Multi-parameter complexity analysis for constrained size graph problems: using greediness for parameterization
Edouard Bonnet, Bruno Escoffier, Vangelis Th. Paschos, Emeric Tourniaire
MMulti-parameter complexity analysis forconstrained size graph problems: using greedinessfor parameterization ∗ ´E. Bonnet B. Escoffier V. Th. Paschos ( a ) ´E. TourniairePSL Research University, Universit´e Paris-Dauphine, LAMSADECNRS, UMR 7243, France { bonnet,escoffier,paschos,tourniaire } @lamsade.dauphine.fr October 29, 2018
Abstract
We study the parameterized complexity of a broad class of problemscalled “local graph partitioning problems” that includes the classical fixedcardinality problems as max k -vertex cover , k -densest subgraph ,etc. By developing a technique “greediness-for-parameterization”, we ob-tain fixed parameter algorithms with respect to a pair of parameters k ,the size of the solution (but not its value) and ∆, the maximum degree ofthe input graph. In particular, greediness-for-parameterization improvesasymptotic running times for these problems upon random separation(that is a special case of color coding) and is more intuitive and sim-ple. Then, we show how these results can be easily extended for gettingstandard-parameterization results (i.e., with parameter the value of theoptimal solution) for a well known local graph partitioning problem. A local graph partitioning problem is a problem defined on some graph G =( V, E ) with two integers k and p . Feasible solutions are subsets V ⊆ V ofsize exactly k . The value of their solutions is a linear combination of sizes ofedge-subsets and the objective is to determine whether there exists a solutionof value at least or at most p . Problems as max k -vertex cover , k -densestsubgraph , k -lightest subgraph , max ( k, n − k ) -cut and min ( k, n − k ) -cut ,also known as fixed cardinality problems, are local graph partitioning problems. ∗ Research supported by the French Agency for Research under the program TODO, ANR-09-EMER-010 ( a ) Institut Universitaire de France a r X i v : . [ c s . CC ] J un hen dealing with graph problems, several natural parameters, other than thesize p of the optimum, can be of interest, for instance, the maximum degree ∆ ofthe input graph, its treewidth, etc. To these parameters, common for any graphproblem, in the case of local graph partitioning problem handled here, one morenatural parameter of very great interest can be additionally considered, thesize k of V . For instance, the most of these problems have mainly been studiedin [4, 8], from a parameterized point of view, with respect to parameter k , andhave been proved W[1]-hard. Dealing with standard parameterization, the onlyproblems that, to the best of our knowledge, have not been studied yet, are the max ( k, n − k ) -cut and the min ( k, n − k ) -cut problems.In this paper we develop a technique for obtaining multi-parameterized re-sults for local graph partitioning problems. Informally, the basic idea behind itis the following. Perform a branching with respect to a vertex chosen upon somegreedy criterion. For instance, this criterion could be to consider some vertex v that maximizes the number of edges added to the solution under construction.Without branching, such a greedy criterion is not optimal. However, if at eachstep either the greedily chosen vertex v , or some of its neighbors (more precisely,a vertex at bounded distance from v ) are a good choice (they are in an optimalsolution), then a branching rule on neighbors of v leads to a branching treewhose size is bounded by a function of k and ∆, and at least one leaf of whichis an optimal solution. This method, called “greediness-for-parameterization”,is presented in Section 2 together with interesting corollaries about particularlocal graph partitioning problems.The results of Section 2 can sometimes be easily extended to standard pa-rameterization results. In Section 3 we study standard parameterization of thetwo still unstudied fixed cardinality problems max and min ( k, n − k ) -cut .We prove that the former is fixed parameter tractable (FPT), while, unfortu-nately, the status of the latter one remains still unclear. In order to handle max ( k, n − k ) -cut we first show that when p (cid:54) k or p (cid:54) ∆, the problem ispolynomial. So, the only “non-trivial” case occurs when p > k and p > ∆, casehandled by greediness-for-parameterization. Unfortunately, this method con-cludes inclusion of min ( k, n − k ) -cut in FPT only for some particular cases.Note that in a very recent technical report by [11], Fomin et al., the follow-ing problem is considered: given a graph G and two integers k, p , determinewhether there exists a set V ⊂ V of size at most k such that at most p edgeshave exactly one endpoint in V . They prove that this problem is FPT withrespect to p . Let us underline the fact that looking for a set of size at most k seems to be radically different that looking for a set of size exactly k (as in min ( k, n − k ) -cut ). For instance, in the case k = n/
