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<ul><li><p>ers 98</p><p>omtefa, RWT</p><p>d in r</p><p>ne 9 F</p><p>F. Me</p><p>ctrice methis problem can be solved in linear time for graphs of bounded treewidth and establish bounds on its parameterized complexity ifthe number of PMUs is the parameter. 2006 Elsevier B.V. All rights reserved.</p><p>Keywords: Graph algorithms; Computational complexity</p><p>1. Introduction</p><p>Electrical networks can be interpreted as undirectedgraphs in a very natural fashion. Each node representsa measuring point and edges denote electrical connec-tions between them. In this paper, we discuss a graphproblem known as POWER DOMINATING SET whichhas been introduced by Haynes et al. [9]. In order toobserve the voltage and phase angle at some measur-ing point, we may place a Phase Measurement Unit(PMU) on it, and by a law resembling Kirchhoffs cur-rent law, we can observe measuring points indirectly aswell. More precisely, we have the following rules thattell us how nodes and edges can be measured:</p><p> Supported by the DFG under grant RO 927/6-1 (TAPI).* Corresponding author.</p><p>E-mail address: rossmani@informatik.rwth-aachen.de(P. Rossmanith).</p><p>(1) A PMU on a node v observes v, all incident edges,and all adjacent nodes.</p><p>(2) If a vertex v of degree d 2 is incident to d 1observed edges, all edges of v are observed.</p><p>(3) If an edge e = {v,w} is observed, v and w are ob-served as well.</p><p>(4) If two adjacent nodes v and w are observed, theedge {v,w} is observed as well.</p><p>The POWER DOMINATING SET problem is definedby the following question:</p><p>Given an undirected graph G = (V ,E) and a non-negative integer k, is there a subset M V with|M| k such that all nodes from V would be observedaccording to the four rules, if we placed a PMU on everyv M?POWER DOMINATING SET is similar to the well-knownproblem DOMINATING SET. In this, we look for a domi-nating set D of k nodes, such that every node either is anInformation Processing Lett</p><p>Parameterized power dJoachim Kneis, Daniel Mlle, S</p><p>Department of Computer ScienceReceived 10 April 2005; receive</p><p>Available onli</p><p>Communicated by</p><p>Abstract</p><p>The optimization problem of measuring all nodes in an elepossible is known as POWER DOMINATING SET. Nodes can b0020-0190/$ see front matter 2006 Elsevier B.V. All rights reserved.doi:10.1016/j.ipl.2006.01.007(2006) 145149www.elsevier.com/locate/ipl</p><p>ination complexity </p><p>n Richter, Peter Rossmanith </p><p>H Aachen University, Germany</p><p>evised form 25 November 2005ebruary 2006</p><p>yer auf der Heide</p><p>al network by placing as few measurement units (PMUs) asasured indirectly according to Kirchhoffs law. We show that</p></li><li><p>ocessi146 J. Kneis et al. / Information Pr</p><p>element of D or has a neighbor in D. Dominating set isitself a classical NP-complete problem [7]. What makesPOWER DOMINATING SET seemingly harder to solveis its non-local structure: For dominating set the place-ment of a dominating node cannot influence other partsin the graph that are far away, while this is perfectly pos-sible for POWER DOMINATING SET. A path of arbitrarylength, for example, can be measured by a single PMUat one of its ends.</p><p>It is well known that DOMINATING SET can besolved efficiently for graphs of bounded treewidth [11].The notion of treewidth was introduced by Robertsonand Seymour [10] and measures how treelike a graphis. Graphs of treewidth k are also called partial k-trees.introduction to this concept. Many graph problems thatare hard in general can be efficiently solved, i.e., in poly-nomial and often linear time, for graphs of boundedtreewidth, e.g., HAMILTONIAN PATH, MAX-CUT, IN-DEPENDENT SET and VERTEX COVER [11]. A notableexception is bandwidth minimization [6].</p><p>Because of its non-local structure, standard tech-niques do not easily lead to efficient algorithms forPOWER DOMINATING SET even for graphs of boundedtreewidth. The first polynomial algorithm for POWERDOMINATING SET works only on trees, i.e., on graphswith treewidth one [9]. Haynes et al. left the generaliza-tion to bounded treewidth as an open question. A smallstep in this direction is a new algorithm by Guo et al. [8]that solves POWER DOMINATING SET in polynomialtime for trees with a constant number of additionaledges. This class of graphs is a subset of all graphs withbounded treewidth, but does not even contain all graphswith treewidth two. E.g., a 2n-grid has treewidth two,but is a tree with n 1 additional edges.</p><p>In what follows, we achieve the general result thatPOWER DOMINATING SET can be solved in linear timefor any fixed treewidth k, thereby solving the open prob-lem posed in the article that incited research in thisarea [9].