3 and 3.5 yields tight upper and lower bounds if we disregardthe constants in the exponent, namely F k (∆) = (cid:0) − exp( − Θ( n ∆)) (cid:1) Θ( k ) . Lemma 3.5.
For all v ∈ V , k ∈ [ n ] , and ∆ ≥ , we have F k (∆) ≥ (cid:0) − exp( − ( n − / (cid:1) n − and F k (∆) ≥ (cid:0) − exp( − ( n − / (cid:1) ( k − . Proof. As τ k ( v ) is monotonically increasing in k , we have F k (∆) ≥ F k +1 (∆) for all k . Thus,we have to prove the claim only for k = n . In this case, τ n ( v ) ∼ P n − i =1 Exp( λ i ), with λ i = i ( n − i ) = λ n − i . Setting m = ⌈ n/ ⌉ and exploiting the symmetry around m yields τ n ( v ) ≤ m X i =1 Exp( λ i ) + m X i =1 Exp( λ i ) = τ m ( v ) + τ m ( v ) . Here, “ ≤ ” means stochastic dominance, “=” means equal distribution, and “+” means addingup two independent random variables. Hence, F n (∆) = P (cid:0) τ n ( v ) ≤ ∆ (cid:1) ≥ P (cid:0) τ m ( v ) + τ m ( v ) ≤ ∆ (cid:1) ≥ P (cid:0) τ m ( v ) ≤ ∆ / (cid:1) . By Lemma 3.3, and using m ≤ ( n + 1) /
2, this is bounded by F n (∆) ≥ (1 − exp( − ( n − m )∆ / m − ≥ (1 − exp( − ( n − / n − . For the second inequality, we use the first inequality of Lemma 3.5 for k − ≥ ( n −
1) andLemma 3.3 for k − < ( n −
1) as then n − k ≥ ( n − / .3 Tail Bounds for | B ∆ ( v ) | and ∆ max Our first tail bound for | B ∆ ( v ) | , which is the number of vertices within distance ∆ of a givenvertex v , follows directly from Lemma 3.3. From this lemma we derive the following corollary,which is a crucial ingredient for the existence of good clusterings and, thus, for the analysis ofheuristics in the remainder of this paper. Corollary 3.6.
Let n ≥ and fix ∆ ≥ and a vertex v ∈ V . Then we have P (cid:18) | B ∆ ( v ) | < min (cid:26) exp (∆ n/ , n + 12 (cid:27)(cid:19) ≤ exp ( − ∆ n/ . Proof.
Lemma 3.4 yields P (cid:18) | B ∆ ( v ) | < min (cid:26) exp (cid:18) ∆ n − (cid:19) , n + 12 (cid:27)(cid:19) ≤ − (cid:18) − exp (cid:18) − n −
12 ∆ (cid:19)(cid:19) exp(∆( n − / ≤ exp (cid:18) − ∆ n − (cid:19) , where the last inequality follows from (1 − x ) y ≥ − xy for y ≥ x ≥
0. Using ( n − / ≥ n/ n ≥ Corollary 3.7.
Fix ∆ ≥ , a vertex v ∈ V , and any c > . Then P (cid:0) | B ∆ ( v ) | ≥ exp( c ∆ n ) (cid:1) < exp (cid:0) − ( c − n (cid:1) . Proof.