2, the former becomes the mincut problem that is polynomial, while the latter becomes the min bisection problem that is NP-hard..In Section 4.1, we mainly revisit the parameterization by k but we handleit from an approximation point of view. Given a problem Π parameterized byparameter ‘ and an instance I of Π, a parameterized approximation algorithmwith ratio g ( . ) for Π is an algorithm running in time f ( ‘ ) | I | O (1) that either findsan approximate solution of value at least/at most g ( ‘ ) ‘ , or reports that there is2o solution of value at least/at most ‘ . We prove that, although W[1]-hard forthe exact computation, max ( k, n − k ) -cut has a parameterized approximationschema with respect to k and min ( k, n − k ) -cut a randomized parameter-ized approximation schema. These results exhibit two problems which are hardwith respect to a given parameter but which become easier when we relax ex-act computation requirements and seek only (good) approximations. To ourknowledge, the only other problem having similar behaviour is another fixedcardinality problem, the max k -vertex cover problem, where one has to findthe subset of k vertices which cover the greatest number of edges [15]. Note thatthe existence of problems having this behaviour but with respect to the standardparameter is an open (presumably very difficult to answer) question in [15]. Letus note that polynomial approximation of min ( k, n − k ) -cut has been studiedin [9] where it is proved that, if k = O (log n ), then the problem admits a ran-domized polynomial time approximation schema, while, if k = Ω(log n ), then itadmits an approximation ratio (1 + εk log n ), for any ε >
0. Approximation of max ( k, n − k ) -cut has been studied in several papers and a ratio 1/2 is achievedin [1] (slightly improved with a randomized algorithm in [10]), for all k .Finally, in Section 4.2, we handle parameterization of local graph partition-ing problems by the treewidth tw of the input graph and show, using a standarddynamic programming technique, that they admit an O ∗ (2 tw )-time FPT algo-rithm, when the O ∗ ( · ) notation ignores polynomial factors. Let us note that theinterest of this result, except its structural aspect (many problems for the priceof a single algorithm), lies also in the fact that some local partitioning problems(this is the case, for instance, of max and min ( k, n − k ) -cut ) do not fit Cour-celle’s Theorem [7]. Indeed, max and min bisection are not expressible in MSOsince the equality of the cardinality of two sets is not MSO-definable. In fact, ifone could express that two sets have the same cardinality in MSO, one would beable to express in MSO the fact that a word has the same number of a’s and b’s,on a two-letter alphabet, which would make that the set E = { w : | w | a = | w | b } is MSO-definable. But we know that, on words, MSO-definability is equivalentto recognizability; we also know by the standard pumping lemma (see, for in-stance, [13]) that E is not recognizable [14], a contradiction. Henceforth, max and min ( k, n − k ) -cut are not expressible in MSO; consequently, the fact thatthose two problems, parameterized by tw are FPT cannot be obtained by Cour-celle’s Theorem. Furthermore, even several known extended variants of MSOwhich capture more problems [16], does not seem to be able to express theequality of two sets either. We first formally define the class of local graph paritioning problems.
Definition 1.
A local graph partitioning problem is a problem having as inputa graph G = ( V, E ) and two integers k and p . Feasible solutions are subsets V ⊆ V of size exactly k . The value of a solution, denoted by val( V ), is a3inear combination α m + α m where m = | E ( V ) | , m = | E ( V , V \ V ) | and α , α ∈ R . The goal is to determine whether there exists a solution ofvalue at least p (for a maximization problem) or at most p (for a minimizationproblem).Note that α = 1, α = 0 corresponds to k -densest subgraph and k -sparsest subgraph , while α = 0, α = 1 corresponds to ( k, n − k ) -cut , and α = α = 1 gives k -coverage . As a local graph partitioning problem is en-tirely defined by α , α and goal ∈ { min , max } we will unambiguously denoteby L (goal , α , α ) the corresponding problem. For conciseness and when noconfusion is possible, we will use local problem instead. In the sequel, k alwaysdenotes the size of feasible subset of vertices and p the standard parameter, i.e.,the solution-size. Moreover, as a partition into k and n − k vertices, respec-tively, is completely defined by the subset V of size k , we will consider it to bethe solution. A partial solution T is a subset of V with less than k vertices.Similarly to the value of a solution, we define the value of a partial solution,and denote it by val ( T ).Informally, we devise incremental algorithms for local problems that addvertices to an initially empty set T (for “taken” vertices) and stop when T becomes of size k , i.e., when T itself becomes a feasible solution. A vertexintroduced in T is irrevocably introduced there and will be not removed later. Definition 2.
Given a local graph partitioning problem L (goal , α , α ), the contribution of a vertex v within a partial solution T (such that v ∈ T ) isdefined by δ ( v, T ) = α | E ( { v } , T ) | + α | E ( { v } , V \ T ) | Note that the value of any (partial) solution T satifies val ( T ) = Σ v ∈ T δ ( v, T ).One can also remark that δ ( v, T ) = δ ( v, T ∩ N ( v )), where N ( v ) denotes the(open) neighbourhood of the vertex v . Function δ is called the contributionfunction or simply the contribution of the corresponding local problem. Definition 3.
Given a local graph partitioning problem L (goal , α , α ), a con-tribution function is said to be degrading if for every v , T and T such that v ∈ T ⊆ T , δ ( v, T ) (cid:54) δ ( v, T ) for goal = min (resp., δ ( v, T ) (cid:62) δ ( v, T ) forgoal = max).Note that it can be easily shown that for a maximization problem, a con-tribution function is degrading if and only if α (cid:62) α / α (cid:54) α / max k -vertex cover , k -sparsestsubgraph and max ( k, n − k ) -cut have a degrading contribution function. Theorem 4.
Every local partitioning problem having a degrading contributionfunction can be solved in O ∗ (∆ k ). Proof.
