A Approximation Algorithm for a Generalization of the Weighted EdgeDominating Set Problem


 Alice Shaw
 1 years ago
 Views:
Transcription
1 Journal of Combinatorial Optimization, 5, , 2001 c 2001 Kluwer Academic Publishers. Manufactured in The Netherlands. A2 1 Approximation Algorithm for a Generalization of the Weighted EdgeDominating Set Problem ROBERT CARR Sandia National Laboratory, P.O. Box 5800, Albuquerque, NM 87185, USA TOSHIHIRO FUJITO Department of Electronics, Nagoya University Furo, Chikusa, Nagoya, , Japan GORAN KONJEVOD OJAS PAREKH Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh PA, , USA Received August 14, 2000; Revised August 14, 2000; Accepted August 30, 2000 Abstract. We study the approximability of the weighted edgedominating set problem. Although even the unweighted case is NPComplete, in this case a solution of size at most twice the minimum can be efficiently computed due to its close relationship with minimum maximal matching; however, in the weighted case such a nice relationship is not known to exist. In this paper, after showing that weighted edge domination is as hard to approximate as the well studied weighted vertex cover problem, we consider a natural strategy, reducing edgedominating set to edge cover. Our main result is a simple 2 1 approximation algorithm for the weighted edgedominating set problem, improving the existing ratio, due to a simple reduction to weighted vertex cover, of 2r WVC, where r WVC is the approximation guarantee of any polynomialtime weighted vertex cover algorithm. log log V The best value of r WVC currently stands at 2 2 log V. Furthermore we establish that the factor of 2 1 is tight in the sense that it coincides with the integrality gap incurred by a natural linear programming relaxation of the problem. Keywords: approximation algorithm, edgedominating set, vertex cover, edge cover 1. Introduction In an undirected graph G = (V, E), E is a set of edges, {u,v}, where u,v belong to the set of vertices, V. An edge e dominates all f E such that e f. A set of edges is an edgedominating set (eds) if its members collectively dominate all the edges in E. The edgedominating set problem (EDS) is then that of finding a minimumcardinality edgedominating set, or if edges are weighted by a function w : E Q +, an edgedominating set of minimum total weight. Work supported in part by the United States Department of Energy under Contract DEAC0494AL Supported in part by an NSF CAREER Grant CCR
2 318 CARR ET AL Notation A vertex v dominates u V if {u,v} E. A vertex v also covers all edges incident upon v,or more formally, v covers an edge e if v e. We overload terminology once again and say that an edge e covers a vertex v if v e. We denote the set of edges that v covers by δ(v). When we wish to discuss the vertices of an edge set S E,wedefine V (S) = e S e.amatching is a set of edges M, such that distinct edges e, f in M do not intersect. A maximal matching is one which is not properly contained in any other matching. For S V, we denote the set {e E e S = 1} by δ(s), and we denote the set {e E e S = 2} by E(S). When given a subset S E and a vector x Q E whose components correspond to the edges in E,we use x(s), as a shorthand for e S x e. Analogously in the case of a function w v : V Q or w : E Q we write w v (S) = u S wv (u) or w(s) = e S w(e), where S V or S E, respectively Related problems Yannakakis and Gavril showed that EDS and the minimum maximal matching problem, whose connection to EDS will be presented later, are NPcomplete even on graphs which are planar or bipartite of maximum degree 3 (Yannakakis and Gavril, 1980). This result was later extended by Horton and Kilakos to planar bipartite, line, total, perfect clawfree, and planar cubic graphs (Horton and Kilakos, 1993). On the other hand polynomially solvable special cases have been discovered. Chronologically by discovery, efficient exact algorithms for trees (Mitchell and Hedetniemi, 1977), clawfree chordal graphs, locally connected clawfree graphs, the line graphs of total graphs, the line graphs of chordal graphs (Horton and Kilakos, 1993), bipartite permutation graphs, cotriangulated graphs (Srinivasan et al., 1995), and other classes are known. Although EDS has important applications in areas such as telephone switching networks, very little is known about the weighted version of the problem. In fact, all the polynomialtime solvable cases listed above apply only to the cardinality case, although we should note that PTAS s are known for weighted planar (Baker, 1994) and λprecision unit disk graphs (Hunt et al., 1994). In particular, while it is a simple matter to compute an edgedominating set of size at most twice the minimum, as any maximal matching will do, such a simple reduction easily fails when arbitrary weights are assigned to edges. In fact the only known approximability result, which follows from a simple reduction to vertex cover, does not seem to have appeared in the literature. The edgedominating set problem, especially the weighted version, seems to be the least studied among the other basic specializations of the set cover problem for graphs. The others are called the (weighted) edge cover (EC ), vertex cover (VC ), and (vertex) dominating set problems in which we seek to obtain a minimumcardinality (weight) set which covers vertices by edges, edges by vertices, and vertices by vertices respectively. Of these only the weighted edge cover problem is known to be solvable in polynomial time (Edmonds and Johnson, 1970; Murty and Perin, 1982; Pulleyblank, 1995). Better known and studied is the dominating set problem. EDS for G is equivalent to the vertexdominating set problem for the line graph of G. The dominating set problem for general graphs is, unfortunately,
3 WEIGHTED EDGEDOMINATING SET PROBLEM 319 equivalent to the set cover problem under an approximation preserving reduction. Although the polynomialtime approximability of set cover is well established and stands at a factor of ln V +1 (Chvátal, 1979; Johnson, 1974; Lovász, 1975), it cannot be efficiently approximated better than ln V unless NP DTIME( V O(log log V ) ) (Feige, 1996). The vertex cover problem seems to be the best studied of the bunch and boasts a vast literature. Most known facts and relevant references can be found in the survey by Hochbaum (1997). The log log V 2 log V best known approximation ratio is 2, and it has been conjectured (see the above survey) that 2 is the best constant approximation factor possible in polynomial time. In this paper we consider a natural strategy of reducing weighted EDS to the related weighted edge cover problem and establish the approximability of EDS within a factor of 2 1. We also obtain the same ratio for the extension in which only a subset of the edges need be dominated. Furthermore the factor of 2 1 is tight in the sense that it coincides with the integrality gap incurred by a natural linear programming relaxation of EDS. 2. Approximation hardness Yannakakis and Gavril proved the NPhardness of EDS by reducing VC to it (Yannakakis and Gavril, 1980). Although their reduction can be made to preserve approximation quality within some constant factor and thus imply the MAX SNPhardness of (unweighted) EDS and the nonexistence of a polynomialtime approximation scheme (unless P = NP) (Arora et al., 1992; Papadimitriou and Yannakakis, 1991), it does not preclude the possibility of better approximation of EDS than that of VC. On the other hand, it is quite straightforward to see that the approximation of weighted EDS is as hard as that of weighted VC. Theorem 1. Weighted VC can be approximated as well as weighted EDS. Proof: Let G = (V, E) be an instance graph for VC with weight function w v : V Q +. Let s be a new vertex not in V, and construct a new graph G = (V {s}, E E ) by attaching s to each vertex of G, that is, E ={{s, u} u V }. Assign a weight function w : E Q + to the edges of G by defining w (e) = w v (u) if e ={s, u} E, and w (e) = w v (u) + w v (v) if e ={u, v} E. By the definition of w, if an edgedominating set D for G contains {u, v} E, it can be replaced by the two edges {u, s}, {v, s} E without increasing the weight of D, so we may assume D E. In this case, however, there exists a onetoone correspondence between vertex covers in G and edgedominating sets in G, namely C def = V (D)\{s} in G and D in G, such that w v (C) = w (D). 3. Previous work 3.1. Cardinality EDS: Reduction to maximal matching Obtaining a 2approximation for the minimumcardinality edgedominating set is easy; the following proposition also demonstrates the equivalence of cardinality EDS and minimummaximal matching.
