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Perfect, Ideal and Balanced Matrices: a Survey

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Perfect, Ideal and Balanced Matrices: a Survey Perfect Ideal and Balanced Matrices Michele Conforti y Gerard Cornuejols Ajai Kap o or z and Kristina Vuskovic December Abstract In this pap er we survey results and op en problems on p erfect ideal and balanced matrices These matrices arise in connection with the set packing and set covering problems two integer programming mo dels with a wide range of applications We concentrate on some of the b eautiful p olyhedral results that have b een obtained in this area in the last thirty years This survey rst app eared in Ricerca Operativa Dipartimento di Matematica Pura ed Applicata Universitadi Padova Via Belzoni Padova Italy y Graduate School of Industrial administration Carnegie Mellon University Schenley Park Pittsburgh PA USA z Department of Mathematics University of Kentucky Lexington Ky USA This work was supp orted in part by NSF grant DMI and ONR grant N Introduction The integer programming mo dels known as set packing and set covering have a wide range of applications As examples the set packing mo del o ccurs in the phasing of trac lights Stoer in pattern recognition Lee Shan and Yang and the set covering mo del in scheduling crews for railways airlines buses Caprara Fischetti and Toth lo cation theory and vehicle routing Sometimes due to the sp ecial structure of the constraint matrix the natural linear programming relaxation yields an optimal solution that is integer thus solving the problem We investigate conditions under which this integrality prop erty holds Let A b e a matrix This matrix is perfect if the fractional set packing p olytop e fx Ax g only has integral extreme p oints It is ideal if the fractional set covering p olyhedron fx Ax g only has integral extreme p oints It is balanced if no square submatrix of o dd order contains exactly two s p er row and p er column The concepts of p erfection and bal ancedness are due to Berge The concept of idealness was introduced by Lehman under the name of widthlength prop erty Berge proved that a matrix is balanced if and only if it and all its submatrices are p erfect or equivalently if and only if it and all its submatrices are ideal n When a p olyhedron Q R only has integral extreme p oints the linear + n program max fcx x Qg has an integral optimal solution x for all c R for which it has an optimal solution Therefore p erfect ideal and balanced matrices give rise to integer programs that can b e solved as linear programs for all ob jective functions Another interesting situation o ccurs when the dual linear program has an integral optimal solution for all integral ob jective functions for which it has an optimal solution let B b e a matrix with integral entries A linear system B x b x is total ly dual integral TDI if the linear program max fcx B x b x g has an integral optimal dual solution y for n all c Z for which it has an optimal solution Edmonds and Giles proved that if the linear system B x b x is TDI then the p olyhedron fx B x b x g only has integral extreme p oints Perfect Matrices Theorem Lovasz For a matrix A the fol lowing statements are equivalent i the linear system Ax x is TDI ii the matrix A is perfect iii max fcx Ax x g has an integral optimal solution x for al l n c R iv max fcx Ax x g has an integral optimal solution x for al l n c f g Clearly i implies ii implies iii implies iv where the rst implication is the EdmondsGiles prop erty and the other two are immediate What is surprising is that iv implies i and in fact that ii implies i Graph theory is very relevant to the study of p erfect matrices In fact his torically this is where the motivation for the study of p erfection originated In Berge prop osed a conjecture ab out graphs In the same pap er Berge also prop osed an elegant weakening of this conjecture to entice re searchers to investigate the topic The weaker conjecture was proved in by Lovasz and is known as the p erfect graph theorem This result implies Theorem and in fact is equivalent to it since the key step of the pro of amounts to showing that iv implies i ab ove Berges conjecture known as the strong p erfect graph conjecture has b een thoroughly investigated but is still unsettled In a graph a clique is a set of pairwise adjacent no des The chromatic number is the smallest number of colours needed to colour the no des so that adjacent no des have distinct colours Since all no des of a clique must have a distinct colour the chromatic number is always at least as large as the size of a largest clique A graph G is perfect if for every no de induced subgraph of G the chromatic number equals the size of a largest clique The p erfect graph theorem states