Lemma 3.4 with k = c ∆ n yields P (cid:0) | B ∆ ( v ) | ≥ exp( c ∆ n ) (cid:1) ≤ (cid:0) − exp( − n ∆) (cid:1) exp( c ∆ n ) − . Using 1 + x ≤ e x , we get P (cid:0) | B ∆ ( v ) | ≥ exp( c ∆ n ) (cid:1) ≤ exp (cid:0) exp( − n ∆) − exp (cid:0) ( c − · ∆ n (cid:1)(cid:1) . Now, we bound exp( − n ∆) ≤ (cid:0) ( c − · ∆ n (cid:1) ≥ c − · ∆ n , which yields theclaimed inequality.Janson [24] derived the following tail bound for the diameter ∆ max . A qualitatively similarbound can be proved using Lemma 3.4 and can also be derived from Hassin and Zemel’s anal-ysis [21]. However, Janson’s bound is stronger with respect to the constants in the exponent. Lemma 3.8 (Janson [24, p. 352]) . For any fixed c > , we have P (∆ max > c ln( n ) /n ) ≤ O ( n − c log n ) . .4 Balls and Clusters In this section, we show our main structural contribution, which is a global property of randomshortest path metrics. We show that such instances can be divided into a small number ofclusters of any given diameter.From now on, let s ∆ = min { exp(∆ n/ , ( n + 1) / } , as in Corollary 3.6. If | B ∆ ( v ) | , thenumber of vertices within distance ∆ of v , is at least s ∆ , then we call the vertex v a dense ∆ -center , and we call the set B ∆ ( v ) of vertices within distance ∆ of v (including v itself) the∆ -ball of v . Otherwise, if | B ∆ ( v ) | < s ∆ , and v is not part of any ∆-ball, we call the vertex v a sparse ∆ -center . Any two vertices in the same ∆-ball have a distance of at most 2∆ becauseof the triangle inequality.If ∆ is clear from the context, then we also speak about centers and balls without parameter.We can bound, by Corollary 3.6, the expected number of sparse ∆-centers to be at most O ( n/s ∆ ).We want to partition the graph into a small number of clusters, each of diameter at most 6∆.For this purpose, we put each sparse ∆-center in its own cluster (of size 1). Then the diameterof each such cluster is 0, which is trivially upper-bounded by 6∆, and the number of theseclusters is expected to be at most O ( n/s ∆ ).We are left with the dense ∆-centers, which we cluster using the following algorithm: Con-sider an auxiliary graph whose vertices are all dense ∆-centers. We draw an edge betweentwo dense ∆-centers u and v if B ∆ ( u ) ∩ B ∆ ( v ) = ∅ . Now consider any maximal independentset of this auxiliary graph (for instance, a greedy independent set), and let t be the numberof its vertices. Then we form initial clusters C ′ , . . . , C ′ t , each containing one of the ∆-ballscorresponding to the vertices in the independent set. By the independence, all these t ∆-ballsare disjoint, which implies t ≤ n/s ∆ . The ball of every remaining center v has at least onevertex in one of the C ′ i . We add all remaining vertices of B ∆ ( v ) to such a C ′ i to form the finalclusters C , . . . , C t . By construction, the diameter of each C i is at most 6∆: Consider any twovertices u, v ∈ C i . The distance of u towards its closest neighbor in the initial ball C ′ i is atmost 2∆. The same holds for v . Finally, the diameter of the initial ball C ′ i is also at most 2∆.With this partitioning, we have obtained the following structure: We have an expectednumber of O ( n/s ∆ ) clusters of size 1 and diameter 0, and a number of O ( n/s ∆ ) clusters of sizeat least s ∆ and diameter at most 6∆. Thus, we have O ( n/s ∆ ) = O (1 + n/ exp(∆ n/ Lemma 3.9.
Consider a random shortest path metric and let ∆ ≥ . If we partition theinstance into clusters, each of diameter at most , then the expected number of clustersneeded is O (1 + n/ exp(∆ n/ . Finding minimum-length perfect matchings in metric instances is the first problem that weconsider. This problem has been widely considered in the past and has applications in, e.g.,optimizing the speed of mechanical plotters [35, 38]. The worst-case running-time of O ( n ) forfinding an optimal matching is prohibitive if the number n of points is large. Thus, simple7euristics are often used, with the greedy heuristic being probably the simplest one: at everystep, choose an edge of minimum length incident to the unmatched vertices and add it to thepartial matching. Let GREEDY denote the cost of the matching output by this greedy matchingheuristic, and let MM denote the optimum value of the minimum-length perfect matching. Theworst-case approximation ratio for greedy matching on metric instances is Θ( n log (3 / ) [35],where log (3 / ≈ .
58. In the case of Euclidean instances, the greedy algorithm has anapproximation ratio of O (1) with high probability on random instances [5]. For independentrandom edge weights (without the triangle inequality), the expected weight of the matchingcomputed by the greedy algorithm is Θ(log n ) [14] whereas the optimal matching has a weightof Θ(1) with high probability, which gives an O (log n ) approximation ratio.We show that greedy matching finds a matching of constant expected length on randomshortest path metrics. Theorem 4.1. E ( GREEDY ) = O (1) .Proof. Let ∆ i = in . We divide the run of GREEDY in phases as follows: we say that
GREEDY is in phase i if edges { u, v } are inserted such that d ( u, v ) ∈ (6∆ i − , i ]. Lemma 3.8 allows toshow that the expected sum of all edges longer than ∆ Ω(log n ) is o (1), so we can ignore them. GREEDY goes through phases i with increasing i (phases can be empty). We now estimatethe contribution of phase i to the matching computed by GREEDY . Using Lemma 3.9, afterphase i − i − using an expectednumber of O (1+ n/e ( i − / ) clusters. Each such cluster can have at most one unmatched vertex.Thus, we have to add at most O (1 + n/e ( i − / ) edges in phase i . Each such edge connectsvertices at a distance of at most 6∆ i . Hence, the contribution of phase i is O ( in · (1+ n/e ( i − / ))in expectation. Summing over all phases yields the desired bound: E (cid:0) GREEDY (cid:1) = o (1) + O (log n ) X i =1 O (cid:18) ie ( i − / + in (cid:19) = O (1) . Careful analysis allows us to bound the expected approximation ratio.