4 320 CARR ET AL. Proposition 1 (Harary (1969)). which is a maximal matching. There exists a minimumcardinality edgedominating set Proof: For a set E of edges, let adj(e ) denote the number of (unordered) pairs of adjacent edges in E, that is adj(e ) = 1 2 {(e, f ) e, f E and e f }. Let D E be a minimumcardinality edgedominating set. Suppose D is not a matching and let e, f D be two adjacent edges, i.e. e f. Since D is minimal, D\ f is not an edgedominating set. Therefore there exists an edge g E adjacent to f, but not to any other member of D. Now the set D = D\{ f } {g} is another minimumcardinality edgedominating set and adj(d )<adj(d). By repeating this exchange procedure on D,we eventually find a minimum edgedominating set D which is a (maximal) matching. Proposition 2. Every maximal matching M gives a 2approximation for the edgedominating set problem. Proof: Let M 1 and M 2 be maximal matchings. The symmetric difference M 1 M 2 consists of disjoint paths and cycles in which edges alternate between those from M 1 and those from M 2. This implies an equal number of edges from M 1 and M 2 in every cycle. By the maximality of M 1 and M 2, every path must contain an edge from each of M 1 and M 2, hence every path contains at most twice as many edges from one as from the other. Letting k i = M i (M 1 M 2 ) for i = 1, 2, we now have k 1 2k 2 and k 2 2k 1. Since M i = M 1 M 2 +k i, it follows that M 1 2 M 2 and M 2 2 M Weighted EDS: Reduction to vertex cover Weighted EDS may be reformulated as finding a set of edges D of minimum weight such that V (D) is a vertex cover of G. This idea leads to a well known 2r WVC approximation algorithm, where r WVC is the approximation guarantee of any polynomialtime weighted vertex cover algorithm. Theorem 2 (Folklore). The weighted edgedominating set problem can be approximated to within a factor of 2r WVC. Proof: Given an instance of weighted EDS, G with weight function w : E Q +,define a vertexweight function w v : V Q + by setting w v (u) = min e δ(u) {w(e)} for every u V. Let D be a minimumweight EDS with respect to w, and let C be a minimumweight vertex cover with respect to w v. Since V (D ) is a vertex cover for G,
5 WEIGHTED EDGEDOMINATING SET PROBLEM 321 w v (C ) w v (V (D )). By the construction of w v, for each u V (D ) Hence, w v (u) min e δ(u) D {w(e)}. w v (C ) w v (V (D )) = u V (D ) w v (u) 2w(D ). (1) Suppose we use an r WVC approximation algorithm to obtain a vertex cover C such that w v (C) r WVC w v (C ). We can construct an edgedominating set D C from C by selecting a minimumweight edge in δ(u) for each u C. Thus w(d C ) w v (C). Combining this with (1) we have w(d C ) r WVC w v (C ) 2r WVC w(d ), which establishes the theorem. As mentioned earlier, the smallest value of r WVC currently known for general weighted log log V graphs is 2 (Hochbaum, 1997), yielding an EDS approximation ratio of 4 2 log V log log V. Of course, for special classes we can do better. For instance exact polynomialtime algorithms exist for weighted VC on bipartite graphs, yielding a 2approximation for log V weighted EDS on bipartite graphs. 4. A 2 1 approximation: Reduction to edge cover 4.1. Polyhedra Given an instance G = (V, E) and a corresponding cost vector c Q E +, we may formulate the weighted edgedominating set problem as an integer program min c e x e (EDS(G)) subject to: e E x(δ(u)) + x(δ(v)) x uv 1 {u,v} E x e {0, 1} e E. The constraints of (EDS(G)) ensure that each edge is covered by at least one edge. Relaxing the 01 constraints yields min c e x e (FEDS(G)) subject to: e E x(δ(u)) + x(δ(v)) x uv 1 {u,v} E x e 0 e E.
6 322 CARR ET AL. We henceforth assume without loss of generality that G has no isolated vertices, since deleting such vertices does not affect an edgedominating set. In our reduction to edge cover we will also be interested in min c e x e (FEC(G)) subject to: e E x(δ(u)) 1 u V x e 0 e E. It is easy to see that the incidence vector of any edge cover for G satisfies all the constraints in (FEC(G)), hence is feasible for it. However, (FEC) may not have integral optimal solutions in general, to which a unitweighted triangle attests. The optimal solution for (FEC) has x e = 1/2, for all e E, for a total weight of 3/2, while the weight of an integral solution must be at least 2. Thus the inequalities (FEC) are not sufficient to define (EC), the convex hull of the incidence vectors of edge covers. Fortunately, due to a result of Edmonds and Johnson (1970), the complete set of linear inequalities describing (EC) is in fact known. Proposition 3 (Edmonds and Johnson (1970)). The edge cover polytope (EC(G)) can be described by the set of linear inequalities of (FEC(G)) in addition to x(e(s)) + x(δ(s)) S +1 2 S V, S odd. (2) 4.2. Algorithm Let x be a feasible solution for (FEDS(G)). Since for each {u,v} E, x(δ(u)) + x(δ(v)) 1 + x uv,wehavemax{x(δ(u)), x(δ(v))} 1+x uv 1. We use this criterion to define a 2 2 vertex set V + as follows. For each edge {u,v} E we select the endpoint whose fractional degree achieves max{x(δ(u)), x(δ(v))} to be in V + ; in the case of a tie, we choose one endpoint arbitrarily. We let V = V \V +. Proposition 4. V + is a vertex cover of G. Since an edge cover of a vertex cover is an edgedominating set, we have reduced the problem at hand to that of finding a good edge cover of the set of vertices V +. This is not quite the standard edge cover problem, yet a fair amount is known about it. For instance one can reduce this problem to the maximum weight capacitated bmatching problem (see Grötschel et al., 1988, p. 259). In fact a complete polynomialtime separable linear description of the associated polytope is also known (Pulleyblank, 1995). Rather than trouble ourselves with the technicalities that dealing directly with the V + edge cover problem imposes, we show how to reduce an instance of this problem to a bona fide instance of weighted edge cover. We construct a new instance Ḡ = ( V, Ē) such that there is a onetoone cost preserving correspondence between V + edge covers in G and edge covers of Ḡ. Recall that V + and V partition V.