</p><p>Parameterized complexity</p><p>In parameterized complexity every input instance hasan associated natural number, called the parameter. Of-ten, the parameter is part of the input, otherwise itis ausually simplefunction of the input. The timecomplexity of a parameterized problem is measured asa function in both the input length n and the parame-ter k. We say that a problem is fixed parameter tractableor in the complexity class FPT if there is an algorithmthat can solve the problem in f (k)nc steps, where c is</p><p>a constant and f is an arbitrary function. The idea be-ng Letters 98 (2006) 145149</p><p>hind parameterized complexity theory is that hard prob-lems can be easy in practice if hard instances do notoccur. The parameter measures how hard an instance isand the complexity explodes only in terms of the pa-rameter. See the influential monograph by Downey andFellows [4] for an introduction to parameterized com-plexity.</p><p>We can parameterize POWER DOMINATING SET inseveral reasonable ways. Two natural parameters arethe treewidth of the graph and the number of neededPMUs:</p><p>Input: A graph G and a number kParameter: tw(G) (the treewidth of G)Question: Is there a power dominating set of</p><p>size k for G?Input: A graph G and a number kParameter: k (the number of PMUs)Question: Is there a power dominating set of</p><p>size k for G?Let t be the treewidth of a graph. If we had two al-</p><p>gorithms that solve POWER DOMINATING SET in ntresp. t tn3 steps, both would imply that the problem canbe solved in polynomial time for graphs of boundedtreewidth, i.e., all graphs whose treewidth is boundedby some constant. The second, but not the first algo-rithm would also imply that POWER DOMINATING SETis fixed parameter tractable. Please note that the secondalgorithm is still feasible if t has a moderate value, butthe first becomes impractical even for small values of t .Hence, parameterized complexity makes a better state-ment than simply polynomial for fixed treewidth.</p><p>In this sense Guo, Hffner and Niedermeier showedthat POWER DOMINATING SET is fixed parametertractable if the number of edges added to a tree is theparameter.</p><p>In conventional complexity there is the class P of ef-ficiently solvable problems, as well as NP, whose com-plete problems are unlikely to be in P. In parameterizedcomplexity, these are paralleled by FPT as opposed toW[1], W[2], . . . and W[P] whose complete problems areunlikely to be fixed parameter tractable. It is known thatDOMINATING SET is W[2]-complete if the size of thedominating set is the parameter [5]. We show in thispaper that POWER DOMINATING SET is in W[P] andis W[2]-hard if the number of PMUs is the parameter.Unfortunately, we cannot establish the complexity ex-actly, but there are several other problems with W[2] asa lower and W[P] as an upper bound. One example is</p><p>MONOCHROME CYCLE COVER.</p></li><li><p>ocess</p><p>MUsdes o</p><p>Thus, by applying the third rule immediately each MSO formula over the vocabulary . LinEMSOL( )</p><p>time we apply the second one, we contract these tworules into one. It then never happens that an observededge is incident to a node that is not yet observed. Thus,we do not have to consider observed edges any longer,and the modification of the first rule as well as the dele-tion of the fourth rule are clearly uncritical. 3. Bounded treewidth tractability</p><p>In this section, we apply a theorem by Courcelle et al.[3, Theorem 27] to show that Power Dominating Set canbe solved in linear time for graphs of bounded treewidth.</p><p>Proposition 1. (Courcelle et al.) Let p and k be fixed</p><p>optimization problems are a bit more complicated, andfor the sake of brevity we only describe the very specialand restricted kind of these problems that is relevant inour context. A graph problem is a LinEMSOL( ) opti-mization problem of this kind if it can be expressed asfollows:</p><p>The input consists of a graph given as a -structureG() = V,E and an evaluation function f :V N.There is a fixed MSO( ) formula that contains a freeset variable X whose universe is V . The problem is tofind a valuation of X that fulfills G(),X |= (X) atminimal cost with respect to f , that is,{ }J. Kneis et al. / Information Pr</p><p>Fig. 1. A graph that is power-dominated by three PMUs. Nodes with Pa circle. All these vertices are observed according to the first rule. Nopoints at the induction source.</p><p>2. A simplified set of rules</p><p>To begin with, we found a way to simplify the prob-lem description by using a smaller set of rules equiva-lent to the four original rules mentioned above.</p><p>Lemma 1. The following set of rules is equivalent to theoriginal set of rules with respect to observed nodes:</p><p>(1) A PMU on a node v observes v and all its neigh-bors.</p><p>(2) If an observed vertex v of degree d 2 is adjacentto d 1 observed nodes, all its neighbors are ob-served.</p><p>Proof. Let us return to the original set of rules. Thereis no rule that creates an observed edge both of whosenodes are not observed. Only the second rule is capa-ble of creating an observed edge exactly one of whosenodes is not observed. By the third rule, however, thisnode becomes observed immediately.integers. Every LinEMSOL(1,p) optimization probleming Letters 98 (2006) 145149 147</p><p>are marked by a square, and neighbors of such nodes are marked bybserved due to the Kirchhoff rule are marked by a comet whose tail</p><p>on the class of partial k-trees can be solved in O(|V |)time and the corresponding algorithm can be derivedconstructively from its LinEMSOL(1,p) definition.