With no loss of generality, we carry out the proof for a minimizationlocal problem L (min , α , α ). We recall that T will be a partial solution andeventually a feasible solution. Consider the following algorithm ALG1 whichbranches upon the closed neighborhood N [ v ] of a vertex v minimizing the greedycriterion δ ( v, T ∪ { v } ). 4 n Tv zN [ v ] \ T Figure 1: Situation of the input graph at a deviating node of the branchingtree. The vertex v can substitute z since, by the hypothesis, N [ v ] \ T and V n are disjoint and the contribution of a vertex can only decrease when we lateradd some of its neighbors in the solution. • if k > – pick the vertex v ∈ V \ T minimizing δ ( v, T ∪ { v } ); – for each vertex w ∈ N [ v ] \ T run ALG1 ( T ∪ { w } , k − • else ( k = 0), store the feasible solution T ; • output the best among the solutions stored.The branching tree of ALG1 has depth k , since we add one vertex at eachrecursive call, and arity at most max v ∈ V | N [ v ] | = ∆ + 1, where N [ v ] denotesthe closed neighbourhood of v . Thus, the algorithm runs in O ∗ (∆ k ).For the optimality proof, we use a classical hybridation technique betweensome optimal solution and the one solution computed by ALG1 .Consider an optimal solution V opt different from the solution V computedby ALG1 . A node s of the branching tree has two characteristics: the partialsolution T ( s ) at this node (denoted simply T if no ambiguity occurs) and thevertex chosen by the greedy criterion v ( s ) (or simply v ). We say that a node s of the branching tree is conform to the optimal solution V opt if T ( s ) ⊆ V opt .A node s deviates from the optimal solution V opt if none of its sons is conformto V opt .We start from the root of the branching tree and, while possible, we moveto a conform son of the current node. At some point we reach a node s whichdeviates from V opt . We set T = T ( s ) and v = v ( s ). Intuitively, T corresponds tothe shared choices between the optimal solution and ALG1 made along the branchfrom the root to the node s of the branching tree. Setting V n = V opt \ T , V n does not intersect N [ v ], otherwise s would not be deviating.Choose any z ∈ V opt \ T and consider the solution induced by the set V e = V opt ∪ { v } \ { z } . We show that this solution is also optimal. Let5 Figure 1: Situation of the input graph at a deviating node of the branchingtree. The vertex v can substitute z since, by the hypothesis, N [ v ] \ T and V n are disjoint and the contribution of a vertex can only decrease when we lateradd some of its neighbors in the solution. Algorithm 5 ( ALG1 ( T , k )) . Set T = ∅ ; • if k > – pick the vertex v ∈ V \ T minimizing δ ( v, T ∪ { v } ); – for each vertex w ∈ N [ v ] \ T run ALG1 ( T ∪ { w } , k − • else ( k = 0), store the feasible solution T ; • output the best among the solutions stored.The branching tree of ALG1 has depth k , since we add one vertex at eachrecursive call, and arity at most max v ∈ V | N [ v ] | = ∆ + 1, where N [ v ] denotesthe closed neighbourhood of v . Thus, the algorithm runs in O ∗ (∆ k ).For the optimality proof, we use a classical hybridation technique betweensome optimal solution and the one solution computed by ALG1 .Consider an optimal solution V opt different from the solution V computedby ALG1 . A node s of the branching tree has two characteristics: the partialsolution T ( s ) at this node (denoted simply T if no ambiguity occurs) and thevertex chosen by the greedy criterion v ( s ) (or simply v ). We say that a node s of the branching tree is conform to the optimal solution V opt if T ( s ) ⊆ V opt .A node s deviates from the optimal solution V opt if none of its sons is conformto V opt .We start from the root of the branching tree and, while possible, we moveto a conform son of the current node. At some point we reach a node s whichdeviates from V opt . We set T = T ( s ) and v = v ( s ). Intuitively, T corresponds tothe shared choices between the optimal solution and ALG1 made along the branchfrom the root to the node s of the branching tree. Setting V n = V opt \ T , V n does not intersect N [ v ], otherwise s would not be deviating.5hoose any z ∈ V opt \ T and consider the solution induced by the set V e = V opt ∪ { v } \ { z } . We show that this solution is also optimal. Let V c = V opt \{ z } . We have val ( V e ) = Σ w ∈ V c δ ( w, V e )+ δ ( v, V e ). Besides, δ ( v, V e ) = δ ( v, V e ∩ N ( v )) = δ ( v, T ∪ { v } ) since V e \ ( T ∪ { v } ) = V n and according tothe last remark of the previous paragraph, N ( v ) ∩ V n = ∅ . By the choiceof v , δ ( v, T ∪ { v } ) (cid:54) δ ( z, T ∪ { z } ), and, since δ is a degrading contribu-tion, δ ( z, T ∪ { z } ) (cid:54) δ ( z, V opt ). Summing up, we get δ ( v, V e ) (cid:54) δ ( z, V opt )and val ( V e ) (cid:54) Σ w ∈ V c δ ( w, V e ) + δ ( z, V opt ). Since v is not in the neighborhoodof V opt \ T = V n only z can degrade the contribution of those vertices, soΣ w ∈ V c δ ( w, V e ) (cid:54) Σ w ∈ V c δ ( w, V opt ), and val ( V e ) (cid:54) Σ w ∈ V c δ ( w, V opt )+ δ ( z, V opt ) =val ( V opt ).Thus, by repeating this argument at most k times, we can conclude that thesolution computed by ALG1 is as good as V opt . Corollary 6. max k -vertex cover , k -sparsest subgraph and max ( k, n − k ) -cut can be solved in O ∗ (∆ k ).As mentioned before, the local problems mentioned in Corollary 6 have adegrading contribution. Theorem 7.
Every local partitioning problem can be solved in O ∗ ((∆ k ) k ). Proof.
