A class of heuristics for the constrained forest problem

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1 Discrete Applied Mathematics 154 (2006) Communication A class of heuristics for the constrained forest problem Michael Laszlo, Sumitra Mukherjee Nova Southeastern University, Graduate School of Computer and Information Sciences 3301 College Avenue, Ft. Lauderdale, FL 33314, USA Received 6 June 2005; accepted 29 June 2005 Available online 31 August 2005 Communicated by F. Maffioli Abstract The constrained forest problem seeks a minimum-weight spanning forest in an undirected edgeweighted graph such that each tree spans at least a specified number of vertices. We present a structured class of greedy heuristics for this NP-hard problem, and identify the best heuristic Elsevier B.V. All rights reserved. Keywords: Constrained forest problem; Tree partitioning; Minimum spanning forests; Greedy heuristics 1. Introduction Given a graph with positive edge weights, the constrained forest problem (CFP) seeks a minimum-weight spanning subgraph each of whose components contains at least m vertices. Such as subgraph is known as an m-forest. In their survey of graph partitioning problems, Cordone and Maffioli [1] take note of the work of Imelieńska et al. [3] which shows that CFP is NP-hard for m 4, and presents a greedy heuristic for CFP with an approximation ratio of two [3]. Recently, Laszlo and Mukherjee [4] present a second greedy heuristic that produces solutions at least as good as, and often better than, those produced by [3]. The current paper characterizes a class of heuristics to which those presented in [3,4] belong as particular cases. Corresponding author. address: mjl@nova.edu (M. Laszlo) X/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi: /j.dam

2 M. Laszlo, S. Mukherjee / Discrete Applied Mathematics 154 (2006) Section 2 presents a class of algorithms for constructing m-forests, and Section 3 characterizes these algorithms with respect to the size of solutions they produce. Section 4 employs these algorithms as heuristics for CFP, and evaluates them in light of the results of Section A class of greedy algorithms for constructing m-forests A forest is called an m-forest if each of its trees contains at least m>0 vertices. This section presents a class of algorithms each of which takes a tree T = (V, E) and constructs a spanning m-forest of T. The input to each algorithm is an ordered set E of the n edges of tree T and a positive integer m n + 1, and the output is a subset of E representing a spanning m-forest. The algorithms are greedy insofar as they attempt to include only edges deemed essential. Given two edges e and f of an ordered edge set in which e precedes f, we say that e is lighter than f and that f is heavier than e. We also refer to the first and last edges of the ordered set as its lightest and heaviest elements, respectively. A tree is said to be large if it contains at least m vertices, and small otherwise. Each algorithm iteratively selects and removes a lightest or heaviest edge from edge set E and either includes the edge in the graph F under construction or rejects the edge. We can describe each algorithm as an edge-coloring process where an edge s color indicates its status: An edge is blue if it belongs to edge set E. An edge is green if it has been added to F. An edge is red if it belongs to neither E nor F. In short, green edges have been included, red edges have been excluded, and blue edges have not yet been considered. At any given time, we refer to the set of green edges as the green forest, which represents a subgraph of the solution that the algorithm eventually produces. A green tree is a (maximal) component of the green forest. Similarly, we refer to the set of edges that are either blue or green as the blue-green forest, which represents a supergraph of the solution that will eventually be produced, and to each component of this forest as a blue-green tree. An edge is said to connect the two green (or blue-green) trees to which it is incident. The edge-coloring process is as follows, where the procedure m-forest is called with an ordered edge set E of tree T: m-forest(e) { for every edge e E, paint e blue; while (E = ) { e the lightest or heaviest edge in E; E E e; if (e is the lightest edge) { if (e connects two large trees in the green forest) then paint e red;