7 WEIGHTED EDGEDOMINATING SET PROBLEM 323 Let the vertex set V be a copy of V, where v V corresponds to v V. We set V = V V and Ē = E E, where E consists of zerocost edges, one between each v V its copy v V.Nowif D is an edge cover of Ḡ, then D E must be an edge set of equal cost covering all the vertices in V +. Conversely if D + is an edge set covering all the vertices in V +, then D + E is an edge cover of Ḡ of equal cost, since the edges in E cost nothing. We are now in a position to describe the algorithm, which may be stated quite simply as 1. Compute an optimal solution x for (FED(G)). 2. Compute V Compute and output a minimumweight set of edges D covering V +. The algorithm clearly runs in polynomial time as the most expensive step is solving a compact linear program. Note that steps 2 and 3 may be implemented by the transformation above or by any method the reader fancies; however, the true benefit of the transformation may not be fully apparent until we analyze the approximation guarantee of the algorithm Analysis As before suppose we are given an instance graph G = (V, E) with no isolated vertices and a nonnegative cost vector c. Let x be some feasible fractional solution for (FEDS(G)). Along the lines of the algorithm, suppose we have computed V + and the resulting transformed instance, Ḡ = ( V = V V, Ē = E E ). Let x = (x, 1 E ) Q Ē + ; that is, x corresponds to the augmentation of the fractional edgedominating set x by E, a zerocost set of edges. Note that by construction, x is feasible for (FEDS(Ḡ)). Similarly we extend c to c = (c, 0 E ) Q Ē +. Note that we have c x = c x. We may now proceed to show that there is an integral edge cover of Ḡ which does not cost too much more than our fractional edgedominating solution, x. Theorem 3. The point 21 x is feasible for (EC(Ḡ)). Proof: Let ȳ = 2 x. Suppose u is a vertex in V.Ifu V +,wehave x(δ(u)) 1 2 ; otherwise u V V, and we have x e = 1 for all e E, so in either case ȳ(δ(u)) 1, (3) hence ȳ is feasible for (FEC(Ḡ)). Yet this is not quite good enough as (FEC(Ḡ)) does not have integral extreme points in general, so we extend this by showing that increasing ȳ by a 1 20 fraction places it in (EC(Ḡ)). To accomplish this we use the fact that x is a fractional edgedominating set of Ḡ, hence ȳ satisfies ȳ(δ(u)) +ȳ(δ(v)) 2 +ȳ uv. (4) Armed with this and the constraints of (FEC(Ḡ)), we proceed to show that ȳ also satisfies (2) with respect to Ḡ.
8 324 CARR ET AL. Suppose S is a subset of V of odd cardinality; let s = S. When s = 1, the constraints (2) are trivially satisfied by ȳ, so suppose s 3. By combining (3) and (4) we see { 2 +ȳuv if uv Ē, ȳ(δ(u)) +ȳ(δ(v)) 2 otherwise. Summing the appropriate inequality above for each pair {u, v} in S S, where u v, we get (s 1)ȳ(δ(S)) + 2(s 1)ȳ(Ē(S)) = (s 1) ȳ(δ(u)) u S = ȳ(δ(u)) +ȳ(δ(v)) {{u,v} S S u v} s(s 1) +ȳ(ē(s)). Isolating the desired left hand side yields ȳ(δ(s)) +ȳ(ē(s)) Using standard optimization techniques, { } s+1 2 max = 21 s 3,odd 20, s(s 1) 2s 3 which is achieved when s = 5. s(s 1) + (s 2)ȳ(δ(S)) 2s 3 s(s 1), for s 3. 2s 3 The theorem implies that 21 x is a convex combination of integral edge covers of Ḡ, which by the construction of Ḡ are also integral edgedominating sets of Ḡ containing the edges in E. Equivalently the theorem implies that 21 x is a convex combination of integral V + edge covers in G, which since V + is a vertex cover of G, are also integral edgedominating sets of G. Thus there must be an integral edgedominating set D of G of cost at most 21 c x. In particular, when x is an optimal fractional edgedominating set of G and z EDS the cost of an optimal integral solution, we find an integral solution of cost at most 21 c x 21 z EDS. Corollary 1. The point 21 x is feasible for (EDS(G)) when x (FEDS(G)). Corollary 2. The algorithm of Section 4.2 generates a solution of cost at most 2 1 times the optimal. Note that when G is bipartite, Ḡ is bipartite as well. In this case (FEC(G)) forms a totally unimodular constraint set, hence 2 x is feasible for (EC(Ḡ)). Proposition 5. The algorithm of Section 4.2 generates a solution of cost at most 2 times the optimal on a bipartite instance graph. This is in fact asymptotically tight as figure 1 demonstrates.