Once again, with no loss of generality, we prove the theorem in the case ofminimization, i.e., L (min , α , α ). The proof of Theorem 7 involves an algorithmfairly similar to ALG1 but instead of branching on a vertex chosen greedily and itsneighborhood, we will branch on sets of vertices inducing connected components(also chosen greedily) and the neighborhood of those sets.Let us first state the following straightforward lemma that bounds the num-ber of induced connected components and the running time to enumerate them.
Lemma 8.
One can enumerate the connected induced subgraphs of size up to k in time O ∗ (∆ k ). Proof of Lemma 8.
One can easily enumerate with no redundancy all the con-nected induced subgraph of size k which contains a vertex v . Indeed, one canlabel the vertices of a graph G with integers from 1 to n , and at each step, takethe vertex in the built connected component with the smaller label and decideonce and for all which of its neighbors will be in the component too. That way,you get each connected induced component in a unique manner.Now, it boils down to counting the number of connected induced subgraphof size k which contains a given vertex v . We denote that set of componentsby C k,v . Let us show that there is an injection from C k,v to the set B k d log ∆ e ofthe binary trees with k d log ∆ e nodes.Recall that the vertices of G are labeled from 1 to n . Given a component C ∈ C k,v , build the following binary tree. Start from the vertex v . From thecomplete binary tree of height d log ∆ e , owning a little more than ∆ orderedleaves, place in those leaves the vertices of N ( v ) according to the order (cid:54) , andkeep only the branches leading to vertices in C ∩ N ( v ). Iterate this process until6 n \ HS TH H c Figure 2: Illustration of the proof, with filled vertices representing the optimalsolution V opt and dotted vertices representing the set S = S | H | computed by ALG2 which can substitute H , since V n does not interact with H c nor with S .not keep the corresponding branch. That way, you get for each vertex of C abranch of size d log ∆ e , and hence there are k d log ∆ e nodes in the tree.Recall that |B k d log ∆ e | is given by the Catalan numbers, so |B k d log ∆ e | = (2 k d log ∆ e )!( k d log ∆ e )!( k d log ∆ e +1)! = O ∗ (4 k log ∆ ) = O ∗ (∆ k ). So, Σ v ∈ V |C k,v | = O ∗ (∆ k ).The proof of Lemma 8 is now completed.Consider now the following algorithm. Algorithm 9 ( ALG2 ( T , k )) . set T = ∅ ; ALG2 ( T , k ) • if k > i from 1 to k , – find S i ∈ V \ T minimizing val ( T ∪ S i ) with S i inducing a connectedcomponent of size i . – for each i , for each v ∈ S i , run ALG2 ( T ∪ { v } , k − • else ( k = 0), stock the feasible solution T .output the stocked feasible solution T minimizing val ( T ).The branching tree of ALG2 has size O ( k k ). Computing the S i in eachnode takes time O ∗ (∆ k ) according to Lemma 8. Thus, the algorithm runs in O ∗ ((∆ k ) k ).For the optimality of ALG2 , we use the following lemma.7
Figure 2: Illustration of the proof, with filled vertices representing the optimalsolution V opt and dotted vertices representing the set S = S | H | computed by ALG2 which can substitute H , since V n does not interact with H c nor with S .you get all the vertices of C exactly once. When a vertex of C reappears, donot keep the corresponding branch. That way, you get for each vertex of C abranch of size d log ∆ e , and hence there are k d log ∆ e nodes in the tree.Recall that |B k d log ∆ e | is given by the Catalan numbers, so |B k d log ∆ e | = (2 k d log ∆ e )!( k d log ∆ e )!( k d log ∆ e +1)! = O ∗ (4 k log ∆ ) = O ∗ (∆ k ). So, Σ v ∈ V |C k,v | = O ∗ (∆ k ).The proof of Lemma 8 is now completed.Consider now the following algorithm. Algorithm 9 ( ALG2 ( T , k )) . set T = ∅ ; ALG2 ( T , k ) • if k > i from 1 to k , – find S i ∈ V \ T minimizing val ( T ∪ S i ) with S i inducing a connectedcomponent of size i . – for each i , for each v ∈ S i , run ALG2 ( T ∪ { v } , k − • else ( k = 0), stock the feasible solution T .output the stocked feasible solution T minimizing val ( T ).The branching tree of ALG2 has size O ( k k ). Computing the S i in eachnode takes time O ∗ (∆ k ) according to Lemma 8. Thus, the algorithm runs in O ∗ ((∆ k ) k ).For the optimality of ALG2 , we use the following lemma.7 emma 10.
Let A , B , X , Y be pairwise disjoint sets of vertices such that val ( A ∪ X ) (cid:54) val ( B ∪ X ), N [ A ] ∩ Y = ∅ and N [ B ] ∩ Y = ∅ . Then, val ( A ∪ X ∪ Y ) (cid:54) val ( B ∪ X ∪ Y ). Proof of Lemma 10.