Theorem 4.2.
The greedy algorithm for minimum-length perfect matching has constant ap-proximation ratio on random shortest path metrics, i.e., E (cid:0) GREEDYMM (cid:1) = O (1) . We will use the following tail bound to estimate the approximation ratios of the greedyheuristic for matching as well as the nearest-neighbor and insertion heuristics for the TSP.
Lemma 4.3.
Let α ∈ [0 , . Let S m be the sum of the lightest m edge weights, where m ≥ αn .Then, for all c ∈ [0 , , we have P ( S m ≤ c ) ≤ (cid:18) e c α (cid:19) αn . Furthermore,
TSP ≥ MM ≥ S n/ , where TSP and MM denote the length of the shortest TSPtour and the minimum-weight perfect matching, respectively, in the corresponding shortest pathmetric. roof. Let X ∼ P mi =1 Exp(1), and let Y be the sum of m independent random variables drawnuniformly from [0 , X stochastically dominates Y , and P ( Y ≤ c ) = c m /m !.The probability that S m ≤ c is at most the probability that there exists a subset of the edgesof cardinality m whose total weight is at most c . By a union bound and using (cid:0) ab (cid:1) ≤ ( ae/b ) b , (cid:0) n (cid:1) ≤ n /
2, and a ! > ( a/e ) a , we obtain P ( S m ≤ c ) ≤ (cid:18)(cid:0) n (cid:1) m (cid:19) · c m m ! ≤ (cid:18) n e c m (cid:19) m ≤ (cid:18) e c α (cid:19) m . We can replace m by its lower bound αn in the exponent [2, Fact 2.1] to obtain the first claim.It remains to prove TSP ≥ MM ≥ S n/ . The first inequality is trivial. For the secondinequality, consider a minimum-weight perfect matching in a random shortest path metric.We replace every edge by the corresponding paths. If we disregard multiple edges, then we arestill left with at least n/ n/ MM and at least S n/ . Proof of Theorem 4.2.
The worst-case approximation ratio of
GREEDY for minimum-weightperfect matching is n log (3 / [35]. Let c > GREEDY on random shortest path instances is E (cid:18) GREEDYMM (cid:19) ≤ E (cid:18) GREEDY c (cid:19) + P ( MM < c ) · n log (3 / . By Theorem 4.1, the first term is O (1). Since c is sufficiently small, Lemma 4.3 shows that thesecond term is o (1). A greedy analogue for the traveling salesman problem (TSP) is the nearest neighbor heuristic:(1) Start with some starting vertex v as the current vertex v . (2) At every iteration, choosethe nearest yet unvisited neighbor u of the current vertex v (called the successor of v ) as thenext vertex in the tour, and move to the next iteration with the new vertex u as the currentvertex v . (3) Go back to the first vertex v if all vertices are visited. Let NN denote boththe nearest-neighbor heuristic itself and the cost of the tour computed by it. Let TSP denotethe cost of an optimal tour. The nearest-neighbor heuristic NN achieves a worst-case ratio of O (log n ) for metric instances and also an average-case ratio (for independent, non-metric edgelengths) of O (log n ) [4]. We show that NN achieves a constant approximation ratio on randomshortest path instances. Theorem 4.4.
For random shortest path instances we have E ( NN ) = O (1) and E (cid:0) NNTSP (cid:1) = O (1) .Proof. The proof is similar to the proof of Theorem 4.2. Let ∆ i = i/n for i ∈ N . Let Q = O (log n/n ) be sufficiently large.Consider the clusters obtained with parameter ∆ i as in the discussion preceding Lemma 3.9.These clusters have diameters of at most 6∆ i . We refer to these clusters as the i -clusters . Let v be any vertex. We call v bad at i , if v is in some i -cluster and NN chooses a vertex at adistance of more than 6∆ i from v for leaving v . Hence, if v is bad at i , then the next vertex9ies outside of the cluster to which v belongs. (Note that v is not bad at i if the outgoing edgeat v leads to a neighbor outside of the cluster of v but at a distance of at most 6∆ i from v .)In the following, let the cost of a vertex v be the distance from v to its successor u . Thelength of the tour produced by NN is equal to the sum of costs over all vertices. Claim 4.5.
