On Totally Unimodularity in Edge-Edge Adjacency Relationships

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1 - 1, ,., -., On Totally Unimodularity in Edge-Edge Adjacency Relationships Yusuke Matsumoto, 1, 3 Naoyuki Kamiyama 2 and Keiko Imai 2 We consider totally unimodularity for edge-edge adjacency matrices which represent a relationship between two edges in a graph. The matrices appear in an integer programming formulation for the edge dominating set problem. We consider a problem of characterizing graphs having totally unimodular matrices as their edge-edge adjacency matrices, and give a necessary and sufficient condition for the characterization. The condition is the first characterization for edge-edge adjacency matrices. Moreover, we introduce graph classes obtained an optimal integer solution using the linear programming relaxation for the above problem. 1 IBM, Japan 2 Department of Information and System Engineering, Chuo University 3 Graduate School of Science and Engineering, Chuo University 1.,,.,,.,.,. 1,.,, 1, 0, 1. 1, 0, 1., 5).,.,,,., 2 5)., -. A. Berger, O. Parekh 1), -.,., , NP- -,. 2. V, E, G = (V, E)., 2. e, f E e f, e f. E e i, e j, - M(G) = (m ei,e j ) 1 e i e j, m ei,e j = 0, 1 c 2011 Information Processing Society of Japan

2 (a) (b) (c) 1 (a) T S, (b) C 4, (c) G T C., m ei,e i = 1. 2 e, f E e f, e f. E D, E \ D, D, D.. Z. e x(e) = 1, x(e) = 0 x Z E,. minimize subject to x(e), (1) e E x(e ) 1 e E, (2) e δ(e) x(e) {0, 1} e E. (3), X V, δ(x) = {e E X e }. (IP). (IP) (2) -. K 1,3 2 T S ( 1(a)), t C t ( 1(b))., C 3 G T C ( 1(c)) ) A Z p q, b Z q {x Ax b, x 0}. 1,, -, (IP)., -.,. 2. -, T S, C t (t 4), G T C., T S, C t(t 4), G T C. 2,, -., -., , -. 2,. 3. 2) A = (a i,j), j, i R 1 a i,j i R 2 a i,j { 1, 0, 1} A R R 1 R G = (V, E) - M(G), ψ G : E {0, 1}, 1 ψ G (e i )π G (e i ) 1 (4) e i δ(e j ) π G : E { 1, 1}, e j E. 4, ψ G (4) π G, M(G)., t 4 M(C t ).. C 3, M(C 3 ) 1., M(C 3)., C 4. C 4 1,, e 3, ( 2)., e 3, 1, 0 ψ C4. 2, a/b a ψ C4 (e), b π C4 (e)., π C4 ( ) = 1. (4), π C4 (e 3) = 1., ψ C4 0 π C4 0 2 c 2011 Information Processing Society of Japan

3 2 e 3 M(C 4 ) ( 1 1, 1 1.) e 1 e 2 e 3 e 4 3 P E 1 (4)., (4) π C4 ()., C 4 (4) π C4., M(C 4)., M(C 8 )., ( 3). 4 P E. P E 2 v 1, v 2, v 1 e 1., P E e 1 v 2 e 2, e 3, e 4., e 1 e 3 1, e 2 e 4 0 ψ PE. C 4 C 4,, C 4. C 4 P E, = v 1 = v 2 C 4+4 = C 8 ( 4). C 4+4, ψ C4+4. ψ C4 (e) ψ C4+4 (e) = ψ PE (e) e C 4, e P E. ψ C4+4,, π C4+4 () = 1., (4) π PE e 1 1, e 3 1., π C4+4 π C4 (e) e C 4, π C4+4 (e) = π PE (e) e P E, 4 = e 1 e 3 e 2 e 3 e 4 = v 2 M(C 4+4 )., π C4+4 (4) 1 1. p M(C 4+4p) ( 5(a))., p., M(C 5+4p ), M(C 6+4p ), M(C 7+4p ) ( 5(b)- (d)) M(T S). 7. M(G T C ). 5, 6, 7,. 8. G, M(G). 3.2, -.,, T S -., -. G, G 1 P C, G ( 8)., P C ), T S.,. 3 c 2011 Information Processing Society of Japan

4 e 3 e 3 (a)c 4+4p (b)c 5+4p e 3 e 3 5 e 6 e 7 (c)c 6+4p (d)c 7+4p 4 t M(C t ) 6 M(T S ) 10. 6) 0 1, 1,. - e i R ei , 1. e M(G T C ) ê 1 ê 2 v e 3 (.) P C = (V PC, E PC ). E PC,,, e EPC ( 8). P C, M(P C)., 10, M(P C ). v V PC, e E \ E PC e V PC = {v} E v. v P C, E PC v 2. 2 e i, e i+1. ê E v, δ(ê) = δ(e i ) δ(e i+1 )., R ei R ei+1 E v 1.,., 10, -. v P C, P C., ê E v, δ(ê) δ( )., E v R e1, - 10., v P C, P C e PC, E v R e PC, - 10., -., M(G) 1., -.. G., G. G 11,. 4 c 2011 Information Processing Society of Japan

5 v e i e i+1 e G, G T G = (V, E TG ). G T S, T G T S., 9, T G. T G, P C = (V PC, E PC )., P C,, e EPC. 11, M(T G ) 1., G T G E. E. 1 : V PC ( 9) 2 : T G V PC 3 : T G 1 : V PC E 1 e. G t 4 C t, P C 1. v. E \ E PC v, G G T C, E \ E PC v ( 9)., e i, e i+1 δ(v), δ(e ) = δ(e i) δ(e i+1), - R ei R ei+1 R e, 10, , 10., v,, v,., , T S, C t(t 4), G T C. 4., 2,.,, 1,.,.,.,., -. 5., -,. -,., -,.,, -. 1) Berger,A. and Parekh,O.: Linear time algorithms for generalized edge dominating set problems. Algorithmica 50, no. 2, pp (2008). 2) Ghouila-Houri, A.: Caractérisation des matrices totalement unimodulaires. Comptes Redus Hebdomadaires des Séances de l Académie des Sciences (Paris) 254, pp (1962). 3) Harary, F. and Schwenk, A.: Trees with Hamiltonian square. Mathematika 18, pp (1971). 4) Hoffman, A. J. and Kruskal, J. B.: Integral boundary points of convex polyhedra. Linear inequalities and related systems (H. W. Kuhn and A. W. Tucker, eds.), Princeton University Press, pp (1956). 5) Korte,B. and Vygen,J.: Combinatorial Optimization: Theory and Algorithms(4th edition). Springer-Verlag (2007). 6) Schrijver,A.: Theory of linear and integer programming. J. Wiley & Sons (1986). 5 c 2011 Information Processing Society of Japan