3 8 M. Laszlo, S. Mukherjee / Discrete Applied Mathematics 154 (2006) 6 14 else paint e green; else { // e is the heaviest edge if (e connects two large trees in the blue-green forest) then paint e red; else paint e green; return the green forest; The m-forest algorithm is nondeterministic since edge selection in each iteration is underspecified. Note that the green forest is initially empty and grows over the course of the process, that the blue-green forest is initially the input edge set E and shrinks over the course of the process, and that the green forest is a subgraph of the blue-green forest. When the process terminates, every edge is green or red, and the green forest is the algorithm s solution. Theorem 1. Where E is an ordered set of edges of a tree with at least m vertices, the m-forest algorithm returns an m-forest. Proof. We show that the following loop-invariant is maintained: The blue-green forest is an m-forest. The initial blue-green forest is an m-forest since it is isomorphic to E, a tree of size at least m. For the inductive step, assume that the blue-green forest is an m-forest, and consider the two cases that can arise when an edge e is selected in an iteration of the while loop: Case 1: e is the lightest edge. The blue-green forest remains unchanged unless e is painted red, in which case its removal from the blue-green forest splits the blue-green tree to which e belonged into two blue-green trees T 1 and T 2. Because e was painted red, it is incident to two large green trees, one of which must be a subgraph of T 1 and the other a subgraph of T 2, implying that T 1 and T 2 are both large blue-green trees. Since no other tree in the blue-green forest is affected by removal of edge e, the blue-green forest remains an m-forest. Case 2: e is the heaviest edge. The blue-green forest remains unchanged unless e is painted red, in which case the blue-green tree to which e belonged is replaced by two large blue-green trees. Since no other tree in the blue-green forest is affected by removal of edge e, the blue-green forest remains an m-forest. Hence we have shown the loop invariant that the blue-green forest remains an m-forest. When the algorithm terminates, every edge is green or red, and the green edges, which comprise the final blue-green forest, form an m-forest. 3. Structure of the class of algorithms We can identify a specific heuristic from the class of heuristics represented by m-forest using a string w = w 1 w 2...w n of length n = E over the alphabet {0, 1. The value of bit w i indicates which edge to choose in iteration i of the while loop: the lightest edge if w i = 0, or the heaviest edge if w i = 1. Under this scheme, we obtain a deterministic version

4 M. Laszlo, S. Mukherjee / Discrete Applied Mathematics 154 (2006) of m-forest that gets called with an ordered edge set E and a string w that dictates the edge selection process: deterministic-m-forest(e, w) { for every edge e E, paint e blue; for i = 1, 2,..., E do { if (w i = 0) { e the lightest edge in E; E E e; if (e connects two large trees in the green forest) then paint e red; else paint e green; else { // w i = 1 e the heaviest edge in E; E E e; if (e connects two large trees in the blue-green forest) then paint e red; else paint e green; return the green forest; As shorthand, we sometimes refer to the algorithm picked out by string w as the w heuristic. Moreover, we let G(w) and B(w) denote the green and blue forest, respectively, produced by running deterministic-m-forest on string w, where the ordered edge set E is assumed and w n. The heuristic 1 n, which selects the heaviest edge in every iteration, is called the heaviest edge first (HEF) heuristic. Similarly, the heuristic 0 n, which selects the lightest edge in every iteration, is called the lightest edge first (LEF) heuristic. It is shown in [4] that, on any given input, the green forest produced by HEF is a subgraph of that produced by LEF. Our key theorem generalizes this result by characterizing the solutions produced by the class of all m-forest algorithms. Specifically, given two strings w = α1β and ŵ = α0β that differ in only one bit, the solution produced by the w heuristic is a subgraph of that produced by the ŵ heuristic: Flipped bit theorem. Let w=α1β and ŵ=α0β where w = ŵ = E. Then G(w) G(ŵ). We prove this theorem in the appendix. Intuitively, when we delay processing a light edge, we increase the likelihood of excluding it and other edges from the m-forest under construction. In the precondition of the flipped bit theorem, the light edge processed by the ŵ heuristic through prefix α0 is not processed by the w heuristic until later, increasing the likelihood that this and other edges are omitted from the green forest G(w). The lattice structure. The 2 n binary strings of length n label the vertices of an n- dimensional hypercube in which two vertices are joined by an edge if and only if their