The expected number of vertices with costs in the range (6∆ i , i +1 ] is at most O (1 + n/ exp( i/ .Proof of Claim 4.5. Suppose that the cost of the neighbor chosen by NN for a vertex v is inthe interval (6∆ i , i +1 ]. Then v is bad at i . This happens only if all other vertices of the i -cluster containing v have already been visited. Otherwise, there would be another vertex u in the same i -cluster with a distance of at most 6∆ i to v . By Lemma 3.9, the number of i -clusters is at most O (1 + n/ exp( i/ max ≤ Q , then it suffices to consider i for i ≤ O (log n ). If ∆ max > Q , then we boundthe value of the tour produced by NN by n ∆ max . This failure event, however, contributes only o (1) to the expected value by Lemma 3.8. For the case ∆ max ≤ Q , the contribution to theexpected length of the NN tour is bounded from above by O (log n ) X i =0 i +1 · O (cid:18) n exp( i/ (cid:19) = O (log n ) X i =0 O (cid:18) i + 1 n + i + 1exp( i/ (cid:19) = O (1) . Using the fact that the worst-case approximation ratio of NN is O (log n ), the proof of theconstant expected approximation ratio is similar to the proof of Theorem 4.2. An insertion heuristic for the TSP is an algorithm that starts with an initial tour on a fewvertices and extends this tour iteratively by adding the remaining vertices. In every iteration,a vertex is chosen according to some rule, and this vertex is inserted at the place in the currenttour where it increases the total tour length the least. The approximation ratio achieveddepends on the rule used for selecting the next node to insert. Certain insertion heuristics suchas nearest neighbor insertion (which is different from the nearest neighbor algorithm from theprevious section) achieve constant approximation ratios [36]. The random insertion algorithm,where the next vertex is chosen uniformly at random from the remaining vertices, has a worst-case approximation ratio of Ω(log log n/ log log log n ), and there are insertion heuristics with aworst-case approximation ratio of Ω(log n/ log log n ) [6].A rule R that specifies an insertion heuristic can be viewed as follows: depending on thedistances d , it (1) chooses a set R V of vertices for computing an initial tour and (2) given anytour of vertices V ′ ⊇ R V , describes how to choose the next vertex. Let INSERT R denote thelength of the tour produced with rule R .For random shortest path metrics, we show that any insertion heuristic produces a tourwhose length is expected to be within a constant factor of the optimal tour. This result holdsirrespective of which insertion strategy we actually use. Theorem 4.6.
For every rule R , we have E ( INSERT R ) = O (1) and E (cid:0) INSERT R TSP (cid:1) = O (1) . roof. Let ∆ i = i/n for i ∈ N and Q = O (log n/n ) be sufficiently large. Assume that ∆ max ≤ Q . If ∆ max > Q , then we bound the length of the tour produced by n · ∆ max . This contributesonly o (1) to the expected value of length of the tour produced by Lemma 3.8.Suppose we have a partial tour T and v is the vertex that we have to insert next. If T hasa vertex u such that v and u are in a common i -cluster, then the triangle inequality impliesthat the costs of inserting v into T is at most 12∆ i because the diameters of i -clusters are atmost 6∆ i [36, Lemma 2]. For each i , only the insertion of the first vertex of each i -clustercan possibly cost more than 12∆ i . Thus, the number of vertices whose insertion would incurcosts in the range (12∆ i , i +1 ] is at most O (cid:0) n exp( i/ (cid:1) in expectation. Note that we onlyhave to consider i with i ≤ O (log n ) since ∆ max ≤ Q . The expected costs of the initial tourare at most TSP = O (1) [19]. Summing up the expected costs for all i plus the costs of theinitial tour, we obtain that the expected costs of the tour obtained by an insertion heuristic isbounded from above by E ( INSERT R ) = O (1) + O (log n ) X i =0 ∆ i · O (cid:18) n exp( i/ (cid:19) = O (1) . Note that the above argument is independent of the rule R used.The proof for the approximation ratio is similar to the proof of Theorem 4.2 and uses theworst-case ratio of O (log n ) for insertion heuristics for any rule R [36, Theorem 3]. The 2-opt heuristic for the TSP starts with an initial tour and successively improves thetour by so-called 2-exchanges until no further refinement is possible. In a 2-exchange, a pairof edges e = { v , v } and e = { v , v } , where v , v , v , v appear in this order in theHamiltonian tour, are replaced by a pair of edges e = { v , v } and e = { v , v } to geta shorter tour. The 2-opt heuristic is easy to implement and widely used. In practice, itusually converges quite quickly to close-to-optimal solutions [25]. To explain its performancein practice, probabilistic analyses of its running-time on geometric instances [18, 28, 33] andits approximation performance on geometric instances [18] and with independent, non-metricedge lengths [17] have been conducted. We prove that for random shortest path metrics, theexpected number of iterations that 2-opt needs is bounded by a polynomial. Theorem 4.7.
