Network Construction with Ordered Constraints
NNetwork Construction with Ordered Constraints
Yi Huang, Mano Vikash Janardhanan, and Lev Reyzin (cid:63)
University of Illinois at Chicago, { yhuang89, mjanar2, lreyzin } @uic.edu Abstract.
In this paper, we study the problem of constructing a net-work by observing ordered connectivity constraints, which we defineherein. These ordered constraints are made to capture realistic prop-erties of real-world problems that are not reflected in previous, moregeneral models. We give hardness of approximation results and nearly-matching upper bounds for the offline problem, and we study the onlineproblem in both general graphs and restricted sub-classes. In the onlineproblem, for general graphs, we give exponentially better upper boundsthan exist for algorithms for general connectivity problems. For the re-stricted classes of stars and paths we are able to find algorithms withoptimal competitive ratios, the latter of which involve analysis using apotential function defined over pq-trees.
Keywords: graph connectivity, network construction, ordered connec-tivity constraints, pq-trees
In this paper, we study the problem of recovering a network after observing howinformation propagates through the network. Consider how a tweet (through“retweeting” or via other means) propagates through the Twitter network – wecan observe the identities of the people who have retweeted it and the timestampswhen they did so, but may not know, for a fixed user, via whom he got the originaltweet. So we see a chain of users for a given tweet. This chain is semi-orderedin the sense that, each user retweets from some one before him in the chain,but not necessarily the one directly before him. Similarly, when a virus such asEbola spreads, each new patient in an outbreak is infected from someone whohas previously been infected, but it is often not immediately clear from whom.In a graphical social network model with nodes representing users and edgesrepresenting links, an “outbreak” illustrated above is captured exactly by theconcept of an ordered constraint which we will define formally below. Onecould hope to be able to learn something about the structure of the network byobserving repeated outbreaks, or a sequence of ordered constraints.Formally we call our problem
Network Construction with OrderedConstraints and define it as follows. Let V = { v , . . . , v n } be a set of ver-tices. An ordered constraint O is an ordering on a subset of V of size s ≥ (cid:63) Supported in part by ARO grant 66497-NS. a r X i v : . [ c s . D S ] F e b Yi Huang, Mano Vikash Janardhanan, Lev Reyzin
The constraint O = ( v k , . . . , v k s ) is satisfied if for any 2 ≤ i ≤ s , there exists atleast one 1 ≤ j < i such that the edge e = (cid:8) v k j , v k i (cid:9) is included in a solution.Given a collection of ordered constraints {O , . . . , O r } , the task is to constructa set E of edges among the vertices V such that all the ordered constraints aresatisfied and | E | is minimized.We can see that our newly defined problem resides in a middle ground be-tween path constraints, which are too rigid to be very interesting, and the well-studied subgraph connectivity constraints [7,17,18], which are more relaxed. Theestablished subgraph connectivity constraints problem involves getting an arbi-trary collection of connectivity constraints { S , . . . , S r } where each S i ⊂ V andrequires vertices in a given constraint to form a connected induced subgraph.The task is to construct a set E of edges satisfying the connectivity constraintssuch that | E | is minimized.We want to point out one key observation relating the ordered constraintto the connectivity constraint – an ordered constraint O = ( v k , . . . , v k s ) isequivalent to s − S , . . . , S s , where S i = { v k , . . . , v k i } .We note that this observation plays an important role in several proofs in thispaper which employ previous results on subgraph connectivity constraints – inparticular, upper bounds from the more general case can be used in the orderedcase (with some overhead), and our lower