5 10 M. Laszlo, S. Mukherjee / Discrete Applied Mathematics 154 (2006) 6 14 labels differ in exactly one bit. This hypercube forms a complete lattice under the bitwise Boolean operations and ( ) and or ( ): the greatest lower bound, or meet, of two strings w and ŵ is given by w ŵ, and least upper bound, or join, byw ŵ. The lattice s least element is the string 0 n and its greatest element the string 1 n. When considered under this lattice interpretation, the flipped bit theorem and transitivity of the subset relation imply the following corollary: Corollary 1. Let G and Ĝ be the green forests produced by any two heuristics w and ŵ, respectively, when applied to some edge set. The green forest produced by the w ŵ heuristic contains both G and Ĝ as subgraphs. The green forest produced by the w ŵ heuristic is contained in both G and Ĝ as a subgraph. Since 0 n and 1 n are the lattice s least and greatest elements, respectively, the green forest produced by the LEF heuristic 0 n contains both G and Ĝ as subgraphs, and the green forest produced by the HEF heuristic 1 n is contained in both G and Ĝ as a subgraph. The following result holds: Corollary 2. The solution produced by HEF is a subgraph of the solution produced by the w heuristic for every w. The solution produced by LEF is a supergraph of the solution produced by the w heuristic for every w. 4. Heuristics for the constrained forest problem The constrained forest problem is defined as follows: Let G = (V, E) be an undirected connected graph with positive edge weights. Given a natural number m V, we seek a minimum-weight subgraph spanning the vertex set V such that each of its components spans at least m vertices. Each w heuristic can be used as a heuristic for CFP: Construct the minimum spanning tree (MST) of G and then apply the w heuristic to the MST s edge set ordered by increasing weight. Corollary 2 implies that, of the class of heuristics considered in this paper, the heaviest edge first 1 n heuristic produces the best solutions for CFP. Given an MST with n edges, one can trace a path from 0 n to 1 n through the lattice of heuristics and generate the sequence of green forests G 0, G 1,...,G n produced by the heuristics along this path. (The forest G 0 results from the 0 n heuristic. The heuristic used to produce G i+1 is obtained by setting one of the 0 bits to 1 in the string used to produce G i.) To examine this sequence of forests, we take the input edge set from a tree with positive edge weights and represent each forest G i by its total weight. Since G i+1 G i for each i, the total weight is monotonically nonincreasing. Fig. 1 graphs total weight for three different paths through the lattice from 0 n to 1 n : where 1 s replace 0 s from left to right ( , , ,...), labeled Advancing in the figure; where 1 s replace 0 s from right to left ( , , ,...), labeled Receding; and for a random path from 0 n to 1 n. The edge set used to create these graphs forms the minimum spanning tree of the Creta data set [2] and contains 1080 edges. In Fig. 1, the graphs resulting from the Advancing and Receding paths bound the graph resulting from the random path. This illustrates a more general result that follows from

6 M. Laszlo, S. Mukherjee / Discrete Applied Mathematics 154 (2006) Advancing Total Weight Receding Random Number of 1s Fig. 1. Total weight produced by three different paths in the heuristic lattice (m = 4). Lemma 3 (see the appendix). For any 0 k n, let G A and G R be the graphs produced by the 1 k 0 n k heuristic and 0 n k 1 k heuristic, respectively, and let G be the green graph produced by any string containing exactly k 1 s. Then G A G G R. 5. Conclusion Our work has presented a class of greedy methods for constructing m-forests, and characterized the structure of this class. These methods are applied to the constrained forest problem, and the 1 n heuristic (HEF) is identified as the best of this class of heuristics with approximation ratio two. Appendix: Proof of the flipped bit theorem In this appendix we prove the main theorem cited in this paper: Flipped bit theorem. Let w=α1β and ŵ=α0β where w = ŵ = E. Then G(w) G(ŵ). In what follows, we assume that all strings are over the alphabet {0,1 and have length no greater than E. Given nonempty string w, the last edge processed by running deterministicm-forest on string w is denoted e = edge(w). Where w = w b, edge e is processed as a light edge if bit b = 0, or as a heavy edge if b = 1.