The expected number of iterations that 2-opt needs to find a local optimum isbounded by O ( n log n ) .Proof. The proof is similar to the analysis of 2-opt by Englert et al. [18]. Consider a 2-exchangewhere edges e and e are replaced by edges f and f as described above. The improvementobtained from this exchange is given by δ = δ ( v , v , v , v ) = d ( v , v ) + d ( v , v ) − d ( v , v ) − d ( v , v ).We estimate the probability P ( δ ∈ (0 , ε ]) of the event that the improvement is at most ε for some ε >
0. The distances d ( v i , v j ) correspond to shortest paths with respect to theexponentially distributed edge weights w . Assume for the moment that we know these paths.Then we can rewrite the improvement as δ = X e ∈ E α e w ( e ) (1)11or some coefficients α e ∈ {− , − , , , } . If the exchange considered is indeed a 2-exchange,then δ >
0. Thus, in this case, there exists at least on edge e = { u, u ′ } with α e = 0. Let I ⊆ { e , e , e , e } be the set of edges of the 2-exchange such that the corresponding pathsuse e .For all combinations of I and e , let δ I,eij be the following quantity: • If e ij / ∈ I , then δ I,eij is the length of a shortest path from v i to v j without using e . • If e ij ∈ I , then δ I,eij is the minimum of – the length of a shortest path from v i to u without e plus the length of a shortestpath from u ′ to v j without e and – the length of a shortest path from v i to u ′ without e plus the length of a shortestpath from u to v j without e .Let δ e,I = δ e,I + δ e,I − δ e,I − δ e,I . Claim 4.8.
For every outcome of the random edge weights, there exists an edge e and a set I such that δ = δ e,I + αw ( e ) , where α ∈ {− , − , , } is determined by e and I .Proof of Claim 4.8. Fix the edge weights arbitrarily and consider any four shortest paths.Then there exists some edge e with non-zero α e in (1). We choose this e , an appropriate set I , and we choose α = α e . Then the claim follows from the definition of δ e,I .Claim 4.8 yields that δ ∈ (0 , ε ] implies that there are an e and an I with δ e,I + αw ( e ) ∈ (0 , ε ]. Claim 4.9.