9 WEIGHTED EDGEDOMINATING SET PROBLEM 325 Figure 1. A fractional extreme point of cost k The algorithm chooses the darkened vertices as V +, yielding a solution of cost 2k, while the optimal integral solution costs k An extension Suppose we are given an instance in which we are asked to find a minimumweight edge set which dominates a specified subset F E. We need only modify our algorithm so that only endpoints of edges in F are considered for V +. The analysis of the previous section remains essentially the same, and our solution will cover the vertices in V + which ensures that the edges in F will be dominated Integrality gap Given that weighted EDS is as hard to approximate as weighted VC and that no polynomialtime algorithm with a constant performance guarantee strictly less than 2 is known for the latter, we might indulge in a respite from developing EDS algorithms if the former were shown to be approximable with a factor of 2. Unfortunately, it turns out that as long as our algorithm analysis is based exclusively on the optimal cost of (FEDS) as a lower bound for that of (EDS), we should relinquish such hope. The formulation (FEDS) introduces an integrality gap, { } minx (EDS(G)) c x max G=(V,E), c Q E min + x (FEDS(G)) c x larger than 2. Corollary 1 bounds it above by 2 1, and it will be shown below that this is in fact a tight bound. Consider the complete graph on 5n vertices, and let G 1,...,G n be n vertex disjoint subgraphs, each isomorphic to K 5. Assign to each edge of G i a weight of 1, and assign to any edge not in any of these subgraphs, some large weight. Let x e = 1/7 ife is an edge of some G i and x e = 0 otherwise. Then it can be verified that x(δ(e)) 1 for all e, hence x is a feasible solution for (FEDS(K 5n )) of cost n. On the other hand, any integral solution 7 must cover all but one vertex in the graph. Prohibited to pick an edge outside of some G i,
10 326 CARR ET AL. an integral solution of small cost would choose 3 edges from each of G i s but one for a total cost of 3n 1. Thus the integrality gap of formulation (FEDS) approaches 2 1. Although this example establishes the integrality gap of the formulation we employ, our algorithm may still perform provably better. The class of graphs depicted in figure 1 preclude it from guaranteeing a bound less than 2, even for the unweighted bipartite case, and we offer our gratitude for a proof that it is indeed a 2approximation. The integrality gap of (FEDS(G)) is at most 2 when G is bipartite (Proposition 5); in fact it may grow arbitrarily close to 2. Let G be a complete bipartite graph K n,n with unit weights. Then, x e = 1 2n 1 for all e E is a feasible solution of cost n 2. Any integral solution must 2n 1 contain k edges since it must cover all of the vertices in at least one vertex class of the bipartition, so the integrality gap must be at least n(2n 1) = 2 1 n 2 n. References S. Arora, C. Lund, R. Motwani, M. Sudan, and M. Szegedy, Proof verification and hardness of approximation problems, in Proceedings of the 33rdAnnual IEEE Symposium on Foundations of Computer Science, 1992, pp B. Baker, Approximation algorithms for NPcomplete problems on planar graphs, J. ACM, vol. 41, pp , V. Chvátal, A greedy heuristic for the setcovering problem, Math. Oper. Res., vol. 4, no. 3, pp , J. Edmonds and E. Johnson, Matching, a well solved class of integer linear programs, in Combinatorial Structures and Their Applications, Gordon & Breach: New York, 1970, pp U. Feige, A threshold of ln n for approximating set cover, in Proceedings of the 28th Annual ACM Symposium on Theory of Computing, May 1996, pp M. Grötschel, L. Lovász, and A. Schrijver, Geometric Algorithms and Combinatorial Optimization, Springer, F. Harary, Graph Theory, AddisonWesley: Reading, MA, D.S. Hochbaum (Ed.), Approximation Algorithms for NPhard Problems, PWS Publishing Company: Boston, MA, J. Horton and K. Kilakos, Minimum edge dominating sets, SIAM J. Discrete Math., vol. 6, no. 3, pp , H. Hunt III, M. Marathe, V. Radhakrishnan, S. Ravi, D. Rosenkrantz, and R. Stearns, A unified approach to approximation schemes for NP and PSPACEhard problems for geometric graphs, in Proc. 2nd Ann. European Symp. on Algorithms, 1994, pp D.S. Johnson, Approximation algorithms for combinatorial problems, J. Comput. System Sci., vol. 9, pp , L. Lovász, On the ratio of optimal integral and fractional covers, Discrete Math., vol. 13, pp , S. Mitchell and S. Hedetniemi, Edge domination in trees, in Proc. 8th Southeastern Conf. on Combinatorics, Graph Theory, and Computing, 1977, pp K.G. Murty and C. Perin, A 1matching blossomtype algorithm for edge covering problems, Networks, vol. 12, pp , C. Papadimitriou and M. Yannakakis, Optimization, approximation and complexity classes, J. Comput. System Sci., vol. 43, pp , W.R. Pulleyblank, Matchings and extensions, in Handbook of Combinatorics, vol. 1, Elsevier, 1995, pp A. Srinivasan, K. Madhukar, P. Nagavamsi, C.P. Rangan, and M.S. Chang, Edge domination on bipartite permutation graphs and cotriangulated graphs, Information Processing Letters, vol. 56, pp , M. Yannakakis and F. Gavril, Edge dominating sets in graphs, SIAM J. Appl. Math., vol. 38, no. 3, pp , 1980.
Nan Kong, Andrew J. Schaefer. Department of Industrial Engineering, Univeristy of Pittsburgh, PA 15261, USA
A Factor 1 2 Approximation Algorithm for TwoStage Stochastic Matching Problems Nan Kong, Andrew J. Schaefer Department of Industrial Engineering, Univeristy of Pittsburgh, PA 15261, USA Abstract We introduce
More informationAnalysis of Approximation Algorithms for kset Cover using FactorRevealing Linear Programs
Analysis of Approximation Algorithms for kset Cover using FactorRevealing Linear Programs Stavros Athanassopoulos, Ioannis Caragiannis, and Christos Kaklamanis Research Academic Computer Technology Institute
More informationOn the kpath cover problem for cacti
On the kpath cover problem for cacti Zemin Jin and Xueliang Li Center for Combinatorics and LPMC Nankai University Tianjin 300071, P.R. China zeminjin@eyou.com, x.li@eyou.com Abstract In this paper we
More informationApproximation Algorithms
Approximation Algorithms or: How I Learned to Stop Worrying and Deal with NPCompleteness Ong Jit Sheng, Jonathan (A0073924B) March, 2012 Overview Key Results (I) General techniques: Greedy algorithms
More informationApproximated Distributed Minimum Vertex Cover Algorithms for Bounded Degree Graphs
Approximated Distributed Minimum Vertex Cover Algorithms for Bounded Degree Graphs Yong Zhang 1.2, Francis Y.L. Chin 2, and HingFung Ting 2 1 College of Mathematics and Computer Science, Hebei University,
More information2.3 Scheduling jobs on identical parallel machines
2.3 Scheduling jobs on identical parallel machines There are jobs to be processed, and there are identical machines (running in parallel) to which each job may be assigned Each job = 1,,, must be processed
More informationARTICLE IN PRESS. European Journal of Operational Research xxx (2004) xxx xxx. Discrete Optimization. Nan Kong, Andrew J.