7 12 M. Laszlo, S. Mukherjee / Discrete Applied Mathematics 154 (2006) 6 14 Assume G(w), B(w), and R(w) are the green, blue, and red forests, respectively, produced by (running m-forest on) string w. We define the blue-green forest as BG(w) = B(w) G(w). We refer to graph components as trees, and sometimes refer to a green tree to emphasize that it belongs to G(w) for some w, and to blue and blue-green trees similarly. A tree is said to be large if it contains at least m vertices, and small otherwise. Lemma 1. Let w = α01 k and ŵ = α1 k for any k 0. Let e denote the edge processed by the zero that follows the prefix α in w; that is, e=edge(α0). If w paints e red, then G(w)=G(ŵ). Proof of Lemma 1. The proof is by induction on k. [Basis] Assume k = 0. Then G(w) = G(α0) = G(α) = G(ŵ), where the second equality follows from the fact that e is not added to the green forest G(α). [Inductive step] Assume as the inductive hypothesis that the lemma holds for all i<k. Let f denote the heavy edge that is processed last; that is, f = edge(w) = edge(ŵ).bythe inductive hypothesis (IH), assume that the corresponding green forests are identical when f is about to be processed; in other words, G(α01 k 1 ) = G(α1 k 1 ). There are two cases to consider: [case 1] ŵ paints f green: By the IH and the fact that w processes the same edges as ŵ, except for edge e which w paints red, we have BG(α1 k 1 ) = BG(α01 k 1 ) e. Hence the two blue-green trees that f touches in BG(α01 k 1 ) are no larger than their counterparts in BG(α1 k 1 ), implying that w paints f green. [case 2] ŵ paints f red: Since the light edge e is painted red by w, edge e touches two large green trees G 1 and G 2 in G(α). Both trees are subgraphs of the green forest G(α01 k 1 ) and hence of the blue-green forest BG(α01 k 1 ). Because BG(α1 k 1 ) = BG(α01 k 1 ) e, edge e separates two blue-green trees in BG(w), but the two blue-green forests BG(w) and BG(ŵ) are otherwise identical. Let T 1 and T 2 be the two blue-green trees that f touches in BG(α01 k 1 ) when w is about to process edge f, and consider T 1.IfT 1 contains one of the green trees G 1 or G 2 as a subgraph, then T 1 is large since both G 1 and G 2 are large. Alternatively, if T 1 contains neither G 1 nor G 2 as a subgraph, then T 1 is one of the two large blue-green trees due to which ŵ paints f red. In both cases T 1 is large. The argument for T 2 is the same, implying that f touches two large trees in the blue-green forest BG(α01 k 1 ), implying that w paints f red. Lemma 2. Let w = ασ and ŵ = βσ where σ = β. Suppose that G(β) = G(α) e and R(β) = R(α) e for some light edge e painted red by w. Then G(ŵ) = G(w) e. Proof of Lemma 2. The proof is induction on σ. [Basis] Assume σ = 0. Then G(ŵ) = G(β) = G(α) e = G(w) e. [Inductive step] Assume as the inductive hypothesis (IH) that the lemma holds for all proper prefixes of σ, in particular that G(βσ ) = G(ασ ) e where σ = σ b for bit b. Since w paints edge e red, e touches two large green trees in the green forest G(ασ ). Let f =edge(w). There are two cases to show that f G(w) implies f G(ŵ): [case 1] f is a light edge (b = 0).Ifw paints f green, f touches at least one small green tree T in G(ασ ). Since T cannot be one of the green trees that e touches (for otherwise T would be large), IH implies that f touches the same tree T in G(βσ ). Hence ŵ paints f green.