Let e and I be arbitrary with α = α e = 0 . Then P ( δ e,I + αw ( e ) ∈ (0 , ε ]) ≤ ε .Proof of Claim 4.9. We fix the edge weights of all edges except for e . This determines δ e,I .Thus, δ e,I + αw ( e ) ∈ (0 , ε ] if and only of w ( e ) assumes a value in a now fixed interval of size ε/α ≤ ε . Since the density of the exponential distribution is bounded from above by 1, theclaim follows.The number of possible choices for e and I is O ( n ). Thus, P ( δ ∈ (0 , ε ]) = O ( n ε ).Let δ min > n different 2-exchanges, we have P ( δ min ≤ ε ) = O ( n ε ).The initial tour has a length of at most n ∆ max . Let T be the number of iterations that 2-opttakes. Then T ≤ n ∆ max /δ min . Now, T > x implies ∆ max /δ min > x/n . The event ∆ max /δ min >x/n is contained in the union of the events ∆ max > log x ln n/n , and δ min < ln n · log x/x . Thefirst happens with a probability of at most n − Ω(log( x )) by Lemma 3.8. The second happenswith a probability of at most O ( n log( x ) /x ). Thus, we obtain P ( T > x ) ≤ n − Ω(log( x )) + O (cid:0) n ln n · log( x ) /x (cid:1) . Since the number of iterations is at most n !, we obtain an upper bound of E ( T ) ≤ n ! X x =1 (cid:16) n − Ω(log( x )) + O ( n ln n log( x ) /x ) (cid:17) . The sum of the n − Ω(log( x )) is negligible. The sum of the O ( n ln n log( x ) /x ) contributes O ( n ln n log( n !) ) = O ( n log n ). 12 k -Median In the (metric) k -median problem, we are given a finite metric space ( V, d ) and should pick k points U ⊆ V such that P v ∈ V min u ∈ U d ( v, u ) is minimized. We call the set U a k -median. Re-garding worst-case analysis, the best known approximation algorithm for this problem achievesan approximation ratio of 3 + ε [3].In this section, we consider the k -median problem in the setting of random shortest pathmetrics. In particular we examine the approximation ratio of the algorithm TRIVIAL , whichpicks k points independently of the metric space, e.g., U = { , . . . , k } or k random points in V .We show that TRIVIAL yields a (1 + o (1))-approximation for k = O ( n − ε ). This can be seenas an algorithmic result since it improves upon the worst-case approximation ratio, but it isessentially a structural result on random shortest path metrics. It means that any set of k points is, with high probability, a very good k -median, which gives some knowledge about thetopology of random shortest path metrics. For larger, but not too large k , i.e., k ≤ (1 − ε ) n , TRIVIAL still yields an O (1)-approximation.The main insight comes from generalizing the growth process described in Section 3.2. Fixing U = { v , . . . , v k } ⊆ V we sort the vertices V \ U by their distance to U in ascending order, callingthe resulting order v k +1 , . . . , v n . Now we consider δ i = d ( v i +1 , U ) − d ( v i , U ) for k ≤ i < n . Theserandom variables are generated by a simple growth process analogous to the one describedin Section 3.2. This shows that the δ i are independent and δ i ∼ Exp( i · ( n − i )). Since a Exp( b ) ∼ Exp( b/a ), we have cost ( U ) = n − X i = k ( n − i ) · δ i ∼ n − X i = k ( n − i ) · Exp( i · ( n − i )) ∼ n − X i = k Exp( i ) . From this, we can read off the expected cost of U immediately, and thus the expected cost of TRIVIAL . Lemma 5.1.
Fix U ⊆ V of size k . We have E ( TRIVIAL ) = E (cid:0) cost ( U ) (cid:1) = H n − − H k − = ln( n/k ) + Θ(1) . Proof.
We have E ( cost ( U )) = P n − i = k i = H n − − H k − . Using H n = ln( n ) + Θ(1) yields thelast equality.By closely examining the random variable P n − i = k Exp( i ), we can show good tail bounds forthe probability that the cost of U is lower than expected. Together with the union bound thisyields tail bounds for the optimal k -median MEDIAN , which implies the following theorem. Inthis theorem, the approximation ratio becomes 1 + O (cid:0) ln ln( n )ln( n ) (cid:1) for k = O ( n − ε ). Theorem 5.2.
Let k ≤ (1 − ε ) n for some constant ε > . Then E (cid:18) TRIVIALMEDIAN (cid:19) = O (1) . If we have k ≤ κn for some sufficiently small constant κ ∈ (0 , , then E (cid:18) TRIVIALMEDIAN (cid:19) = 1 + O (cid:18) ln ln( n/k )ln( n/k ) (cid:19) . (2)13e need the following lemmas to prove Theorem 5.2. Lemma 5.3.
The density f of P mi = k Exp( i ) is given by f ( x ) = k · (cid:18) mk (cid:19) · exp( − kx ) · (cid:0) − exp( − x ) (cid:1) m − k . Proof.
The distribution P mi = k Exp( i ) corresponds to the k -th largest element of a set of m independent, exponentially distributed random variables with parameter 1. The density ofsuch order statistics is known [37, Example 2.38]. Lemma 5.4.
Let c > be sufficiently large, and let k ≤ c ′ n for c ′ = c ′ ( c ) > be sufficientlysmall. Then P (cid:16) MEDIAN < ln (cid:16) nk (cid:17) − ln ln (cid:16) nk (cid:17) − ln c (cid:17) = n − Ω( c ) . Proof.
