Calculation of typical running time of a branch-and-bound algorithm for the vertex-cover problem

Transcription

1 Calulation of typial running time of a branh-and-bound algorithm for the vertex-over problem Joni Pajarinen, Joni.Pajarinen@iki.fi Otober 21, Introdution The vertex-over problem is one of a olletion of NP-omplete problems studied atively in the siene ommunity. The problem s solving time is in the worst ase exponential, but the running time varies for different vertex-over realizations. From experimental data, different phases an be identified for the running time and the analysis of vertex-over algorithms bak the observed results up. Interestingly the worst-ase typial average solutions are loated at the x () phase boundary, whih is studied analytially in the book [2] and in the paper [3]. This summary is based on these two douments. We start by giving a short desription of the vertex-over problem and introduing symbols related to the problem. We proeed and introdue a branh-and-bound algorithm for finding vertex overs of a given size for a graph. Experimental results are displayed in order to show the running time landsape we are dealing with. The experimental results show different phases, whih an be studied analytially further. The vertex-over algorithm s traversal of the onfiguration tree is explained. The relationship between different kinds of onfiguration tree traversals and the phases shown in the experimental results is desribed. The analysis of the vertex-over algorithm is started by introduing the first-moment method. The method provides us with a lower bound for the x () phase boundary for large values. It also shows a general way of using the average as a bound that an be exploited in similar situations. We start the analysis of the typial running time by examining the behaviour of the algorithm during the first desent into the onfiguration tree. This produes results that an be used to find the nodes in the onfiguration tree, where a solution is unreoverable and a subtree must be baktraked. In the baktraking analysis the typial running time for the largest subtree is approximated. This yields an analytial solution for the typial running time in the baktraking phase. The analytial results are ompared to the experimental results presented before. As the last step in this paper, a summary of the most important results is given. 2 A branh-and-bound algorithm for the vertex-over problem In this setion we will quikly desribe the vertex-over problem and outline a branh-and-bound algorithm for finding a vertex-over of a maximum size for a graph. Typial running time experimental data for the branh-and-bound algorithm from simulation runs is presented and the relationship of the onfiguration tree traversal to the typial running time of the algorithm is disussed. 1

2 2.1 Vertex-over problem A graph is denoted by G = (V, E). V is the set of verties and E is the set of edges. Graphs an be sampled randomly from either a fixed number N of verties and fixed amount of edges M ensemble G(N, M), where eah possible edge has the same probability of appearane, or from an ensemble G(N, p), where N is the fixed amount of verties and p the edge probability. A graph sampled from the ensemble G(N, p) does not have a fixed number of edges. A valid vertex over overs verties so that all edges onnet to at least one overed vertex: V v V : i V v j V v, (i, j) E. The minimum vertex-over is then arg min V v V v. The size of the largest allowed vertex-over is xn, where x [0, 1] and N = V. x is used here in the vertex-over size, beause it is independent of graph size. 2.2 A branh-and-bound vertex-over algorithm The branh-and-bound vertex-over algorithm goes through different partial vertex overs by marking verties as either overed or unovered at eah step. It baktraks if it reahes a state, where all the available overing marks xn have been used. The goal is to find a vertex-over of size at most xn. The algorithm terminates, when it has found a valid vertex-over or if it has gone through all possible onfigurations. The proess of the algorithm is desribed by an onfiguration tree. In the onfiguration tree a node speifies the urrent state of all the verties of the graph. All verties are marked as free at the start of the algorithm. The algorithm proeeds by marking a random free vertie as overed if the vertex has free or unovered neighbours. If the xn overing marks are not all used, the algorithm an go on with the tree traversal, otherwise it has to baktrak. If the algorithm returns to the node by baktraking, then the vertex is unovered and the other branh in the onfiguration tree is taken. If a node s all neighbours are overed, it is first marked as unovered. A simple bound is used to prune the onfiguration tree: don t mark a vertex unovered, if it has unovered neighbours. In the example figure 1 marking a vertex overed is equal to a left branh in the onfiguration tree and as unovered is equal to a right branh. 2.3 Experimental results, traversal of the onfiguration tree Figure 2 displays experimental results from running the branh-and-bound algorithm on the vertex-over problem. The simulations were performed by sampling graphs from the ensemble G(N, N ). Here N is the number of verties, the average vertex degree and p = N the edge probability. For eah x value graphs were sampled from the ensemble and the algorithm run on eah graph. The typial running time was alulated as the normalized and averaged logarithm of the number of onfiguration tree nodes, that the algorithm had to proess. In the figure two exponential phases (C, B) and one linear phase (A) are learly to be seen. The exponential phases C and B are separated by the stati phase boundary x (). x () is the phase boundary value, below whih a random graph almost surely has no vertex over of size xn and above whih a random graph almost surely has a vertex over of size xn, in the thermodynami limit N. The exponential phase B and the linear phase A are separated by the dynamial phase boundary x b (). The worst ase typial running time is observable at the stati phase boundary x (). Figure 1 shows example onfiguration trees for the branh-and-bound algorithm in the three different phases. In the sub figure A, the example solution (blak irle) is found during the first desent into the onfiguration tree. Here we don t have to baktrak at all and the solution time 2