Simply observe that val ( A ∪ X ∪ Y ) = val ( Y ) + val ( A ∪ X ) − α | E ( X, Y ) | + α | E ( X, Y ) | (cid:54) val ( Y ) + val ( B ∪ X ) − α | E ( X, Y ) | + α | E ( X, Y ) | = val ( B ∪ X ∪ Y ), that completes the proof of the lemma.We now show that ALG2 is sound, using again hybridation between an opti-mal solution V opt and the one solution found by ALG2 . We keep the same nota-tion as in the proof of the soundness of
ALG1 . Node s is a node of the branchingtree which deviates from V opt , all nodes in the branch between the root and s are conform to V opt , the shared choices constitute the set of vertices T = T ( s )and, for each i , set S i = S i ( s ) (analogously to v ( s ) in the previous proof, s isnow linked to the subsets S i computed at this node). Set V n = V opt \ T . Takea maximal connected (non empty) subset H of V n . Set S = S | H | and consider V e = V opt \ H ∪ S = ( T ∪ V n ) \ H ∪ S = T ∪ S ∪ ( V n \ H ). Note that, byhypothesis, N [ S ] ∩ V n = ∅ since s is a deviating node. By the choice of S atthe node s , val ( T ∪ S ) (cid:54) val ( T ∪ H ). So, val ( V e ) = val ( T ∪ S ∪ ( V n \ H )) =val ( T ∪ H ∪ ( V n \ H )) = val ( T ∪ V n ) = val ( V opt ) according to Lemma 10, sinceby construction neither N [ H ] nor N [ S ], do intersect V n \ H . Iterating the argu-ment at most k times we get to a leaf of the branching tree of ALG2 which yieldsa solution as good as V opt . The proof of the theorem is now completed. Corollary 11. k -densest subgraph and min ( k, n − k ) -cut can be solved in O ∗ ((∆ k ) k ).Here also, simply observe that the problems mentioned in Corollary 11 arelocal graph partitioning problems.Theorems 4 and 7 improve the O ∗ (2 (∆+1) k ((∆ + 1) k ) log((∆+1) k ) ) time com-plexity for the corresponding problems given in [5] obtained there by the randomseparation technique. Recall that random separation consists of randomly guess-ing if a vertex is in an optimal subset V of size k (white vertices) or if it is in N ( V ) \ V (black vertices). For all other vertices the guess has no importance.As a right guess concerns at most only k + k ∆ vertices, it is done with highprobability if we repeat random guesses f ( k, ∆) times with a suitable function f .Given a random guess, i.e., a random function g : V → { white,black } , a solutioncan be computed in polynomial time by dynamic programming. Although ran-dom separation (and a fortiori color coding [2]) have also been applied to otherproblems than local graph partitioning ones, greediness-for-parameterizationseems to be quite general and improves both running time and easiness of im-plementation since our algorithms do not need complex derandomizations.Let us note that the greediness-for-parameterization technique can be evenmore general, by enhancing the scope of Definition 1 and can be applied toproblems where the objective function takes into account not only edges butalso vertices. The value of a solution could be defined as a function val : P ( V ) → R such that val ( ∅ ) = 0, the contribution of a vertex v in a partial8 V v ?? v S w a p (a) Vertices v ∈ V and v ∈ V (that has at least one neighbor in V ) will be swapped. V V v ?? v (b) With the swapping the cut sizeincreases. Figure 1: Illustration of a swapping ( k , n − k ) -cut In the sequel, we denote by N ( v ) the set of neighbors of v in G = ( V, E ), namely { w ∈ V : { v, w } ∈ E } and define N [ v ] = N ( v ) ∪ { v } . We also use the standard notation G [ U ] for any U ⊆ V to denote the subgraph induced by the vertices of U . In this section, we show that max ( k, n − k ) -cut parameterized by the standard parameter, i.e., by the value p of the solution, isFPT. Using an idea of bounding above the value of an optimal solution by a swapping process(see Figure ?? ), we show that the non trivial case satisfies p > k . We also show that p > ∆holds for non trivial instances and get the situation depicted by Figure ?? . The rest of the proof(see Theorem ?? ) shows that max ( k, n − k ) -cut parameterized by k + ∆ is FPT, by designing aparticular branching algorithm. This branching algorithm is based on the following intuitive idea.Consider a vertex v of maximum degree in the graph. If an optimal solution E ( V , V \ V ) is suchthat no vertex of N ( v ) is in V , then it is always interesting to take v in V (this provides ∆ edgesto the cut, which is the best we can do). This leads to a branching rule with ∆+1 branches, wherein each branch we take in V one vertex from N [ v ]. Lemma 1.
In a graph with minimum degree r , the optimal value opt of a max ( k, n − k ) -cut satisfies opt > min { n − k, rk } .Proof. We divide arbitrarily the vertices of a graph G = ( V, E ) into two subsets V and V of size k and n − k , respectively. Then, for every vertex v ∈ V , we check if v has a neighbor in V . If not, wetry to swap v and a vertex v ∈ V which has strictly less than r neighbors in V (see Figure ?? ). Ifthere is no such vertex, then every vertex in V has at least r neighbors in V , so determining a cutof value at least rk . When swapping is possible, as the minimum degree is r and the neighborhoodof v is entirely contained in V , moving v from V to V will increase the value of the cut by atleast r . On the other hand, moving v from V to V will reduce the value of the cut by at most r −
1. In this way, the value of the cut increases by at least 1.Finally, either the process has reached a cut of value rk (if no more swap is possible), or everyvertex in V has increased the value of the cut by at least 1 (either immediately, or after a swappingprocess), which results in a cut of value at least n − k , and the proof of the lemma is completed. Corollary 2.
In a graph with no isolated vertices, the optimal value for max ( k, n − k ) -cut is atleast min { n − k, k } . Theorem 3.