that a graph G is p erfect if and only if its complement G is p erfect The complement of graph G is the graph G having same no de set and complement edge set The connection b etween p erfect graphs and p erfect matrices was stud ied by Fulkerson See also Chapter in Schrijver The next theorem due to Chvatal gives a crisp statement of this connection It is obtained through the concept of cliqueno de matrix of a graph A cliquenode matrix of a graph G is a matrix whose columns are indexed by the no des of G and whose rows are the incidence vectors of the maximal cliques of G Theorem Chvatal A matrix is perfect if and only if its nonredundant rows form the cliquenode matrix of a perfect graph A ma jor op en question is to characterize the graphs that are not p erfect but all their prop er no de induced subgraphs are These graphs are called minimal ly imperfect A hole is a chordless cycle of length greater than three and it is odd if it contains an o dd number of edges Odd holes are minimally imp erfect since their chromatic number is three and the size of the largest clique is two but all prop er induced subgraphs are bipartite and therefore p erfect If G is minimally imp erfect then so is its complement by Lovaszs p erfect graph theorem In particular complements of o dd holes are minimally imp erfect Conjecture Berge The odd holes and their complements are the only minimal ly imperfect graphs Several classes of graphs that arise in applications such as comparability graphs and interval graphs are p erfect In these sp ecial cases fast combina torial algorithms have b een developed for solving the set packing and related problems A comprehensive collection of pap ers on p erfect graphs can b e found in the b o ok edited by Berge and Chvatal Ideal Matrices In a network with source s and destination t a path from s to t is an st path and an edge set disconnecting s from t is an stcut It is easy to see that the pro duct of the minimum number of edges in an stpath by the minimum number of edges in an stcut is at most equal to the total number of edges in the network This lengthwidth inequality can b e generalized to any nonnegative edge lengths and widths w the minimum length of e e an stpath times the minimum width of an stcut is at most equal to the scalar pro duct w This lengthwidth inequality was observed by Mo ore and Shannon and Dun Construct two matrices A and B as follows the columns are indexed by the edges of the network the rows of A are all the incidence vectors of minimal stpaths and the rows of B are all the incidence vectors of minimal stcuts Mo ore Shannon and Dun show that the widthlength inequality implies that b oth A and B are ideal Lehman showed that ideal matrices always come in pairs and that the widthlength inequality is in fact a characterization of idealness Another imp ortant result of Lehman ab out ideal matrices is the following Theorem Lehman For a matrix A the fol lowing statements are equivalent i the matrix A is ideal ii min fcx Ax x g has an integral optimal solution x for al l n c R iii min fcx Ax x g has an integral optimal solution x for al l n c f g Statements i and ii are equivalent by the denition of idealness and ii implies iii is immediate The dicult part of Lehmans theorem is that iii implies ii Here contrary to the situation for p erfection idealness of A do es not imply TDIness of the linear system Ax x This can b e seen using A Q dened as follows 6 C B C B C Q B 6 A Cho osing c the unique optimal dual solution is y 1 1 1 1 Yet it is easy to check that the p olyhedron fx Ax g 2 2 2 2 only has integral extreme p oints Therefore we are led to dene another class of matrices a matrix A has the max ow min cut property MFMC prop erty if the linear system Ax x is TDI Why is this prop erty called MFMC The reason is that when A is the incidence matrix of stpaths versus edges min fcx Ax x g is the classical minimum stcut problem Its dual is the maximum ow problem The max ow min cut theorem of Ford and Fulkerson states that for all nonnegative integral vectors c there exists a maximum ow that is all integral By analogy Seymour coined the term MFMC prop erty for any matrix A with such a dual integrality prop erty By the theorem of Edmonds and Giles if a matrix has the MFMC prop erty then it is ideal Conjecture Conforti and Cornuejols For a matrix A the fol lowing statements are equivalent i the matrix A has the MFMC property ii min fcx Ax x g has an integral optimal dual solution y for n al l c Z + iii min fcx Ax x g has an integral optimal dual solution y for n al l c f g Statements i and ii are equivalent by the denition of the MFMC prop erty and ii implies iii is immediate The dicult part of the conjec ture is to show that iii implies ii This holds in an imp
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