bounds apply to the general problem.In the offline version of the Network Construction with Ordered Constraintsproblem, the algorithm is given all of the constraints all at once; in the online version of the problem, the constraints are given one by one to the algorithm,and edges must be added to satisfy each new constraint when it is given. Edgescannot be removed.An algorithm is said to be c - competitive if the cost of its solution is lessthan c times OPT, where OPT is the best solution in hindsight ( c is also calledthe competitive ratio). When we restrict the underlying graph in a problem tobe a class of graphs, e.g. trees, we mean all the constraints can be satisfied, in anoptimal solution (for the online case, in hindsight), by a graph from that class. In this paper we study the problem of network construction from ordered con-straints. This is an extension of the more general model where constraints comeunordered.For the general problem, Korach and Stern [17] had some of the initial results,in particular for the case where the constraints can be optimally satisfied by atree, they give a polynomial time algorithm that finds the optimal solution. Insubsequent work, in [18] Korach and Stern considered this problem for the evenmore restricted problem where the optimal solution forms a tree, and all of theconnectivity constraints must be satisfied by stars.Then, Angluin et al. [7] studied the general problem, where there is no re-striction on structure of the optimal solution, in both the offline and onlinesettings. In the offline case, they gave nearly matching upper and lower boundson the hardness of approximation for the problem. In the online case, they give etwork Construction with Ordered Constraints 3 a O ( n / log / n )-competitive algorithm against oblivious adversaries; we showthat this bound can be drastically improved in the ordered version of the prob-lem. They also characterized special classes of graphs, i.e. stars and paths, whichwe are also able to do herein for the ordered constraint case. Independently ofthat work, Chockler et al. [12] also nearly characterized the offline general case.In a different line of work Alon et al. [3] explore a wide range of networkoptimization problems; one problem they study involves ensuring that a networkwith fractional edge weights has a flow of 1 over cuts specified by the constraints.Alon et al. [2] also study approximation algorithms for the Online Set Coverproblem which have been shown by Angluin et al [7] to have connections withNetwork Construction problems.In related areas, Gupta et al. [16] considered a network design problem forpairwise vertex connectivity constraints. Moulin and Laigret [19] studied networkconnectivity constraints from an economics perspective. Another motivation forstudying this problem is to discover social networks from observations. This andsimilar problems have also been studied in the learning context [5,6,14,22].Finally, in query learning, the problem of discovering networks from connec-tivity queries has been much studied [1,4,8,9,15,21]. In active learning of hiddennetworks, the object of the algorithm is to learn the network exactly. Our modelis similar, except the algorithm only has the constraints it is given, and the taskis to output the cheapest network consistent with the constraints. In Section 2, we examine the offline problem, and show that the Network Con-struction problem is NP-Hard to approximate within a factor of Ω (log n ). Anearly matching upper bound comes from Theorem 2 of [7].In Section 3, we study online problem. For problems on n nodes, for r con-straints, we give an O ((log r + log n ) log n ) competitive algorithm against obliv-ious adversaries, and an Ω (log n ) lower bound (Section 3.1).Then, for the special cases of stars and paths (Sections 3.2 and 3.3), we findasymptotic optimal competitive ratios of 3 / n . In this section, we examine the Network Construction with Ordered Constraintsproblem in the offline case. We are able to obtain the same lower bound asAngluin et al. [7] in the general connectivity constraints case.
Theorem 1.