The max ( k, n − k ) -cut problem parameterized by the standard parameter p is FPT. Figure 3: Illustration of a swappingsolution T is δ ( v, T ) = val ( T ∪ v ) − val ( T ). Thus, for any subset T , val ( T ) =val ( T \ { v k } ) + δ ( v k , T \ { v k } ) where k is the size of T and v k is the lastvertex added to the solution. Hence, val ( T ) = Σ (cid:54) i (cid:54) k δ ( v i , { v , . . . , v i − } ) +val ( ∅ ) = Σ (cid:54) i (cid:54) k δ ( v i , { v , . . . , v i − } ). Now, the only hypothesis we need to showTheorem 7 is the following: for each T such that ( N ( T ) \ T ) ∩ ( N ( v ) \ T ) = ∅ , δ ( v, T ∪ T ) = δ ( v, T ).Notice also that, that under such modification, max k -dominating set ,asking for a set V of k vertices that dominate the highest number of vertices in V \ V fulfils the enhancement just discussed. We therefore derive the following. Corollary 12. max k -dominating set can be solved in O ∗ ((∆ k ) k . ( k , n − k ) -cut ( k , n − k ) -cut In the sequel, we use the standard notation G [ U ] for any U ⊆ V to denotethe subgraph induced by the vertices of U . In this section, we show that max ( k, n − k ) -cut parameterized by the standard parameter, i.e., by the value p ofthe solution, is FPT. Using an idea of bounding above the value of an optimalsolution by a swapping process (see Figure 3), we show that the non-trivial casesatisfies p > k . We also show that p > ∆ holds for non trivial instances andget the situation illustrated in Figure 4. The rest of the proof is an immediateapplication of Corollary 6. Lemma 13.
In a graph with minimum degree r , the optimal value opt of a max (k,n-k)-cut satisfies opt (cid:62) min { n − k, rk } .9 V v ?? v S w a p (a) Vertices v ∈ V and v ∈ V (that has at least one neighbor in V ) will be swapped. V V v ?? v (b) With the swapping the cut sizeincreases. Figure 1: Illustration of a swapping ( k , n − k ) -cut In the sequel, we denote by N ( v ) the set of neighbors of v in G = ( V, E ), namely { w ∈ V : { v, w } ∈ E } and define N [ v ] = N ( v ) ∪ { v } . We also use the standard notation G [ U ] for any U ⊆ V to denote the subgraph induced by the vertices of U . In this section, we show that max ( k, n − k ) -cut parameterized by the standard parameter, i.e., by the value p of the solution, isFPT. Using an idea of bounding above the value of an optimal solution by a swapping process(see Figure ?? ), we show that the non trivial case satisfies p > k . We also show that p > ∆holds for non trivial instances and get the situation depicted by Figure ?? . The rest of the proof(see Theorem ?? ) shows that max ( k, n − k ) -cut parameterized by k + ∆ is FPT, by designing aparticular branching algorithm. This branching algorithm is based on the following intuitive idea.Consider a vertex v of maximum degree in the graph. If an optimal solution E ( V , V \ V ) is suchthat no vertex of N ( v ) is in V , then it is always interesting to take v in V (this provides ∆ edgesto the cut, which is the best we can do). This leads to a branching rule with ∆+1 branches, wherein each branch we take in V one vertex from N [ v ]. Lemma 1.
In a graph with minimum degree r , the optimal value opt of a max ( k, n − k ) -cut satisfies opt > min { n − k, rk } .Proof. We divide arbitrarily the vertices of a graph G = ( V, E ) into two subsets V and V of size k and n − k , respectively. Then, for every vertex v ∈ V , we check if v has a neighbor in V . If not, wetry to swap v and a vertex v ∈ V which has strictly less than r neighbors in V (see Figure ?? ). Ifthere is no such vertex, then every vertex in V has at least r neighbors in V , so determining a cutof value at least rk . When swapping is possible, as the minimum degree is r and the neighborhoodof v is entirely contained in V , moving v from V to V will increase the value of the cut by atleast r . On the other hand, moving v from V to V will reduce the value of the cut by at most r −
1. In this way, the value of the cut increases by at least 1.Finally, either the process has reached a cut of value rk (if no more swap is possible), or everyvertex in V has increased the value of the cut by at least 1 (either immediately, or after a swappingprocess), which results in a cut of value at least n − k , and the proof of the lemma is completed. Corollary 2.
In a graph with no isolated vertices, the optimal value for max ( k, n − k ) -cut is atleast min { n − k, k } . Theorem 3.
The max ( k, n − k ) -cut problem parameterized by the standard parameter p is FPT. n nk n − kp ∆Figure 4: Location of parameter p , relatively to k and ∆.10 Figure 4: Location of parameter p , relatively to k and ∆. Proof.
We divide arbitrarily the vertices of a graph G = ( V, E ) into two sub-sets V and V of size k and n − k , respectively. Then, for every vertex v ∈ V ,we check if v has a neighbor in V . If not, we try to swap v and a vertex v ∈ V which has strictly less than r neighbors in V (see Figure 3). If there is no suchvertex, then every vertex in V has at least r neighbors in V , so determining acut of value at least rk . When swapping is possible, as the minimum degree is r and the neighborhood of v is entirely contained in V , moving v from V to V will increase the value of the cut by at least r . On the other hand, moving v from V to V will reduce the value of the cut by at most r −
1. In this way, thevalue of the cut increases by at least 1.Finally, either the process has reached a cut of value rk (if no more swap ispossible), or every vertex in V has increased the value of the cut by at least 1(either immediately, or after a swapping process), which results in a cut of valueat least n − k , and the proof of the lemma is completed. Corollary 14.