3 Figure 1: Example onfiguration trees. The three sub figures A, B and C show examples of onfiguration trees in the three different phases. In sub figure A the blak irle is an example solution in the easy reoverable phase, where only a straight or almost straight desent is neessary. The blak irle in sub figure B is an example solution in the hard reoverable phase, where we have to baktrak to the irle pointed to by the arrow. In sub figure C, in the hard unreoverable phase, we have to traverse the tree until we know a solution an t be found (to the long dashed line). is obviously linear. The probability for the algorithm to find a solution in linear time is larger, the larger the maximum vertex-over size xn is. This linear running time an be observed as phase A in figure 2. The example solution (blak irle) in the sub figure B had to be found by baktraking. The algorithm had to baktrak the whole left subtree of the node pointed to by the arrow. This is the exponential overable phase B shown in figure 2. The long dashed line in sub figure C shows the depth of the traversal of the onfiguration tree, when the maximum vertex-over size xn is so small that a solution an t be found from the graph. This ase an be seen in figure 2 as the unreoverable exponential phase C in terms of typial running time. 3 First-moment method In this setion we will show a method for alulating a lower bound on the stati phase boundary x (). We will use a graph ensemble G(N, M) with fixed vertex N and edge number M = 2 N. This should give a reasonable result, beause even if we would use the graph ensemble G(N, p), with the speified edge probability p = N, the edge number would onentrate around the edge average M = 2N in the thermodynami limit. An upper bound for the probability that graph G has a vertex-over an be obtained by using the average number of vertex-overs. This also explains the name of the method. P( V v (G), V v (G) = xn) N v (G, xn) This bound holds, beause a graph with a vertex-over ontributes 1 on the left hand side and its number of vertex-overs on the right hand side. Another way to look at it is, that the average is the probability weighted with the number of vertex-overs ( 1). 3

4 Figure 2: Experimental results for typial running time and the analytial solution. τ(x ) = lim N 1 N ln t bt( G, x) is the normalized and averaged logarithm of running time. C and B are the exponential phases and A is the linear phase. Dynami phase boundary x b () and the stati phase boundary x () are also shown. Original figure is from [3]. 4