If P (cid:54) = NP, the approximation ratio of the
Network Constructionwith Ordered Constraints problem is Ω (log n ) .Proof. We prove the theorem by reducing from the Hitting Set problem. Let( U, S ) be a hitting set instance, where U = { u , . . . , u n } is the universe, and Yi Huang, Mano Vikash Janardhanan, Lev Reyzin S = { S , . . . , S m } is a set of subsets of U . A subset H ⊂ U is called a hittingset if H ∩ S i (cid:54) = ∅ . The objective of the Hitting Set problem is to minimize | H | .We know from [13,20] that the Hitting Set problem cannot be approximated byany polynomial time algorithm within a ratio of o (log n ) unless P=NP. Here weshow that the Network Construction problem is inapproximable better than an O (log n ) factor by first showing that we can construct a corresponding NetworkConstruction instance to any given Hitting Set instance, and then showing thatif there is a polynomial time algorithm that can achieve an approximation ratio o (log n ) to the Network Construction problem, then the Hitting Set problem canalso be approximated within in a ratio of o (log n ), which is a contradiction.We first define a Network Construction instance, corresponding to a givenHitting Set instance ( U, S ), with vertex set U ∪ W , where W = { w , . . . , w n c } for some c >
2. Note that we use the elements of the universe of hitting set in-stance as a part of the vertex set of Network Construction instance. The orderedconstraints are the union of the following two sets: – { ( u i , u j ) } ≤ i There is a polynomialtime O (log r +log n ) -approximation algorithm for the Network Construction withOrdered Constraintsproblem on n nodes and r constraints.Proof. Observing that r ordered constraints imply at most nr unordered con-straints on a graph with n nodes, we can use the O (log r ) upper bound fromAngluin et al. [7]. (cid:117)(cid:116) Here, we study the online problem, where constraints come in one at a time, andthe algorithm must satisfy them by adding edges as the constraints arrive. The competitive ratio for Online Network Construction withOrdered Constraints problem on n nodes and r ordered constraints has anupper bound of O ((log r + log n ) log n ) against an oblivious adversary.Proof. To prove the statement, we first define the Fractional Network Con-struction problem, which has been shown by Angluin et al. [7] to have an O (log n )-approximation algorithm. The upper bound is then obtained by ap-plying a probabilistic rounding scheme to the fractional solution given by theapproximation. The proof heavily relies on arguments developed by Buchbinderand Noar [11], and Angluin et al. [7].In the Fractional Network Construction problem, we are also given aset of vertices and a set of constraints { S , . . . , S r } where each S i is a subsetof the vertex set. Our task is to assign weights w e to each edge e so that themaximum flow between each pair of vertices in S i is at least 1. The optimizationproblem is to minimize (cid:80) w e . Since subgraph connectivity constraint is equiva-lent to requiring a maximum flow of 1 between each pair of vertices with edgeweight w e ∈ { , } , the fractional network construction problem is the linearrelaxation of the subgraph connectivity problem. Lemma 2 of Angluin et al. [7]gives an algorithm that multiplicatively updates the edge weights until all theflow constraints are satisfied. It also shows that the sum of weights given by thealgorithm is upper bounded by O (log n ) times the optimum.As we pointed out in the introduction, an ordered constraint O is equivalentto a sequence of subgraph connectivity constraints. So in the first step, we feed Yi Huang, Mano Vikash Janardhanan, Lev Reyzin the r sequences of connectivity constraints, each one is equivalent to an orderedconstraint, to the approximation algorithm to the fractional network construc-tion problem and get the edge weights. Then we apply a rounding scheme similarto the one considered by Buchbinder and Noar [11] to the weights. For each edge e , we choose t random variables X ( e, i ) independently and uniformly from [0 , T ( e ) = min ti =1 X ( e, i ). We add e to the graph if w e ≥ T ( e ).Since the rounding scheme has no guarantee to produce a feasible solution,the first thing we need to do is to determine how large t should be to make allthe ordered constraints satisfied with high probability.We note that an ordered constraint O i = { v i , v i , . . . , v is i } is