In a graph with no isolated vertices, the optimal value for max ( k, n − k ) -cut is at least min { n − k, k } .Then, Corollary 6 suffices to conclude the proof of the the following theorem. Theorem 15.
The max ( k, n − k ) -cut problem parameterized by the standardparameter p is FPT. ( k , n − k ) -cut Unfortunately, unlike what have been done for max ( k, n − k ) -cut , we have notbeen able to show until now that the case p < k is “trivial”. So, Algorithm ALG2 in Section 2 cannot be transformed into a standard FPT algorithm for thisproblem.However, we can prove that when p (cid:62) k , then min ( k, n − k ) -cut parame-terized by the value p of the solution is FPT. This is an immediate corollary ofthe following proposition. Proposition 16. min ( k, n − k ) -cut parameterized by p + k is FPT. Proof.
Each vertex v such that | N ( v ) | (cid:62) k + p has to be in V \ V (of size n − k ).Indeed, if one puts v in V (of size k ), among its k + p incident edges, at least p +1 leave from V ; so, it cannot yield a feasible solution. All the vertices v suchthat | N ( v ) | (cid:62) k + p are then rejected. Thus, one can adapt the FPT algorithm10n k + ∆ of Theorem 7 by considering the k -neighborhood of a vertex v not inthe whole graph G , but in G [ T ∪ U ]. One can easily check that the algorithmstill works and since in those subgraphs the degree is bounded by p + k we getan FPT algorithm in p + k .In [9], it is shown that, for any ε >
0, there exists a randomized (1 + εk log n )-approximation for min ( k, n − k ) -cut . From this result, we can easily derivethat when p < log nk then the problem is solvable in polynomial time (by arandomized algorithm). Indeed, fixing ε = 1, the algorithm in [9] is a (1+ k log( n ) )-approximation. This approximation ratio is strictly better than 1 + p . Thismeans that the algorithm outputs a solution of value lower than p + 1, hence atmost p , if there exists a solution of value at most p .We now conclude this section by claiming that, when p (cid:54) k , min ( k, n − k ) -cut can be solved in time O ∗ ( n p ). Proposition 17. If p (cid:54) k , then min ( k, n − k ) -cut can be solved in time O ∗ ( n p ). Proof.
Since p (cid:54) k , there exist in the optimal set V , p (cid:54) p vertices incident tothe p outgoing edges. So, the k − p remaining vertices of V induce a subgraphthat is disconnected from G [ V \ V ].Hence, one can enumerate all the p (cid:54) p subsets of V . For each such subset e V ,the graph G [ V \ e V ] is disconnected. Denote by C = ( C i ) (cid:54) i (cid:54) | C | the connectedcomponents of G [ V \ e V ] and by α i the number of edges between C i and e V . Wehave to pick a subset C ⊂ C among these components such that P C i ∈ C | C i | = k − p and maximizing P C i ∈ C α i . This can be done in polynomial time usingstandard dynamic programming techniques. k and approximation of max andmin ( k , n − k ) -cut Recall that both max and min ( k, n − k ) -cut parameterized by k are W[1]-hard [8, 4]. In this section, we give some approximation algorithms working inFPT time with respect to parameter k . Proposition 18. max ( k, n − k ) -cut , parameterized by k has a fixed-parameterapproximation schema. On the other hand, min ( k, n − k ) -cut parameterizedby k has a randomized fixed-parameter approximation schema. Proof.
We first handle max (k,n-k)-cut. Fix some ε >
0. Given a graph G =( V, E ), let d (cid:54) d (cid:54) . . . (cid:54) d k be the degrees of the k largest-degree vertices v , v , . . . v k in G . An optimal solution of value opt is obviously bounded fromabove by B = Σ ki =1 d i . Now, consider solution V = { v , v , . . . , v k } . As thereexist at most k ( k − / (cid:54) k / V is a k -clique) inner edges, solution V has a value sol at least B − k . Hence, the approximation ratio is at least11 − k B = 1 − k B . Since, obviously, B (cid:62) d = ∆, an approximation ratio at least1 − k ∆ is immediately derived.If ε (cid:62) k ∆ then V is a (1 − ε )-approximation. Otherwise, if ε (cid:54) k ∆ , then ∆ (cid:54) k ε . So, the branching algorithm of Theorem 15 with time-complexity O ∗ (∆ k )is in this case an O ∗ ( k k ε k )-time algorithm.For min (k,n-k)-cut, it is proved in [9] that, for ε >
0, if k < log n , then thereexists a randomized polynomial time (1 + ε )-approximation. Else, if k > log n ,the exhaustive enumeration of the k -subsets takes time O ∗ ( n k ) = O ∗ ((2 k ) k ) = O ∗ (2 k ).Finding approximation algorithms that work in FPT time with respect toparameter p is an interesting question. Combining the result of [9] and an O (log . ( n ))-approximation algorithm in [10] we can show that the problemis O ( k / ) approximable in polynomial time by a randomized algorithm. But,is it possible to improve this ratio when allowing FPT time (with respect to p )? When dealing with parameterization of graph problems, some classical param-eters arise naturally. One of them, very frequently used in the fixed parameterliterature is the treewidth of the graph.It has already been proved that min and max ( k, n − k ) -cut , as well as k -densest subgraph can be solved in O ∗ (2 tw ) [3, 12]. We show here that thealgorithm in [3] can be adapted to handle the whole class of local problems,deriving so the following result. Proposition 19.