Fix U ⊆ V of size k and consider cost ( U ) ∼ P n − i = k Exp( i ). In the following we set m := n − f ( x ) from above at x = ln (cid:0) mak (cid:1) for asufficiently large a with 1 ≤ a ≤ m/k (such an a exists since k is small enough). Plugging inthis particular x and using (cid:0) mk (cid:1) ≤ m k e k /k k yields f ( x ) = k · (cid:18) mk (cid:19) · a k k k ( m − ak ) m − k m m ≤ k ( ea ) k (cid:18) − akm (cid:19) m − k . Using 1 + x ≤ e x and m − k = Ω( m ), so that ( m − k ) /m = Ω(1), yields f ( x ) ≤ k ( ea ) k exp( − Ω( ak )) . Since a is sufficiently large, the first two factors are lower order terms that we can hide by theΩ. Thus, we can simplify this further to f ( x ) ≤ exp( − Ω( ak )) . Rearranging this using a = mk e − x yields f ( x ) = exp( − Ω( m exp( − x )) , (3)which holds for any x ∈ [0 , ln (cid:0) mαk (cid:1) ] for any sufficiently large α ≥
1. Now we can bound theprobability that cost ( U ) < ln (cid:0) mαk (cid:1) . This probability is equal to Z ln( mαk )0 f ( x ) d x = Z ln( mαk )0 f (cid:16) ln (cid:16) mαk (cid:17) − x (cid:17) d x = Z ln( mαk )0 exp (cid:0) − Ω( αk exp( x )) (cid:1) d x using (3) ≤ Z ∞ exp (cid:0) − Ω( αk (1 + x )) (cid:1) d x ≤ exp (cid:0) − Ω( αk ) (cid:1) since R ∞ exp( − Ω( αkx )) d x = O (1 / ( αk )) ≤ α is sufficiently large.14n order for MEDIAN to be less than ln (cid:0) mαk (cid:1) , one of the subsets U ⊆ V of size k has to havecost less than ln (cid:0) mαk (cid:1) . We bound the probability of the latter using the union bound and get P (cid:16) MEDIAN < ln (cid:16) mαk (cid:17)(cid:17) = P (cid:16) ∃ U ⊆ V, | U | = k : cost ( U ) < ln (cid:16) mαk (cid:17)(cid:17) ≤ (cid:18) nk (cid:19) · P (cid:16) cost ( U ) < ln (cid:16) mαk (cid:17)(cid:17) ≤ (cid:18) nk (cid:19) · exp (cid:0) − Ω( αk ) (cid:1) . By setting α = c ln (cid:0) nk (cid:1) for sufficiently large c ≥
1, we fulfill all conditions on α . This yields P (cid:16) MEDIAN < ln (cid:16) nk (cid:17) − ln ln (cid:16) nk (cid:17) − ln c (cid:17) ≤ (cid:16) enk (cid:17) k · (cid:16) nk (cid:17) − Ω( ck ) . Since k is sufficiently smaller than n , we have enk ≤ ( nk ) . Thus, for sufficiently large c , theright hand side simplifies to ( nk ) − Ω( ck ) . Since k is at least 1 and sufficiently smaller than n , wehave ( nk ) k ≥ n . Thus, the probability is bounded by n − Ω( c ) , which finishes the proof.To bound the expected value of the quotient TRIVIAL / MEDIAN , we further need to boundthe probabilities that
TRIVIAL is much too large or
MEDIAN is much too small. This is achievedby the following two lemmas.
Lemma 5.5.
Let k ≤ (1 − ε ) n for some constant ε > . Then, for any c > , we have P ( MEDIAN < c ) = O ( c ) Ω( n ) . Proof.
Since n − k vertices have to be connected to the k -median, the cost of the k -median isthe sum of n − k shortest path lengths. Thus, the cost of the minimal k -median is at least thesum of the smallest n − k edge weights w ( e ). We use Lemma 4.3 with α = ε . Lemma 5.6.
For any c ≥ , we have P ( TRIVIAL > n c ) ≤ exp( − n c/ ) .Proof. We can bound very roughly
TRIVIAL ≤ n max e { w ( e ) } . As max e { w ( e ) } is the maximumof (cid:0) n (cid:1) independent exponentially distributed random variables, we have P (cid:0) TRIVIAL ≤ n c (cid:1) ≥ (1 − exp( − n c − ))( n ) ≥ − (cid:18) n (cid:19) · exp( − n c − ) ≥ − exp (cid:0) − n c − (cid:1) ≥ − exp (cid:0) − n c/ (cid:1) . Proof of Theorem 5.2.