satisfied if andonly if the ( s − 1) connectivity constraints { v i , v i } , . . . , { v i , . . . , v is i − , v is i } are satisfied which is equivalent, in turn, to the fact that there is an edgethat goes across the ( { v i , . . . , v ij − } , { v ij } ) cut, for 2 ≤ j ≤ s i . For anyfixed cut C , the probability the cut is not crossed equals (cid:81) e ∈ C (1 − w e ) t ≤ exp (cid:0) − t (cid:80) e ∈ C w e (cid:1) . By the max-flow min-cut correspondence, we know that (cid:80) c ∈ C w e ≥ C = ( { v i , . . . , v ij − } , { v ij } ), 1 ≤ i ≤ r , 2 ≤ j ≤ s i , and hence theprobability that there exists at least one unsatisfied O i is upper bounded by rn exp ( − t ). So t = c (log n + log r ), for any c > 1, makes the probability that therounding scheme fails to produce a feasible solution approaches 0 as n increases.Because the probability that e is added equals the probability that at leastone X ( e, i ) is less than w e , and hence is upper bounded by w e t , we get theexpected number of edges added is upper bounded by t (cid:80) w e by linearity ofexpectation. Since the fractional solution is upper bounded by O (log n ) timesthe optimum of the fractional problem, which is upper bounded by any integralsolution, our rounding scheme gives a solution that is O ((log r + log n ) log n )times the optimum. (cid:117)(cid:116) Corollary 2. If the number of ordered constraints r = poly( n ) , then the algo-rithm above gives a O (cid:0) (log n ) (cid:1) upper bound for the competitive ratio against anoblivious adversary.Remark 1. We can generalise theorem 2 to the weighted version of the OnlineNetwork Construction with Ordered Constraints problem. In the weighted ver-sion, each edge e = ( u, v ) is associated with a cost c e and the task is to selectedges such that the connectivity constraints are satisfied and (cid:80) c e w e is min-imised where w e ∈ { , } is a variable indicating whether an edge is picked ornot and c e is the cost of the edge. The same approach in the proof of Theorem 2gives an upper bound of O ((log r + log n ) log n ) for the competitive ratio of theweighted version of the Online Network Construction with Ordered Constraintsproblem. Theorem 3. This is a Ω (log n ) lower bound for the competitive ratio for the Online Network Construction with Ordered Constraints problem againstan oblivious adversary.Proof. The adversary divides the vertex set into two parts U and V , where | U | = √ n and | V | = n − √ n , and gives the constraints as follows. Firstly, it etwork Construction with Ordered Constraints 7 forces a complete graph in U by giving the constraint { u i , u j } for each pair ofvertices u i , u j ∈ U . At this stage both the algorithm and optimal solution willhave a clique in U , which costs Θ ( n ).Then, for each v ∈ V , first fix a random permutation π v on U and give theordered constraint O ( v,i ) = ( π v (1) , π v (2) , . . . , π v ( i ) , v ) . First note that all these constraints can be satisfied by adding e v = { π v (1) , v } for each v ∈ V which costs Θ ( n ). However, the adversary gives constraints inthe following order: O ( v, √ n ) , O ( v, √ n − , . . . , O ( v, . We now claim that for each v ∈ V , the algorithm will add Ω (log n ) edges inexpectation. This is because each edge added by the algorithm is a randomguess for π v (1) and this edge cuts down the number of unsatisfied O ( v,i ) by halfin expectation. This means the algorithm adds Ω ( n + n log n ) edges. This givesus the desired result because OPT = O ( n ). (cid:117)(cid:116) Now we study the online problem when it is known that an optimal graphcan be a star or a path. These special cases are challenging in their own rightand are often studied in the literature to develop more general techniques [7]. The optimal competitive ratio for the Online Network Con-struction with Ordered Constraints problem when the algorithm knows thatan optimal solution forms a star is asymptotically / .Proof. For lower bound, we note that the adversary can simply give O i =( v , v , v i ), 3 ≤ i ≤ n obliviously for the first n − { v , v } in the first round, can only choose from addingeither { v , v i } or { v , v i } , or both in each round. After the first n − v and v ’s neighbors, and chooses the one withfewer neighbors, say v , to be the center by adding ( v , v i ) for some 3 ≤ i ≤ n .Since the algorithm has to add at least (cid:100) ( n − / (cid:101) edges that are unnecessaryin the hindsight, we get