8 M. Laszlo, S. Mukherjee / Discrete Applied Mathematics 154 (2006) [case 2] f is a heavy edge (b=1).ifwpaints f green, f touches at least one small blue-green tree T in BG(ασ ). Since T cannot contain either green tree that e touches, IH implies that f touches the same tree T in BG(βσ ), hence ŵ paints f green. It follows from both cases that f G(w) implies f G(ŵ). The other direction, that f G(ŵ) implies f G(w), follows from the fact that G(ασ ) G(βσ ), which obtains from IH. Hence f G(w) if and only if f G(ŵ), and the lemma follows. Lemma 3. Let w = α1 j 0β and let ŵ = α01 j β. Let the edges processed by these two strings be labeled as follows: e f 1 f 2 f 3... f k 1 f k ŵ : α β w : α β f 1 f 2 f 3 f 4... f k e Then the following conditions hold: (a) if e G(w), then e G(ŵ); and (b) each f i is painted the same color by w and ŵ; and (c) either G(ŵ) = G(w) or G(ŵ) = G(w) e. Lemma 3 expresses the idea that by postponing the processing of a light edge e, we obtain a green graph G(w) that is either the same as the green graph G(ŵ) we would otherwise obtain, or that excludes edge e which G(ŵ) includes. Proof of Lemma 3. (a) We show that if e is painted red by ŵ, then e is painted red by w. Assume e is painted red by ŵ. Then e touches two large green trees in graph G(α). Processing of the heavy edges f 1 through f k can only add more green edges to G(α), hence G(α) G(α1 k ). It follows that e touches two large green trees in G(α1 k ) and is painted red by w. (b) There are two cases. First, assume ŵ paints edge e green. Then e remains in the blue-green forest implying that w and ŵ paint each edge f i the same color. Second, assume ŵ paints edge e red. By Lemma 1, w and ŵ paint each edge f i the same color. (c) Part (a) of this lemma implies that one of three cases hold: e belongs to both G(w) and G(ŵ); e belongs to neither G(w) and G(ŵ);ore belongs to G(ŵ) G(w). In the first two cases, part (b) of this lemma implies that G(w) = G(ŵ). In the last case, Lemma 2 implies that G(w) = G(ŵ) or G(ŵ) = G(w) e. Lemma 4. Let w = α11 k and ŵ = α01 k where w = ŵ = E, and let e = edge(α0). Then either G(ŵ) = G(w) or G(ŵ) = G(w) e. Proof of Lemma 4. We can rewrite w as w = α11 k 1 1 = α11 k 1 0 since the last bit corresponds to the last remaining blue edge in G, which can be regarded as light or heavy. We can also rewrite ŵ as ŵ = α01 k 1 1. The lemma follows from Lemma 3. Flipped bit theorem. Let w=α1β and ŵ=α0β where w = ŵ = E. Then G(w) G(ŵ).

9 14 M. Laszlo, S. Mukherjee / Discrete Applied Mathematics 154 (2006) 6 14 Proof of the flipped bit theorem. Proof by induction on the number of zeros in string β. [Basis] Where β = 1 k for some k, the basis is established by Lemma 4. [Inductive step] Suppose β = 1 k 0σ, sow = α11 k 0σ = α1 k+1 0σ and ŵ = α01 k 0σ. Let string w = α01 k 1σ. By Lemma 3 we have G(w ) = G(w) or G(w ) = G(w) e, hence G(w) G(w ). By the induction hypothesis we have G(w ) G(ŵ). It follows that G(w) G(ŵ) by transitivity of the subset relation. References [1] R. Cordone, F. Maffioli, On the complexity of graph tree partition problems, Discrete Appl. Math. 134 (2004) [2] R.A. Dandekar, J. Domingo-Ferrer, F. Sebe, LHS-based hybrid microdata vs rank swapping and microaggregation for numeric microdata protection, in: J. Domingo-Ferrer (Ed.), Inference Control in Statistical Databases, Lecture Notes in Computer Science, vol. 2316, Springer, Berlin, [3] C. Imelieńska, B. Kalantari, L. Khachiyan, A greedy heuristic for a minimum weight forest problem, Oper. Res. Lett. 14 (1993) [4] M. Laszlo, S. Mukherjee, Another greedy heuristic for the constrained forest problem, Oper. Res. Lett. 33(6) (2005), in press, /j.orl