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Incorporating negative-weight vertices in certain vertex-search graph algorithms
Information Processing Letters 42 (1992) 293-294 North-Holland
24 July 1992
Incorporating negative-weight vertices in certain vertex-search graph algorithms Glenn K. Manacher and Terrance A. Mankus Department
of Mathematics,
Statistics, and Computer
Science, University of Illinois at Chicago, Chicago, IL 60680, USA
Communicated by D. Gries Received 6 April 1992 Revised 7 May 1992 Keywords: Algorithms, vertex-search graph algorithms
Consider a graph whose vertices have real weights. Consider a vertex-search problem that yields some minimum-weight subset of the vertices satisfying certain conditions. Many published algorithms for such problems, e.g. [81 and [3], depend crucially on the properties of positive-weight vertices and therefore restrict vertex weights to nonnegative numbers. These algorithms are for the most part related to, though not restricted to, domination. The purpose of this note is to demonstrate that such algorithms can be made to accommodate negative-weight vertices without loss of efficiency. The result was first announced and proven in [6], but the literature indicates that it is not widely known. A vertex-search problem of this kind is an extendible-search problem, written ESP, if every vertex superset of a solution is also a solution, albeit not necessarily of minimum weight. Examples of ESPs are the dominating set problem [4], the vertex cover problem [4], the connected dominating set problem, and the total dominating set problem [1,21. Many other natural problems over intersection models are also ESPs, for instance the spatial cover problem [5].
Correspondence to: G.K. Manacher, Department of Mathematics, Statistics and Computer Science (M/C 249), University of Illinois at Chicago, 322 Science and Engineering Offices, Box 4348, Chicago, IL 60680, USA.
Suppose we have an ESP and an algorithm that solves it when the vertex weights are nonnegative. We extend the algorithm to work on graphs that include negative-weight vertices as follows.
(1) Let G be the graph and let N, Z, and P be the sets of negative-weight, zero-weight, and positive-weight vertices. Let G * = ((P u No U Z), E > be the graph formed from G by setting the weights of the vertices in N to zero. Let No denote the latter set. (2) Use any algorithm to obtain D *, a minimum-weight set of vertices that solves the ESP for the graph G *. (3) Then (D * fl P) U N U Z is a minimum-weight solution for the problem on graph G. Proof (sketch). Suppose there exists a smallerweight solution, D’, to the given ESP for the graph G. Since all negative-weight vertices are present in the solution (D* n P) UN U Z, this implies that wt(D’ n P) < wt(D* n PI. But then (D’ n PI U No U Z would be a solution of smaller weight than D* found in step 2 above. Hence D’ does not exist. q
Ramalingam and Rangan independently rediscovered this result for the dominating set problem in [7], without proof.
0020-0190/92/%05.00 0 1992 - Elsevier Science Publishers B.V. All rights reserved
293
Volume
42, Number
6
INFORMATION
PROCESSING
References [l] A.A. Bertossi and A. Gori, Total domination and irredundance in weighted interval graphs, SL4MJ. Discrete Math. l(1988) 317-327. [2] E.J. Cockayne, R.M. Dawes and ST. Hedetniemi, Total domination in graphs, Networks 10 (1980) 211-219. [3] M. Farber and J.M. Keil, Domination in permutation graphs, J. Algorithms 6 (1985) 309-321. [4] M.R. Carey and D.S. Johnson, Computers and Intractability, a Guide to the Theory of NP-completeness (Freeman, San Francisco, CA, 1979).
294
LETTERS
24 July 1992
[5] C.C. Lee and D.T. Lee, On a circle-cover minimization problem, Inform. Process. Left. 18 (1984) 109-115. [6] G.K. Manacher and C.J. Smith, Efficient algorithms for new problems on interval graphs and interval models, Manuscript, 1984. [7] G. Ramalingam and C. Pandu Rangan, A unified approach to domination problems on interval graphs, Inform. Process. Lett. 27 (1988) 271-274. [8] C. Rhee, Y.D. Liang, S.K. Dhall and S. Lakshmivarahan, An O(m + n) time algorithm for finding a minimum weight dominating set in permutation graphs, School of Electrical Engineering and Computer Science, The University of Oklahoma, Norman, OK, 1991.