A factor 1 European Journal of Operational Research xxx (00) xxx xxx Discrete Optimization approximation algorithm for twostage stochastic matching problems Nan Kong, Andrew J. Schaefer * Department of
More informationTopic: Greedy Approximations: Set Cover and Min Makespan Date: 1/30/06
CS880: Approximations Algorithms Scribe: Matt Elder Lecturer: Shuchi Chawla Topic: Greedy Approximations: Set Cover and Min Makespan Date: 1/30/06 3.1 Set Cover The Set Cover problem is: Given a set of
More informationDefinition 11.1. Given a graph G on n vertices, we define the following quantities:
Lecture 11 The Lovász ϑ Function 11.1 Perfect graphs We begin with some background on perfect graphs. graphs. First, we define some quantities on Definition 11.1. Given a graph G on n vertices, we define
More informationTHE PROBLEM WORMS (1) WORMS (2) THE PROBLEM OF WORM PROPAGATION/PREVENTION THE MINIMUM VERTEX COVER PROBLEM
1 THE PROBLEM OF WORM PROPAGATION/PREVENTION I.E. THE MINIMUM VERTEX COVER PROBLEM Prof. Tiziana Calamoneri Network Algorithms A.y. 2014/15 2 THE PROBLEM WORMS (1)! A computer worm is a standalone malware
More information! Solve problem to optimality. ! Solve problem in polytime. ! Solve arbitrary instances of the problem. #approximation algorithm.
Approximation Algorithms 11 Approximation Algorithms Q Suppose I need to solve an NPhard problem What should I do? A Theory says you're unlikely to find a polytime algorithm Must sacrifice one of three
More informationChapter 11. 11.1 Load Balancing. Approximation Algorithms. Load Balancing. Load Balancing on 2 Machines. Load Balancing: Greedy Scheduling
Approximation Algorithms Chapter Approximation Algorithms Q. Suppose I need to solve an NPhard problem. What should I do? A. Theory says you're unlikely to find a polytime algorithm. Must sacrifice one
More informationprinceton univ. F 13 cos 521: Advanced Algorithm Design Lecture 6: Provable Approximation via Linear Programming Lecturer: Sanjeev Arora
princeton univ. F 13 cos 521: Advanced Algorithm Design Lecture 6: Provable Approximation via Linear Programming Lecturer: Sanjeev Arora Scribe: One of the running themes in this course is the notion of
More informationOn Integer Additive SetIndexers of Graphs
On Integer Additive SetIndexers of Graphs arxiv:1312.7672v4 [math.co] 2 Mar 2014 N K Sudev and K A Germina Abstract A setindexer of a graph G is an injective setvalued function f : V (G) 2 X such that
More information! Solve problem to optimality. ! Solve problem in polytime. ! Solve arbitrary instances of the problem. !approximation algorithm.
Approximation Algorithms Chapter Approximation Algorithms Q Suppose I need to solve an NPhard problem What should I do? A Theory says you're unlikely to find a polytime algorithm Must sacrifice one of
More informationDiscrete Applied Mathematics. The firefighter problem with more than one firefighter on trees
Discrete Applied Mathematics 161 (2013) 899 908 Contents lists available at SciVerse ScienceDirect Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam The firefighter problem with
More informationSmall independent edge dominating sets in graphs of maximum degree three
Small independent edge dominating sets in graphs of maimum degree three Grażna Zwoźniak Grazna.Zwozniak@ii.uni.wroc.pl Institute of Computer Science Wrocław Universit Small independent edge dominating
More informationPolytope Examples (PolyComp Fukuda) Matching Polytope 1
Polytope Examples (PolyComp Fukuda) Matching Polytope 1 Matching Polytope Let G = (V,E) be a graph. A matching in G is a subset of edges M E such that every vertex meets at most one member of M. A matching
More informationPermutation Betting Markets: Singleton Betting with Extra Information
Permutation Betting Markets: Singleton Betting with Extra Information Mohammad Ghodsi Sharif University of Technology ghodsi@sharif.edu Hamid Mahini Sharif University of Technology mahini@ce.sharif.edu
More informationOn the Relationship between Classes P and NP
Journal of Computer Science 8 (7): 10361040, 2012 ISSN 15493636 2012 Science Publications On the Relationship between Classes P and NP Anatoly D. Plotnikov Department of Computer Systems and Networks,
More informationEvery tree contains a large induced subgraph with all degrees odd
Every tree contains a large induced subgraph with all degrees odd A.J. Radcliffe Carnegie Mellon University, Pittsburgh, PA A.D. Scott Department of Pure Mathematics and Mathematical Statistics University
More informationSHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH
31 Kragujevac J. Math. 25 (2003) 31 49. SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH Kinkar Ch. Das Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, W.B.,
More informationGeneralized Induced Factor Problems
Egerváry Research Group on Combinatorial Optimization Technical reports TR200207. Published by the Egrerváry Research Group, Pázmány P. sétány 1/C, H 1117, Budapest, Hungary. Web site: www.cs.elte.hu/egres.
More informationMean RamseyTurán numbers
Mean RamseyTurán numbers Raphael Yuster Department of Mathematics University of Haifa at Oranim Tivon 36006, Israel Abstract A ρmean coloring of a graph is a coloring of the edges such that the average
More information3. Linear Programming and Polyhedral Combinatorics
Massachusetts Institute of Technology Handout 6 18.433: Combinatorial Optimization February 20th, 2009 Michel X. Goemans 3. Linear Programming and Polyhedral Combinatorics Summary of what was seen in the
More informationJUSTINTIME SCHEDULING WITH PERIODIC TIME SLOTS. Received December May 12, 2003; revised February 5, 2004
Scientiae Mathematicae Japonicae Online, Vol. 10, (2004), 431 437 431 JUSTINTIME SCHEDULING WITH PERIODIC TIME SLOTS Ondřej Čepeka and Shao Chin Sung b Received December May 12, 2003; revised February
More informationCycle transversals in bounded degree graphs
Electronic Notes in Discrete Mathematics 35 (2009) 189 195 www.elsevier.com/locate/endm Cycle transversals in bounded degree graphs M. Groshaus a,2,3 P. Hell b,3 S. Klein c,1,3 L. T. Nogueira d,1,3 F.