Let T = TRIVIAL and C = MEDIAN for short. We have for any m ≥ E (cid:18) TC (cid:19) ≤ E (cid:18) Tm (cid:19) + P ( C < m ) · E (cid:18) TC (cid:12)(cid:12) C < m (cid:19) . (4) Case 1 ( k ≤ c ′ n , c ′ sufficiently small): Using Lemma 5.4, we can pick c > P h C < ln (cid:16) nk (cid:17) − ln ln (cid:16) nk (cid:17) − ln c i ≤ n − . m = ln (cid:0) nk (cid:1) − ln ln (cid:0) nk (cid:1) − ln c . Then, by Lemma 5.1, we have E (cid:18) Tm (cid:19) ≤ ln( n/k ) + O (1) m ≤ O (cid:18) ln ln( n/k )ln( n/k ) (cid:19) . We show that the second summand of inequality (4) is O (1 /n ) in the current situation, whichshows the claim. We have P ( C < m ) · E (cid:18) TC (cid:12)(cid:12) C < m (cid:19) = P ( C < m ) · Z ∞ P (cid:18) TC ≥ x (cid:12)(cid:12) C < m (cid:19) d x ≤ P ( C < m ) · (cid:18) n + Z ∞ n P (cid:18) TC ≥ x (cid:12)(cid:12) C < m (cid:19) d x (cid:19) ≤ n − + Z ∞ n P (cid:18) TC ≥ x and C < m (cid:19) d x ≤ n − + Z ∞ n P (cid:18) TC ≥ x (cid:19) d x ≤ n − + Z ∞ n (cid:26) P (cid:0) T ≥ √ x (cid:1) , P (cid:18) C ≤ √ x (cid:19)(cid:27) d x since T /C ≥ x implies T ≥ √ x or C ≤ / √ x . Using Lemmas 5.5 and 5.6, this yields P ( C < m ) · E (cid:18) TC (cid:12)(cid:12) C < m (cid:19) ≤ n − + Z ∞ n ( exp (cid:0) − x / (cid:1) , O (cid:18) √ x (cid:19) Ω( n ) ) d x = O (1 /n ) . Case 2 ( c ′ n < k ≤ (1 − ε ) n ): We repeat the proof above, now choosing m to be a sufficientlysmall constant. Then P ( C < m ) = O ( m ) Ω( n ) ≤ O ( n − ) by Lemma 5.5, and we have E (cid:18) Tm (cid:19) = ln( n/k ) + O (1) m = O (1) , since k > c ′ n . Together with the first case, this shows the first claim. Using a coupling argument, Janson [24, Section 3] proved that the results about the lengthof a fixed edge and the longest edge carry over if the exponential distribution is replaced bya probability distribution with the following property: the probability that an edge weight issmaller than x is x + o ( x ). This property is satisfied, e.g., by the exponential distribution withparameter 1 and by the uniform distribution on the interval [0 , O (log n/n ) = o (1), only the behavior of the distribution in asmall, shrinking interval [0 , o (1)] is relevant and the o ( x ) term becomes irrelevant.We believe that also all of our results carry over to such probability distributions. In fact,we started our research using the uniform distribution and only switched to exponential dis-tributions because they are technically easier to handle. However, we decided not to carry outthe corresponding proofs because, first, they seem to be technically very tedious and, second,we feel that they do not add much. 16 .2 Open Problems To conclude the paper, let us list the open problems that we consider most interesting:1. While the distribution of distances in asymmetric instances does not differ much fromthe symmetric case, an obstacle in the application of asymmetric random shortest pathmetrics seems to be the lack of clusters of small diameter (see Section 3). Is there anasymmetric counterpart for this?2. Is it possible to prove an 1 + o (1) approximation ratio (like Dyer and Frieze [15] for thepatching algorithm) for any of the simple heuristics that we analyzed?3. What is the approximation ratio of 2-opt in random shortest path metrics? In theworst case on metric instances, it is O ( √ n ) [12]. For independent, non-metric edgelengths drawn uniformly from the interval [0 , O ( √ n · log / n ) [17]. For d -dimensional geometric instances, the smoothed approximationratio is O ( φ /d ) [18], where φ is the perturbation parameter.We easily get an approximation ratio of O (log n ) based on the two facts that the lengthof the optimal tour is Θ(1) with high probability and that ∆ max = O (log n/n ) with highprobability. Can we prove that the expected ratio of 2-opt is o (log n )? References [1] Louigi Addario-Berry, Nicolas Broutin, and G´abor Lugosi. The longest minimum-weightpath in a complete graph.
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