an asymptotic lower bound 3 / O is ( v , v , . . . ),the algorithm works as follows:1. It adds { v , v } in the first round.2. Then for any constraint that starts with v and v , it splits the remainingvertices in the constraint (other than v and v ) into two sets of sizes differingby at most 1, and connects each vertex in first set to v and each vertex inthe other set to v .3. Upon seeing a constraint that does not start with v and v , which revealsthe center of the star, it connects the center to all vertices that are not yetconnected to the center.Since the algorithm adds, at most n/ − / 2, which matches the lower bound. (cid:117)(cid:116) Yi Huang, Mano Vikash Janardhanan, Lev Reyzin In the next two theorems, we give matching lower and upper bounds (in thelimit) for path graphs. Theorem 5. The competitive ratio for the Online Network Constructionwith Ordered Constraints problem when the algorithm knows that the optimalsolution forms a path has an asymptotic lower bound of .Proof. Fix an arbitrary ordering of the vertices { v , v , v , . . . , v n } . For 3 ≤ i ≤ n , define the pre-degree of a vertex v i to be the number of neighbors v i has in { v , v , v , . . . , v i − } . Algorithm 1 below is a simple strategy the adversary cantake to force v , . . . , v n to all have pre-degree at least 2. Since any algorithm willadd at least 2 n − Algorithm 1 Forcing pre-degree to be at least 2 Give ordered constraint O = ( v , v , v , . . . , v n ) to the algorithm; for i = 3 to n doif the pre-degree of v i is at least 2 then continue; else pick up at random a path (say P i ) that satisfies all the constraints up to thisround and an endpoint u of the path that is not connected to v i , and gives thealgorithm the constraint ( v i , u ); end ifend for P i was the path picked in round i (i.e. P i satisfies all constraints upto round i ). Then, P i along with the edge ( v i , u ) is a path that satisfies all constraintsupto round i + 1. Hence by induction, for all i , there is a path that satisfies allconstraints given by the adversary upto round i . (cid:117)(cid:116) Theorem 6. The competitive ratio for the Online Network Constructionwith Ordered Constraints problem when the algorithm knows that the optimalsolution forms a path has an asymptotic upper bound of .Proof. For our algorithm matching the upper bound, we use the pq-trees, intro-duced by Booth and Lueker [10], which keep track all consistent permutationsof vertices given contiguous intervals of vertices. Our analysis is based on ideasfrom Angluin et al. [7], who also use pq-trees for analyzing the general problem. A pq-tree is a tree whose leaf nodes are the vertices and each internal nodeis either a p-node or a q-node . – A p-node has a two or more children of any type. The children of a p-nodeform a contiguous interval that can be in any order. Angluin et al. [7] have a small error in their argument because their potential functionfails to explicitly consider the number of p-nodes, which creates a problem for someof the pq-tree updates. We fix this, without affecting their asymptotic bound. Forthe ordered constraints case, we are also able to obtain a much finer analysis.etwork Construction with Ordered Constraints 9 – A q-node has three or more children of any type. The children of a q-nodeform a contiguous interval, but can only be in the given order of its inverse.Every time a new interval constraint comes, the tree update itself by identifyingany of the eleven patterns, P0, P1, . . . , P6, and Q0, Q1, Q2, Q3, of the arrange-ment of nodes and replacing it with each correspondent replacement. The updatefails when it cannot identify any of the patterns, in which case the contiguousintervals fail to produce any consistent permutation. We refer readers to Section2 of Booth and Lueker [10] for a more detailed description of pq-trees.The reason we can use a pq-tree to guide our algorithm is because of an ob-servation made in Section 1 that each ordered constraint ( v , v , v , . . . , v k − , v k )is equivalent to k interval constraints { v , v } , { v , v , v } , · · · , { v , . . . , v k − } , { v , . . . , v k − , v k } . So upon seeing one ordered constraints, we reduce the pq-tree with the equivalent interval constraints, in order . Then what our algorithmdoes is simply to add edge(s) to the graph every time a pattern is identifiedand replaced with its replacement, so that the graph satisfies all the seen con-straints. Note that