5 ( N The number of potential vertex overs Vv is xn ), beause we try to over xn verties with the available overing marks from the total of N verties in the graph. In Vv the probability that both verties of an edge are unovered is (1 x) 2 and that at least one is overed is 1 (1 x) 2 = x(2 x). If we onsider that a potential vertex-over must over all edges, for it to be a valid vertex-over, then we get the average vertex-over number as This gives us ( ) N N v (G, xn) = xn N v (G, xn) = ( N xn [x(2 x)] }{{} P(edge overed) # edges {}}{ 2 N ) [x(2 x)] 2 N = N! [x(2 x)] 2 N (xn)! [(1 x) N]! e {N[ x ln x (1 x)ln(1 x)+ 2 ln{x(2 x)}]} (1) Now we have in the exponent a value, whih determines in the thermodynami limit (N ), whether the average is zero or above zero. The exponent hanges sign at x an () < x (). x an () is a lower bound on x (), beause for values smaller than x an () the average N v (G, xn) goes in the thermodynami limit to 0 and there is no vertex-over. 0 = x an ()lnx an () (1 x an ())ln (1 x an ()) + 2 ln {x an ()(2 x an ())} (2) The asymptoti for x an () for large average degree is given by x an () = 1 2 ln() ( ) ln {ln()} + O and the preise asymptoti is alulated in [1]: x () = 1 2 (ln() ln {ln()} ln(2) + 1) + o( 1 ) x an () [0, 0.5] plotted using the equation 2 is shown in figure 3. Comparing x an () with x () shown in figure 4 hints that x an () really is a lower bound for x (). = e {ln(n!)+ 2 N ln[x(2 x)] ln[(xn)!] ln[{(1 x)n}!]} Stirling s approximation ln(n!) N lnn N 4 Typial running time analysis 4.1 Analysis of first desent into the onfiguration tree This subsetion deals with analysing the first desent into the onfiguration tree, i.e. the linear phase, where no baktraking ours. When searhing for a vertex-over in the graph G G(N, N ), we onsider overed verties and edges as removed from the graph. Thus at time step T, the urrent graph is G G(N T, N ). The graph at time step T remains a random graph, beause the order of vertex removal is random. For some heuristis this is not true. In eah step exatly one vertex is removed from the graph and at time step T the average edge degree is 5

6 0.5 x an () x an () Figure 3: x an() [0,0.5] plotted using equation 0 = x an () ln x an () (1 x an ()) ln (1 x an ()) + ln {xan () (2 xan ())} 2 (T) (N T) N = (1 T N ), beause the edge probability p = N remains the same over time. We use resaled time t = T N, whih is ontinuous in the thermodynami limit. Using a resaled time simplifies the alulations. The urrent graph is then G G((1 t)n, N ). Beause overing marks are put on all verties exept isolated ones, we are interested in the probability that a vertex is not isolated. This an be alulated with the stati edge probability and with the number of possible edge endpoints at time t. For a single vertex at time t: ( P(vertex is not isolated) = 1 1 ) }{{ N } P(no edge) # possible edge endpoints {}}{ (1 t)n 1 = 1 e {((1 t)n 1)ln(1 N )} 1 e {(1 t) lim N N ln(1 N )} = 1 e {(1 t) lim N N N } = 1 e { (1 t)} With the probability that a vertex is not isolated, i.e. that a vertex is overed, we an alulate the amount of available overing marks by subtrating from the original overing mark amount the amount used at time t: X(t) = xn N t 0 P(vertex is not isolated)dt 6

7 = xn N t 0 ( 1 e { (1 t)}) dt = xn Nt + N e (1 t) e x(t) is the amount of available overing marks at time t divided by the amount of verties at time t. The average vertex degree dereases linearly with time. Putting this together gives us the trajetory for the first desent: (t) = (1 t) x(t) = X(t) N(t) = x t 1 t + e (1 t) e (1 t) Figure 4 displays trajetories in the (x, ) plane. Phase boundary x b (), displayed in the figure, an be alulated by setting x(t ) = 1 and then t = 1: x(t ) = 1 1 = x b() t 1 t + e (1 t ) e (1 t ) 1 t = x b () t + e (1 t ) e t = 1 0 = x b () e x b () = 1 + e 1 If x starts below this boundary, an exponential running time is to be expeted, beause the first desent does not find a solution. 4.2 Baktraking analysis The overable exponential running time phase B is shown in figure 2. In this phase baktraking ours, when there are not enough overing marks available to over the remaining random subgraph. Baktraking analysis an be diffiult, beause of onfiguration tree node dependenies. The problem is approahed level by level of the onfiguration tree and the number of nodes at eah level is taken into aount. This is feasible, beause the omputation time depends on the number of nodes, but not on the order they are gone through. At time t the remaining overing marks xñ of the first desent are not enough. From the point of view of the onfiguration tree, the onfiguration tree node at time t is pointed to by the blue arrow in the example figure 1. x () and the first desent ross at time t at ( x, ). At time t there are xñ overing marks remaining. As disussed in the previous setion, we always have a random graph, when first desending the onfiguration tree and removing randomly seleted verties and edges. The subgraph at time t is G G(Ñ, ) and it must be baktraked in Ñ exponential time, beause it has no solution. The subgraph G dominates the running time, whih an be approximated with the amount of nodes in the subgraph. 7