More informationAlgorithm Design and Analysis
Algorithm Design and Analysis LECTURE 27 Approximation Algorithms Load Balancing Weighted Vertex Cover Reminder: Fill out SRTEs online Don t forget to click submit Sofya Raskhodnikova 12/6/2011 S. Raskhodnikova;
More informationBOUNDARY EDGE DOMINATION IN GRAPHS
BULLETIN OF THE INTERNATIONAL MATHEMATICAL VIRTUAL INSTITUTE ISSN (p) 04874, ISSN (o) 04955 www.imvibl.org /JOURNALS / BULLETIN Vol. 5(015), 19704 Former BULLETIN OF THE SOCIETY OF MATHEMATICIANS BANJA
More information5.1 Bipartite Matching
CS787: Advanced Algorithms Lecture 5: Applications of Network Flow In the last lecture, we looked at the problem of finding the maximum flow in a graph, and how it can be efficiently solved using the FordFulkerson
More informationAn Approximation Algorithm for Bounded Degree Deletion
An Approximation Algorithm for Bounded Degree Deletion Tomáš Ebenlendr Petr Kolman Jiří Sgall Abstract Bounded Degree Deletion is the following generalization of Vertex Cover. Given an undirected graph
More informationApplied Algorithm Design Lecture 5
Applied Algorithm Design Lecture 5 Pietro Michiardi Eurecom Pietro Michiardi (Eurecom) Applied Algorithm Design Lecture 5 1 / 86 Approximation Algorithms Pietro Michiardi (Eurecom) Applied Algorithm Design
More informationApproximating Minimum Bounded Degree Spanning Trees to within One of Optimal
Approximating Minimum Bounded Degree Spanning Trees to within One of Optimal ABSTACT Mohit Singh Tepper School of Business Carnegie Mellon University Pittsburgh, PA USA mohits@andrew.cmu.edu In the MINIMUM
More informationFairness in Routing and Load Balancing
Fairness in Routing and Load Balancing Jon Kleinberg Yuval Rabani Éva Tardos Abstract We consider the issue of network routing subject to explicit fairness conditions. The optimization of fairness criteria
More informationExponential time algorithms for graph coloring
Exponential time algorithms for graph coloring Uriel Feige Lecture notes, March 14, 2011 1 Introduction Let [n] denote the set {1,..., k}. A klabeling of vertices of a graph G(V, E) is a function V [k].
More informationCOMBINATORIAL PROPERTIES OF THE HIGMANSIMS GRAPH. 1. Introduction
COMBINATORIAL PROPERTIES OF THE HIGMANSIMS GRAPH ZACHARY ABEL 1. Introduction In this survey we discuss properties of the HigmanSims graph, which has 100 vertices, 1100 edges, and is 22 regular. In fact
More informationA Note on Maximum Independent Sets in Rectangle Intersection Graphs
A Note on Maximum Independent Sets in Rectangle Intersection Graphs Timothy M. Chan School of Computer Science University of Waterloo Waterloo, Ontario N2L 3G1, Canada tmchan@uwaterloo.ca September 12,
More informationConnected Identifying Codes for Sensor Network Monitoring
Connected Identifying Codes for Sensor Network Monitoring Niloofar Fazlollahi, David Starobinski and Ari Trachtenberg Dept. of Electrical and Computer Engineering Boston University, Boston, MA 02215 Email:
More informationThe Open University s repository of research publications and other research outputs
Open Research Online The Open University s repository of research publications and other research outputs The degreediameter problem for circulant graphs of degree 8 and 9 Journal Article How to cite:
More informationResource Allocation with Time Intervals
Resource Allocation with Time Intervals Andreas Darmann Ulrich Pferschy Joachim Schauer Abstract We study a resource allocation problem where jobs have the following characteristics: Each job consumes
More informationA Turán Type Problem Concerning the Powers of the Degrees of a Graph
A Turán Type Problem Concerning the Powers of the Degrees of a Graph Yair Caro and Raphael Yuster Department of Mathematics University of HaifaORANIM, Tivon 36006, Israel. AMS Subject Classification:
More informationFinding and counting given length cycles
Finding and counting given length cycles Noga Alon Raphael Yuster Uri Zwick Abstract We present an assortment of methods for finding and counting simple cycles of a given length in directed and undirected
More informationMinimize subject to. x S R
Chapter 12 Lagrangian Relaxation This chapter is mostly inspired by Chapter 16 of [1]. In the previous chapters, we have succeeded to find efficient algorithms to solve several important problems such
More informationCompletely Positive Cone and its Dual
On the Computational Complexity of Membership Problems for the Completely Positive Cone and its Dual Peter J.C. Dickinson Luuk Gijben July 3, 2012 Abstract Copositive programming has become a useful tool
More informationBicolored Shortest Paths in Graphs with Applications to Network Overlay Design
Bicolored Shortest Paths in Graphs with Applications to Network Overlay Design Hongsik Choi and HyeongAh Choi Department of Electrical Engineering and Computer Science George Washington University Washington,
More informationTenacity and rupture degree of permutation graphs of complete bipartite graphs
Tenacity and rupture degree of permutation graphs of complete bipartite graphs Fengwei Li, Qingfang Ye and Xueliang Li Department of mathematics, Shaoxing University, Shaoxing Zhejiang 312000, P.R. China
More informationCSC2420 Fall 2012: Algorithm Design, Analysis and Theory
CSC2420 Fall 2012: Algorithm Design, Analysis and Theory Allan Borodin November 15, 2012; Lecture 10 1 / 27 Randomized online bipartite matching and the adwords problem. We briefly return to online algorithms
More informationApproximation Algorithms: LP Relaxation, Rounding, and Randomized Rounding Techniques. My T. Thai
Approximation Algorithms: LP Relaxation, Rounding, and Randomized Rounding Techniques My T. Thai 1 Overview An overview of LP relaxation and rounding method is as follows: 1. Formulate an optimization
More informationLecture 3: Linear Programming Relaxations and Rounding
Lecture 3: Linear Programming Relaxations and Rounding 1 Approximation Algorithms and Linear Relaxations For the time being, suppose we have a minimization problem. Many times, the problem at hand can
More informationWeek 5 Integral Polyhedra
Week 5 Integral Polyhedra We have seen some examples 1 of linear programming formulation that are integral, meaning that every basic feasible solution is an integral vector. This week we develop a theory
More informationPermutation Betting Markets: Singleton Betting with Extra Information
Permutation Betting Markets: Singleton Betting with Extra Information Mohammad Ghodsi Sharif University of Technology ghodsi@sharif.edu Hamid Mahini Sharif University of Technology mahini@ce.sharif.edu
More informationAnalysis of Algorithms, I
Analysis of Algorithms, I CSOR W4231.002 Eleni Drinea Computer Science Department Columbia University Thursday, February 26, 2015 Outline 1 Recap 2 Representing graphs 3 Breadthfirst search (BFS) 4 Applications
More informationOn the independence number of graphs with maximum degree 3
On the independence number of graphs with maximum degree 3 Iyad A. Kanj Fenghui Zhang Abstract Let G be an undirected graph with maximum degree at most 3 such that G does not contain any of the three graphs
More informationCSC 373: Algorithm Design and Analysis Lecture 16
CSC 373: Algorithm Design and Analysis Lecture 16 Allan Borodin February 25, 2013 Some materials are from Stephen Cook s IIT talk and Keven Wayne s slides. 1 / 17 Announcements and Outline Announcements
More informationA simpler and better derandomization of an approximation algorithm for Single Source RentorBuy
A simpler and better derandomization of an approximation algorithm for Single Source RentorBuy David P. Williamson Anke van Zuylen School of Operations Research and Industrial Engineering, Cornell University,
More informationCombinatorial 5/6approximation of Max Cut in graphs of maximum degree 3
Combinatorial 5/6approximation of Max Cut in graphs of maximum degree 3 Cristina Bazgan a and Zsolt Tuza b,c,d a LAMSADE, Université ParisDauphine, Place du Marechal de Lattre de Tassigny, F75775 Paris
More informationScheduling to Minimize Power Consumption using Submodular Functions
Scheduling to Minimize Power Consumption using Submodular Functions Erik D. Demaine MIT edemaine@mit.edu Morteza Zadimoghaddam MIT morteza@mit.edu ABSTRACT We develop logarithmic approximation algorithms
More informationWhy? A central concept in Computer Science. Algorithms are ubiquitous.