Any local graph partitioning problem can be solved in time O ∗ (2 tw ). Proof.
A tree decomposition of a graph G ( V, E ) is a pair (
X, T ) where T is atree on vertex set N ( T ) the vertices of which are called nodes and X = ( { X i : i ∈ N ( T ) } ) is a collection of subsets of V such that: (i) ∪ i ∈ N ( T ) X i = V , (ii) foreach edge ( v, w ) ∈ E , there exist an i ∈ N ( T ) such that { v, w } ∈ X i , and(iii) for each v ∈ V , the set of nodes { i : v ∈ X i } forms a subtree of T . Thewidth of a tree decomposition ( { X i : i ∈ N ( T ) } , T ) equals max i ∈ N ( T ) {| X i | − } .The treewidth of a graph G is the minimum width over all tree decompositionsof G . We say that a tree decomposition is nice if any node of its tree that is notthe root is one of the following types: • a leaf that contains a single vertex from the graph; • an introduce node X i with one child X j such that X i = X j ∪ { v } for somevertex v ∈ V ; • a forget node X i with one child X j such that X j = X i ∪ { v } for somevertex v ∈ V ; 12 a join node X i with two children X j and X l such that X i = X j = X l .Assume that the local graph partitioning problem Π is a minimization problem(we want to find V such that val( V ) (cid:54) p ), the maximization case being similar.An algorithm that transforms in linear time an arbitrary tree decomposition intoa nice one with the same treewidth is presented in [12]. Consider a nice treedecomposition of G and let T i be the subtree of T rooted at X i , and G i = ( V i , E i )be the subgraph of G induced by the vertices in S X j ∈ T i X j . For each node X i = ( v , v , . . . , v | X i | ) of the tree decomposition, define a configuration vector ~c ∈ { , } | X i | ; ~c [ j ] = 1 ⇐⇒ v j ∈ X i belongs to the solution. Moreover, for eachnode X i , consider a table A i of size 2 | X i | × ( k + 1). Each row of A i represents aconfiguration and each column represents the number k , 0 (cid:54) k (cid:54) k , of verticesin V i \ X i included in the solution. The value of an entry of this table equalsthe value of the best solution respecting both the configuration vector and thenumber k , and −∞ is used to define an infeasible solution. In the sequel, weset X i,t = { v h ∈ X i : ~c ( h ) = 1 } and X i,r = { v h ∈ X i : ~c ( h ) = 0 } .The algorithm examines the nodes of T in a bottom-up way and fills in thetable A i for each node X i . In the initialization step, for each leaf node X i andeach configuration ~c , we have A i [ ~c, k ] = 0 if k = 0; otherwise A i [ ~c, k ] = −∞ .If X i is a forget node, then consider a configuration ~c for X i . In X j thisconfiguration is extended with the decision whether vertex v is included intothe solution or not. Hence, taking into account that v ∈ V i \ X i we get: A i [ ~c, k ] = min { A j [ ~c × { } , k ] , A j [ ~c × { } , k − } for each configuration ~c and each k , 0 (cid:54) k (cid:54) k .If X i is an introduce node, then consider a configuration ~c for X j . If v is takenin V , its inclusion adds the quantity δ v = α | E ( { v } , X i,t ) | + α | E ( { v } , X i,r ) | to the solution. The crucial point is that δ v does not depend on the k verticesof V i \ X i taken in the solution. Indeed, by construction a vertex in V i \ X i hasits subtree entirely contained in T i . Besides, the subtree of v intersects T i onlyin its root, since v appears in X i , disappears from X j and has, by definition, aconnected subtree. So, we know that there is no edge in G between v and anyvertex of V i \ X i . Hence, A i [ ~c × { } , k ] = A j [ ~c, k ] + δ v , since k counts only thevertices of the current solution in V i \ X i . The case where v is discarded fromthe solution (not taken in V ) is completely similar; we just define δ v accordingto the number of edges linking v to vertices of T i respectively in V and notin V .If X i is a join node, then for each configuration ~c for X i and each k , 0 (cid:54) k (cid:54) k , we have to find the best solution obtained by k j , 0 (cid:54) k j (cid:54) k , verticesin A j plus k − k j vertices in A l . However, the quantity δ ~c = α | E ( X i,t ) | + α | E ( X i,t , X i,r ) | is counted twice. Note that δ ~c depends only on X i,t and X i,r ,since there is no edge between V l \ X i and V j \ X i . Hence, we get: A i [ ~c, k ] = max (cid:54) k j (cid:54) k { A j [ ~c, k j ] + A l [ ~c, k − k j ] } − δ c and the proof of the proposition is completed.13 orollary 20. Restricted to trees, any local graph partitioning problem canbe solved in polynomial time.
Corollary 21. min bisection parameterized by the treewidth of the inputgraph is FPT.It is worth noticing that the result easily extends to the weighted case (whereedges are weighted) and to the case of partitioning V into a constant numberof classes (with a higher running time).Another natural parameter frequently used in the parameterized complexityframework is the size τ of a minimum vertex cover of the input graph. Sinceit always holds that tw (cid:54) τ , the result of Proposition 19 immediately appliesto parameterization by τ . However, the algorithm developed there needs expo-nential space. In what follows, we give a simple parameterization by τ usingpolynomial space. Proposition 22. max and min ( k, n − k ) -cut parameterized by τ can be solvedin FPT O ∗ (2 τ ) time and in polynomial space. Proof.
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