to reduce the pq-tree with one interval constraint, there maybe multiple patterns identified and hence multiple edges may be added.Before running into details of how the patterns determine which edge(s)to add, we note that, without loss of generality, we can assume that the thealgorithm is in either one of the following two stages. – The pq-tree is about to be reduced with { v , v } . – The pq-tree is about to be reduced with { v , . . . , v k } , when the reductionswith { v , v } , · · · , { v , . . . , v k − } have been done.Because of the structure of constraints discussed above, we do not encounterall pq-tree patterns in their full generality, but in the special forms demonstratedin Table 1. Based on this, we make three important observations which can beverified by carefully examining how a pq-tree evolves along with our algorithm.1. The only p-node that can have more than two children is the root.2. At least one of the two children of a non-root p-node is a leaf node.3. For all q-nodes, there must at least one leaf node in any two adjacent children.Hence, Q3 doesn’t appear.Now we describe how the edges are going to be added. Note that a pq-treeinherently learns edges that appear in optimum even when those edges are notforced by constraints. Apart from adding edges that are necessary to satisfy theconstraints, our algorithm will also add any edge that the pq-tree inherentlylearns. For all the patterns except Q2 such that a leaf node v k is about to beadded as a child to a different node, we can add one edge joining v k to v k − .For all such patterns except Q2, it is obvious that this would satisfy the currentconstraint and all inherently learnt edges are also added. For Q2, the pq-treecould learn two edges. The first edge is ( v k , v k − ). The second one is an edgebetween the leftmost subtree of the daughter q-node (call T l ) and the node to itsleft (call v l ). Based on Observation 3, v l is a leaf. But based on the algorithm,one of these two edges is already added. Hence, we only need to add one edgewhen Q2 is applied. For P5, we add the edge as shown in Table 1. v k v k P3P4(1) v k v k P4(2) v k v k P5P6(1) v k v k P6(2) v k v k Q2 v k v k Table 1: Specific patterns and replacements that appear through the algorithm.P4(1) denotes the case of P4 where the top p-node is retained in the replacementand P4(2) denotes the case where the top p-node is deleted. The same is truefor P6. P0, P1, Q0, and Q1 are just relabelling rules, and we have omitted thembecause no edges need to be added. We use the same shapes to represent p-nodes,q-nodes, and subtrees as in Booth and Lueker’s paper [10] for easy reference, andwe use diamonds to represent leaf nodes. etwork Construction with Ordered Constraints 11 Let us denote by P and Q the sets of p-nodes and q-nodes, respectively, andby c ( p ) the number of children node p has. And let potential function φ of a tree T be defined as φ ( T ) = a (cid:88) p ∈ P c ( p ) + b | P | + c | Q | , where a , b , and c are coefficients to be determined later. (cid:80) p ∈ P c ( p ) | P | | Q | − ∆Φ number of edges addedP2 1 1 0 − a − b − − a + b − c − a − − a + b − − a + b − − a + c − − − a + b + c − c − c Table 2: How the terms in the potential function: (cid:80) p ∈ P c ( p ), | P | , and | Q | change according to the updates.We want to upper bound the number of edges added for each pattern bythe drop of potential function. We collect the change in the three terms in thepotential function that each replacement causes in Table 2, and we can solve asimple linear system to get that choosing a = 2, b = − 3, and c = 1 is sufficient.For ease of analysis, we add a dummy vertex v n +1 that does not appear in anyconstraint. Now, the potential function starts at 2 n − n + 1 children) and decreases to 2 when a path is uniquely determined. Hence,the number of edges added by the algorithm is 2 n − 3, which gives the desiredasymptotic upper bound. (cid:117)(cid:116) References 1. Noga Alon and Vera Asodi. Learning a hidden subgraph. SIAM Journal on DiscreteMathematics , 18(4):697–712, 2005.2. Noga Alon, Baruch Awerbuch, and Yossi Azar. The online set cover problem. In Proceedings of the thirty-fifth annual ACM symposium on Theory of computing ,pages 100–105. ACM, 2003.3. Noga Alon, Baruch Awerbuch, Yossi Azar, Niv Buchbinder, and Joseph Seffi Naor.A general approach to online network optimization problems. 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