8 Figure 4: First desent into the onfiguration tree. Dotted line is x b () and long dashed line is x (). The lines start at = 2.0. The symbols are numerial results from a random graph with 10 6 verties. Figure is from [3]. 8

9 From the experimental data we know that exponential solution times( are log-normal ) distributed for large N. The typial solution time is e Nτ(x,). Denote with t bt GÑ,, x the baktraking time of the unoverable GÑ,. The normalized averaged logarithmi running time is Ñ Ñ [ ( )] 1 the quenhed average τ(x, ) = lim N N ln t bt GÑ,, x. Here the logarithm enfores that Ñ we obtain the typial running time, beause the solution times are log-normal distributed. The baktraking time is upper bounded by the number of leaves of the subtree times the number of verties in the subgraph t bt ÑN l. The linear Ñ O(N) ontribution an be disarded and we just use the number of leaves to approximate the exponential subgraph running time: 1 [ ( )] τ(x, ) lim N N ln N l GÑ,, x Ñ The number of leaves ( is upper ) bounded by the distribution of remaining overing marks on Ñ the remaining verties. Similar to the alulations done in equation 1 we get for the xñ unbounded version of the algorithm τ(x, ) { xln x + (1 x)ln (1 x)} For the bounded version of the branh-and-bound algorithm, we have to take into aount the level at whih the bound stops the traversing of the onfiguration tree. We mark with κñ the level in the subtree, where the bound beomes effetive. Omitting a lot of alulations the result for the bounded version is presented here: τ(x, ) max κ= x,...,1 [ κs an ( )] x κ, κ s an (x, ) = x lnx (1 x )ln(1 x ) + 2 ln(x [2 x ]) The typial running time is approximated with the largest subtree by finding the saddle point 3 on the κ value. Figure 2 shows also this analytial result. It is worth noting that the analytial results seem to fit well with the experimental data. 5 Summary A simple branh-and-bound algorithm was introdued for the vertex-over problem. Experimental data of typial running times for the algorithm was shown. Two exponential dynamial phases and one linear phase were disernible from the data. The first-moment method was explained and a lower bound on x () was onstruted by using the average vertex-over amount as an upper bound for the vertex-over probability of a graph. Analysis of the first desent vertex-over algorithm was done and (x, )-trajetories of first desent runs were analysed and experimental data of them displayed. The dynami phase boundary x b (), whih shows where exponential solution times start ourring, was alulated. In the analysis, important steps were the onsidering of the urrent graph with overed verties and edges removed, alulating the probability for a not-isolated vertex and alulating the amount of available overing marks at a time step, using a ontinuous (in the thermodynami limit) resaled time. In the baktraking analysis the typial running time was alulated for the unbounded version of the algorithm using the amount of leaves in the baktraking subtree. For the bounded version (3) 9

10 the typial running time was found by onsidering the largest subgraph that was possible with the bound effetive. Referenes [1] A. M. Frieze. Disr. Math., 81(171), [2] Alexander K. Hartmann and Martin Weigt. Phase Transitions in Combinatorial Optimization Problems. WILEY-VCH, [3] Martin Weigt and Alexander K. Hartmann. Typial solution time for a vertex-overing algorithm on finite-onnetivity random graphs. Physial Review Letters, 86(8), February