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A Polyhedral Approach to Even Factors

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A Polyhedral Approach to Even Factors Egervary´ Research Group on Combinatorial Optimization Technical reportS TR-2003-05. Published by the Egrerv´aryResearch Group, P´azm´any P. s´et´any 1/C, H{1117, Budapest, Hungary. Web site: www.cs.elte.hu/egres . ISSN 1587{4451. A polyhedral approach to even factors M´artonMakai September 2003 EGRES Technical Report No. 2003-05 1 A polyhedral approach to even factors M´artonMakai? Abstract Generalizing path-matchings, W.H. Cunningham and J.F. Geelen intro- duced the notion of even factor in directed graphs. In weakly symmetric di- rected graphs they proved a min-max formula for the maximum cardinality even factor by algebraic method and also discussed a primal-dual method for the weighted case. Later, Gy. Pap and L. Szeg}oproved a simplified formula by purely combinatorial method and derived a Gallai-Edmonds type structure theorem. In this paper, polyhedra related to even factors are considered, inte- grality and total dual integrality of these linear descriptions are proved directly, without using earlier unweighted results. 1 Introduction Cunningham and Geelen [2, 3] introduced the notion of even factor in digraphs as the edge set of vertex disjoint union of dipaths and dicycles of even length. For short, on a digraph we mean directed graph, dicycle and dipath mean directed cycle and directed path, while in a cycle or in a path, there can be forward and backward edges, too. The maximum cardinality even factor problem is NP-hard in general (Wang, see [3]) but there are special classes of digraphs where it can be solved. In the paper throughout digraphs with no loops and with no parallel edges are considered, only oppositely directed edges are permitted. An edge of a digraph is called symmetric, if the reserved edge is in the edge set of the digraph, too. A digraph is symmetric if all its edges are symmetric, while a digraph is said to be weakly symmetric if its strongly connected components are symmetric. Then weak symmetry means that the edges contained in dicycle are symmetric. Pap and Szeg}o [8] introduced the more general class of hardly symmetric digraphs where the problem remains tractable. A digraph is said to be hardly symmetric if the edges contained in odd dicycle are symmetric. Using an algebraic approach, Cunningham and Geelen proved a min-max formula for the maximum cardinality even factor problem in weakly symmetric digraphs [3]. Later, Pap and Szeg}o[8] proved a simpler formula for the same problem with purely ?Department of Operations Research, E¨otv¨osUniversity, P´azm´any P´eters´et´any 1/C, Budapest, Hungary, H-1117. e-mail: [email protected]. The author is a member of the Egerv´aryResearch Group (EGRES). Supported by the Hungarian National Foundation for Scientific Research, OTKA T037547 and OTKA N034040. September 2003 Section 1. Introduction 2 combinatorial methods and they also observed that the same proof and formula works for the hardly symmetric case. As the unweighted problem is tractable only in special graphs, the weight function also has to possess certain symmetries. If G0 = (V 0;A0) is a weakly symmetric graph and c0 : A0 R is a weight function s.t. c0(uv) = c0(vu) if both uv and vu belong to A0, ! then the pair (G0; c0) is called weakly symmetric. Cunningham and Geelen considered weighted generalization in [3] for weakly symmetric pairs. Using their unweighted formula and a primal-dual method they derived integrality of a polyhedron similar to the perfect matching polyhedron. Here we deal with a more general class of digraphs and weight functions which turn out to be general enough to contain the mentioned cardinality and weighted cases. We prove integrality of polyhedra related to even factors and total dual integrality. We do not use earlier unweighted results, which rather follow as consequences. This approach enables us to derive weighted min-max formulas, nevertheless we omit to present them explicitly, since these are direct consequences of the polyhedral theorems. m n m Before, let A Z × be an integer matrix, b Z an integer vector. The poly- hedron defined by2 x : Ax b is said to be integer2 if every face of it contains an integer vector. Thef system≤Axg b is called totally