24 July 1992
Incorporating negative-weight vertices in certain vertex-search graph algorithms Glenn K. Manacher and Terrance A. Mankus Department
of Mathematics,
Statistics, and Computer
Science, University of Illinois at Chicago, Chicago, IL 60680, USA
Communicated by D. Gries Received 6 April 1992 Revised 7 May 1992 Keywords: Algorithms, vertex-search graph algorithms
Consider a graph whose vertices have real weights. Consider a vertex-search problem that yields some minimum-weight subset of the vertices satisfying certain conditions. Many published algorithms for such problems, e.g. [81 and [3], depend crucially on the properties of positive-weight vertices and therefore restrict vertex weights to nonnegative numbers. These algorithms are for the most part related to, though not restricted to, domination. The purpose of this note is to demonstrate that such algorithms can be made to accommodate negative-weight vertices without loss of efficiency. The result was first announced and proven in [6], but the literature indicates that it is not widely known. A vertex-search problem of this kind is an extendible-search problem, written ESP, if every vertex superset of a solution is also a solution, albeit not necessarily of minimum weight. Examples of ESPs are the dominating set problem [4], the vertex cover problem [4], the connected dominating set problem, and the total dominating set problem [1,21. Many other natural problems over intersection models are also ESPs, for instance the spatial cover problem [5].
Correspondence to: G.K. Manacher, Department of Mathematics, Statistics and Computer Science (M/C 249), University of Illinois at Chicago, 322 Science and Engineering Offices, Box 4348, Chicago, IL 60680, USA.
Suppose we have an ESP and an algorithm that solves it when the vertex weights are nonnegative. We extend the algorithm to work on graphs that include negative-weight vertices as follows.
(1) Let G be the graph and let N, Z, and P be the sets of negative-weight, zero-weight, and positive-weight vertices. Let G * = ((P u No U Z), E > be the graph formed from G by setting the weights of the vertices in N to zero. Let No denote the latter set. (2) Use any algorithm to obtain D *, a minimum-weight set of vertices that solves the ESP for the graph G *. (3) Then (D * fl P) U N U Z is a minimum-weight solution for the problem on graph G. Proof (sketch). Suppose there exists a smallerweight solution, D’, to the given ESP for the graph G. Since all negative-weight vertices are present in the solution (D* n P) UN U Z, this implies that wt(D’ n P) < wt(D* n PI. But then (D’ n PI U No U Z would be a solution of smaller weight than D* found in step 2 above. Hence D’ does not exist. q
Ramalingam and Rangan independently rediscovered this result for the dominating set problem in [7], without proof.
0020-0190/92/%05.00 0 1992 - Elsevier Science Publishers B.V. All rights reserved
293
Volume
42, Number
6
INFORMATION
PROCESSING
References [l] A.A. Bertossi and A. Gori, Total domination and irredundance in weighted interval graphs, SL4MJ. Discrete Math. l(1988) 317-327. [2] E.J. Cockayne, R.M. Dawes and ST. Hedetniemi, Total domination in graphs, Networks 10 (1980) 211-219. [3] M. Farber and J.M. Keil, Domination in permutation graphs, J. Algorithms 6 (1985) 309-321. [4] M.R. Carey and D.S. Johnson, Computers and Intractability, a Guide to the Theory of NP-completeness (Freeman, San Francisco, CA, 1979).
294
LETTERS
24 July 1992
[5] C.C. Lee and D.T. Lee, On a circle-cover minimization problem, Inform. Process. Left. 18 (1984) 109-115. [6] G.K. Manacher and C.J. Smith, Efficient algorithms for new problems on interval graphs and interval models, Manuscript, 1984. [7] G. Ramalingam and C. Pandu Rangan, A unified approach to domination problems on interval graphs, Inform. Process. Lett. 27 (1988) 271-274. [8] C. Rhee, Y.D. Liang, S.K. Dhall and S. Lakshmivarahan, An O(m + n) time algorithm for finding a minimum weight dominating set in permutation graphs, School of Electrical Engineering and Computer Science, The University of Oklahoma, Norman, OK, 1991.