Analysis of Algorithms: A Brief Introduction Why? A central concept in Computer Science. Algorithms are ubiquitous. Using the Internet (sending email, transferring files, use of search engines, online
More informationTotal colorings of planar graphs with small maximum degree
Total colorings of planar graphs with small maximum degree Bing Wang 1,, JianLiang Wu, SiFeng Tian 1 Department of Mathematics, Zaozhuang University, Shandong, 77160, China School of Mathematics, Shandong
More informationThe Conference Call Search Problem in Wireless Networks
The Conference Call Search Problem in Wireless Networks Leah Epstein 1, and Asaf Levin 2 1 Department of Mathematics, University of Haifa, 31905 Haifa, Israel. lea@math.haifa.ac.il 2 Department of Statistics,
More informationDiscrete Mathematics & Mathematical Reasoning Chapter 10: Graphs
Discrete Mathematics & Mathematical Reasoning Chapter 10: Graphs Kousha Etessami U. of Edinburgh, UK Kousha Etessami (U. of Edinburgh, UK) Discrete Mathematics (Chapter 6) 1 / 13 Overview Graphs and Graph
More informationWeighted Sum Coloring in Batch Scheduling of Conflicting Jobs
Weighted Sum Coloring in Batch Scheduling of Conflicting Jobs Leah Epstein Magnús M. Halldórsson Asaf Levin Hadas Shachnai Abstract Motivated by applications in batch scheduling of jobs in manufacturing
More informationShort Cycles make Whard problems hard: FPT algorithms for Whard Problems in Graphs with no short Cycles
Short Cycles make Whard problems hard: FPT algorithms for Whard Problems in Graphs with no short Cycles Venkatesh Raman and Saket Saurabh The Institute of Mathematical Sciences, Chennai 600 113. {vraman
More informationBest Monotone Degree Bounds for Various Graph Parameters
Best Monotone Degree Bounds for Various Graph Parameters D. Bauer Department of Mathematical Sciences Stevens Institute of Technology Hoboken, NJ 07030 S. L. Hakimi Department of Electrical and Computer
More informationZachary Monaco Georgia College Olympic Coloring: Go For The Gold
Zachary Monaco Georgia College Olympic Coloring: Go For The Gold Coloring the vertices or edges of a graph leads to a variety of interesting applications in graph theory These applications include various
More informationTransportation Polytopes: a Twenty year Update
Transportation Polytopes: a Twenty year Update Jesús Antonio De Loera University of California, Davis Based on various papers joint with R. Hemmecke, E.Kim, F. Liu, U. Rothblum, F. Santos, S. Onn, R. Yoshida,
More informationCMPSCI611: Approximating MAXCUT Lecture 20
CMPSCI611: Approximating MAXCUT Lecture 20 For the next two lectures we ll be seeing examples of approximation algorithms for interesting NPhard problems. Today we consider MAXCUT, which we proved to
More informationDistributed Computing over Communication Networks: Maximal Independent Set
Distributed Computing over Communication Networks: Maximal Independent Set What is a MIS? MIS An independent set (IS) of an undirected graph is a subset U of nodes such that no two nodes in U are adjacent.
More informationImproved Algorithms for Data Migration
Improved Algorithms for Data Migration Samir Khuller 1, YooAh Kim, and Azarakhsh Malekian 1 Department of Computer Science, University of Maryland, College Park, MD 20742. Research supported by NSF Award
More informationCycles in a Graph Whose Lengths Differ by One or Two
Cycles in a Graph Whose Lengths Differ by One or Two J. A. Bondy 1 and A. Vince 2 1 LABORATOIRE DE MATHÉMATIQUES DISCRÉTES UNIVERSITÉ CLAUDEBERNARD LYON 1 69622 VILLEURBANNE, FRANCE 2 DEPARTMENT OF MATHEMATICS
More informationThe Approximability of the Binary Paintshop Problem
The Approximability of the Binary Paintshop Problem Anupam Gupta 1, Satyen Kale 2, Viswanath Nagarajan 2, Rishi Saket 2, and Baruch Schieber 2 1 Dept. of Computer Science, Carnegie Mellon University, Pittsburgh
More informationClique coloring B 1 EPG graphs
Clique coloring B 1 EPG graphs Flavia Bonomo a,c, María Pía Mazzoleni b,c, and Maya Stein d a Departamento de Computación, FCENUBA, Buenos Aires, Argentina. b Departamento de Matemática, FCEUNLP, La
More informationPartitioning edgecoloured complete graphs into monochromatic cycles and paths
arxiv:1205.5492v1 [math.co] 24 May 2012 Partitioning edgecoloured complete graphs into monochromatic cycles and paths Alexey Pokrovskiy Departement of Mathematics, London School of Economics and Political
More informationTreerepresentation of set families and applications to combinatorial decompositions
Treerepresentation of set families and applications to combinatorial decompositions BinhMinh BuiXuan a, Michel Habib b Michaël Rao c a Department of Informatics, University of Bergen, Norway. buixuan@ii.uib.no
More informationA 2factor in which each cycle has long length in clawfree graphs
A factor in which each cycle has long length in clawfree graphs Roman Čada Shuya Chiba Kiyoshi Yoshimoto 3 Department of Mathematics University of West Bohemia and Institute of Theoretical Computer Science
More informationLocal search for the minimum label spanning tree problem with bounded color classes
Available online at www.sciencedirect.com Operations Research Letters 31 (003) 195 01 Operations Research Letters www.elsevier.com/locate/dsw Local search for the minimum label spanning tree problem with
More informationTesting Hereditary Properties of NonExpanding BoundedDegree Graphs
Testing Hereditary Properties of NonExpanding BoundedDegree Graphs Artur Czumaj Asaf Shapira Christian Sohler Abstract We study graph properties which are testable for bounded degree graphs in time independent
More informationApplication Placement on a Cluster of Servers (extended abstract)
Application Placement on a Cluster of Servers (extended abstract) Bhuvan Urgaonkar, Arnold Rosenberg and Prashant Shenoy Department of Computer Science, University of Massachusetts, Amherst, MA 01003 {bhuvan,
More informationAll trees contain a large induced subgraph having all degrees 1 (mod k)