dual integral (TDI) if the dual of the linear program max cx : ≤Ax b has an integer valued optimal solution for every integer valued c suchf that it has≤ ang optimal solution. Polyhedra defined by TDI systems are interesting because of the fact that they are integer polyhedra. To distinguish directed and undirected edges, the directed edge with tail u and head v will be denoted by uv and the undirected edge with endvertices u and v will be denoted by u; v . If G = (V; E = Ed Eu) is a mixed graph, Ed and Eu denote respectively thef set ofg its directed edges and[ the set of its undirected edges and y RE, then for u V and S V we will use the notations 2 d (u) = 2 y( u;⊆ v ) + y(uv) + y(vu), y u;v Eu uv Ed vu Ed Pf g2 f g P 2 P 2 iy(S) = y( u; v ): u; v Eu; u; v S + y(uv): uv Ed; u; v S , P f f g f g 2 2 g P f 2 2 g %y(S) = y(uv): uv Ed; u = S; v S and δy(S) = %y(V S). We do not indicate thePf graph because2 it will2 be clear2 fromg y and from the context.− For any digraph G = (V; A), one can consider the following system of linear in- equalities x RA; x 0 (1) 2 ≥ %x(v) 1 for every v V (2) ≤ 2 δx(v) 1 for every v V (3) ≤ 2 ix(S) S 1 for every S U, S odd, U (G) (4) ≤ j j − ⊆ j j 2 C whose solution set is denoted by EF(G) and (G) denotes the set of vertex sets of the strongly connected components of G. We doC not distinguish a subset of the edges and its characteristic vector, thus we can observe easily, that the integer solutions of EF(G) are exactly the even factors of G. However, as mentioned above, the polyhedron defined by these inequalities is not integer. After some preliminaries we will define a class of graphs and weight functions where we can state that max cx : x EF(G) is attained on an integer vector. f 2 g EGRES Technical Report No. 2003-05 Section 1. Introduction 3 For a digraph G0, G0 denotes its underlying undirected graph. The maximal strongly connected subgraphs of a digraph having 2-vertex-connected underlying undirected graph are called blocks. If U (G), then let (U) denote the collection of vertex- sets of blocks of G[U]. where2G C [U] denotes theD subgraph of G induced by U.A digraph G0 is said to be bipartite if G0 is bipartite. Throughout the paper, we deal with digraphs s.t. for any U (G), S (U), G[S] is symmetric or bipartite. We V 2 C 2 D consider families ; 2 s.t. = , = U (G) (U), for any S , G[S] is symmetricS andB⊆ for any S S\B, G[S]; is bipartite.S[B [ A2C digraphD is said to be evenly2 S symmetric if there exist families2 satisfying B the above criteria. The pair (G; c) is said to be evenly symmetric if G = (V; A) is an evenly symmetric digraph with respect to and , and c : A R is a weight function s.t. c(uv) = c(vu) if uv and vu belong to AS and BS s.t. u;! v S. More precisely we should say that (G; c) is evenly symmetric with9 respect2 S to the families2 and , but the context will make clear everywhere the families we consider. We letS ? beB the family of maximal members of S T : ;T = s.t. the hypergraph (T; ) is connected : f 9T ⊆ S [T T g It is easy to see that EF(G) equals to the solution set of x RA; x 0 (5) 2 ≥ %x(v) 1 for every v V (6) ≤ 2 δx(v) 1 for every v V (7) ≤ 2 ? ix(S) S 1 for every S U, S odd, U : (8) ≤ j j − ⊆ j j 2 S Moreover, with respect to some objective function, this system and (1)-(2)-(3)-(4) have integer dual optimal solution at the same time. A family of subsets of V is said to be laminar if for any X; Y , at least one of X Y , YF X or X Y = holds. Now we are ready to state a2 result F on dual integrality.⊆ ⊆ \ ; Theorem 1.1. If (G; c) is evenly symmetric and c is an integer vector, then the dual of the linear program max cx : (5) (6) (7) (8) has integer optimal solution % δ f − − − g (λ ; λ ; zS). Moreover, there is one for which S : zS > 0 is a laminar family. v v f g To deal with primal integrality and to exploit the theory of TDI systems, the symmetric edges spanned by some members of are considered as undirected edges S and we take the mixed graph with vertex set V and edge set E = Ed Eu, [ Eu = u; v : uv; vu A; u; v U for some U ; ff g 2 2 2 Sg Ed = uv A : u; v = Eu : f 2 f g 2 g We need the description (in the sense of linear inequalities) of the following polyhedron x(uv) + x(vu) if e = u; v Eu SEF(G) = y RE : x EF(G); y(e) = f g 2 : 2 9 2 x(uv) if e = uv Ed 2 EGRES Technical Report No.
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