</p><p>LinEMSOL(1,p) denotes monadic second orderlogic (MSO) with linear evaluation functions. To be-gin with, recall that monadic second order logic is anextension of first order logic that allows us to quantifyover unary predicate variables (that is, subsets of theuniverse). The vocabulary 1 consists of a single binaryrelation E over V V that encodes the edges of a givengraph. Clearly, every graph G can be expressed as a 1-structure V,E.</p><p>For any fixed integer p, the vocabulary 1,p is anextension of 1 that introduces p additional unary pred-icates U1, . . . ,Up to the language (that is, 1 can beinterpreted as 1,0). These extended vocabularies, how-ever, are not going to be required in this section at all,hence we omit further details.</p><p>For a vocabulary , we say that a graph problem is anMSO( ) decision problem if it can be stated as a closedarg minXV xX</p><p>f (x) | G(),X = (X) .</p></li><li><p>ocessi148 J. Kneis et al. / Information Pr</p><p>In the general case, LinEMSOL optimization prob-lems can be a lot more complex, even for the vocabulary1 alone: there can be an arbitrary number of free setvariables Xi , an arbitrary number of evaluation func-tions fj , and fixed integer weights as coefficients foreach fj (Xi).</p><p>Lemma 2. POWER DOMINATING SET can be ex-pressed as an LinEMSOL(1) optimization problem asfollows:arg minXV</p><p>|X|:VM V. VM = observed_nodes(X) VM = V,</p><p>where</p><p>observed_nodes(X) {v V | near_PMU(v) kirchhoff (v)},</p><p>near_PMU(v) v X u X.adj(v,u),kirchhoff (v) w VM.</p><p>(adj(v,w) w V.(</p><p>adj(w,w) w = v w VM))</p><p>.</p><p>Proof. It is easily verified that near_PMU(v) holds ofa node v exactly if it is directly measured by a node inX via the first rule. As well, kirchhoff (v) if and onlyif v is observed using the second rule. Note that VM isconstructed as a fixed point in a quasi iterative manner:If a node w and all of his neighbors save one have beenmeasured (and thus are part of VM ), the last remainingneighbor is also observed, by the Kirchhoff rule.</p><p>Thus, we have that under the premise VM = observed_nodes(X), exactly the measured nodes are containedin VM . That way, the conclusion VM = V is satisfiedif and only if the whole graph G is power-dominatedby X. We achieve the smallest such X by the min opti-mization. Theorem 1. POWER DOMINATING SET can be solvedin linear time for graphs of bounded tree width.</p><p>Notice that Proposition 1 allows for defining theinteger-weighted version of the problem as well, whereplacing a PMU on a node v has cost f (v). By omittingf in the above lemma, we implicitly assigned a weightof 1 to each node.</p><p>4. Hardness for W[2]</p><p>The most natural parameter for the POWER DOM-INATING SET problem may not be the treewidth, but</p><p>the number of PMUs required to power-dominate theng Letters 98 (2006) 145149</p><p>Fig. 2. A dominated graph and its corresponding power-dominatedantenna graph. In the first graph, diamonds and circles denote dom-inators and dominated nodes, respectively. The nodes in the secondgraph are marked according to the notation used in Fig. 1.</p><p>input graph. In this section, we show that POWER DOM-INATING SET is W[2]-hard for this parameterization.It is already known that POWER DOMINATING SET isNP-complete, but the reduction from 3-SAT given byHaynes et al. [9] cannot be used to show W[2]-hardness.Therefore, we take a different approach.</p><p>Theorem 2. POWER DOMINATING SET is W[2]-hardwhen we choose the number of PMUs as the parameter.</p><p>Proof. By reduction from DOMINATING SET, wherethe number of dominators is the parameter. This prob-lem is W[2]-complete [5].</p><p>Given an input graph G = (V ,E) for DOMINATINGSET, construct the graph G = (V ,E) by copying Gand adding a single edge with a new node v to eachnode v V . Let us call such an edge {v, v} the an-tenna of v. We will now prove the following equalityof the two domination numbers: G can be dominated byd nodes if and only if G can be power-dominated by dnodes.</p><p>() Let D V dominate G such that |D| is mini-mal. Clearly, D also dominates all nodes in G exceptfor, maybe, nodes v on antennae {v, v} for some nodesv V . As all neighbors of any such node v, except forv, are dominated by D, the Kirchhoff rule applies, andwe have that D power-dominates the entire graph G.</p><p>() Let D V power-dominate G such that |D|is minimal. If D uses the node v of the antenna {v, v}for some v V , we may replace D by D ({v}) {v}without losing the power domination property, as v isthe only neighbor of v. Since this operation does notchange the size of D, we may assume that D Vwithout loss of generality. It remains to show that theKirchhoff rule never applies to any node from V , whichmeans that D is also a dominating set for G[V ] andthus for G.</p><p>Mark all nodes in D and all their neighbors, and as-sume there exists a node v V that remains unmarked,</p><p>which means that v is not dominated by D. The Kirch-</p></li><li><p>J. Kneis et al. / Information Processing Letters 98 (2006) 145149 149</p><p>hoff rule can only be applied if v has a neighbor w Vall of whose neighbors but v are marked. As there is anantenna {w,w}, w has a neighbor w / V that cannotbe ma...</p></li></ul>