All trees contain a large induced subgraph having all degrees 1 (mod k) David M. Berman, A.J. Radcliffe, A.D. Scott, Hong Wang, and Larry Wargo *Department of Mathematics University of New Orleans New
More informationComputer Algorithms. NPComplete Problems. CISC 4080 Yanjun Li
Computer Algorithms NPComplete Problems NPcompleteness The quest for efficient algorithms is about finding clever ways to bypass the process of exhaustive search, using clues from the input in order
More informationON THE COMPLEXITY OF THE GAME OF SET. {kamalika,pbg,dratajcz,hoeteck}@cs.berkeley.edu
ON THE COMPLEXITY OF THE GAME OF SET KAMALIKA CHAUDHURI, BRIGHTEN GODFREY, DAVID RATAJCZAK, AND HOETECK WEE {kamalika,pbg,dratajcz,hoeteck}@cs.berkeley.edu ABSTRACT. Set R is a card game played with a
More informationGENERATING LOWDEGREE 2SPANNERS
SIAM J. COMPUT. c 1998 Society for Industrial and Applied Mathematics Vol. 27, No. 5, pp. 1438 1456, October 1998 013 GENERATING LOWDEGREE 2SPANNERS GUY KORTSARZ AND DAVID PELEG Abstract. A kspanner
More informationCSC2420 Spring 2015: Lecture 3
CSC2420 Spring 2015: Lecture 3 Allan Borodin January 22, 2015 1 / 1 Announcements and todays agenda Assignment 1 due next Thursday. I may add one or two additional questions today or tomorrow. Todays agenda
More informationEnergy Efficient Monitoring in Sensor Networks
Energy Efficient Monitoring in Sensor Networks Amol Deshpande, Samir Khuller, Azarakhsh Malekian, Mohammed Toossi Computer Science Department, University of Maryland, A.V. Williams Building, College Park,
More informationOffline sorting buffers on Line
Offline sorting buffers on Line Rohit Khandekar 1 and Vinayaka Pandit 2 1 University of Waterloo, ON, Canada. email: rkhandekar@gmail.com 2 IBM India Research Lab, New Delhi. email: pvinayak@in.ibm.com
More informationLarge induced subgraphs with all degrees odd
Large induced subgraphs with all degrees odd A.D. Scott Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, England Abstract: We prove that every connected graph of order
More informationP versus NP, and More
1 P versus NP, and More Great Ideas in Theoretical Computer Science Saarland University, Summer 2014 If you have tried to solve a crossword puzzle, you know that it is much harder to solve it than to verify
More informationMax Flow, Min Cut, and Matchings (Solution)
Max Flow, Min Cut, and Matchings (Solution) 1. The figure below shows a flow network on which an st flow is shown. The capacity of each edge appears as a label next to the edge, and the numbers in boxes
More informationWeighted Sum Coloring in Batch Scheduling of Conflicting Jobs
Weighted Sum Coloring in Batch Scheduling of Conflicting Jobs Leah Epstein Magnús M. Halldórsson Asaf Levin Hadas Shachnai Abstract Motivated by applications in batch scheduling of jobs in manufacturing
More informationarxiv:1112.0829v1 [math.pr] 5 Dec 2011
How Not to Win a Million Dollars: A Counterexample to a Conjecture of L. Breiman Thomas P. Hayes arxiv:1112.0829v1 [math.pr] 5 Dec 2011 Abstract Consider a gambling game in which we are allowed to repeatedly
More informationLabeling outerplanar graphs with maximum degree three
Labeling outerplanar graphs with maximum degree three Xiangwen Li 1 and Sanming Zhou 2 1 Department of Mathematics Huazhong Normal University, Wuhan 430079, China 2 Department of Mathematics and Statistics
More informationOutline. NPcompleteness. When is a problem easy? When is a problem hard? Today. Euler Circuits
Outline NPcompleteness Examples of Easy vs. Hard problems Euler circuit vs. Hamiltonian circuit Shortest Path vs. Longest Path 2pairs sum vs. general Subset Sum Reducing one problem to another Clique
More informationPrivate Approximation of Clustering and Vertex Cover
Private Approximation of Clustering and Vertex Cover Amos Beimel, Renen Hallak, and Kobbi Nissim Department of Computer Science, BenGurion University of the Negev Abstract. Private approximation of search
More informationKey words. multiobjective optimization, approximate Pareto set, biobjective shortest path
SMALL APPROXIMATE PARETO SETS FOR BI OBJECTIVE SHORTEST PATHS AND OTHER PROBLEMS ILIAS DIAKONIKOLAS AND MIHALIS YANNAKAKIS Abstract. We investigate the problem of computing a minimum set of solutions that
More informationON INDUCED SUBGRAPHS WITH ALL DEGREES ODD. 1. Introduction
ON INDUCED SUBGRAPHS WITH ALL DEGREES ODD A.D. SCOTT Abstract. Gallai proved that the vertex set of any graph can be partitioned into two sets, each inducing a subgraph with all degrees even. We prove
More information11. APPROXIMATION ALGORITHMS
11. APPROXIMATION ALGORITHMS load balancing center selection pricing method: vertex cover LP rounding: vertex cover generalized load balancing knapsack problem Lecture slides by Kevin Wayne Copyright 2005
More informationPh.D. Thesis. Judit NagyGyörgy. Supervisor: Péter Hajnal Associate Professor
Online algorithms for combinatorial problems Ph.D. Thesis by Judit NagyGyörgy Supervisor: Péter Hajnal Associate Professor Doctoral School in Mathematics and Computer Science University of Szeged Bolyai
More informationThe Independence Number in Graphs of Maximum Degree Three
The Independence Number in Graphs of Maximum Degree Three Jochen Harant 1 Michael A. Henning 2 Dieter Rautenbach 1 and Ingo Schiermeyer 3 1 Institut für Mathematik, TU Ilmenau, Postfach 100565, D98684
More informationMinimal Cost Reconfiguration of Data Placement in a Storage Area Network
Minimal Cost Reconfiguration of Data Placement in a Storage Area Network Hadas Shachnai Gal Tamir Tami Tamir Abstract VideoonDemand (VoD) services require frequent updates in file configuration on the
More informationData Migration in Heterogeneous Storage Systems
011 31st International Conference on Distributed Computing Systems Data Migration in Heterogeneous Storage Systems Chadi Kari Department of Computer Science and Engineering University of Connecticut Storrs,
More information