On the separation of partition inequalities

Transcription

1 On the separation of partition inequalities ohamed Didi Biha Ali Ridha ahjoub Lise Slama Laboratoire d Analyse non linéaire et Géométrie Université d Avignon 339 chemin des einajariès Avignon Cedex 9 LIOS-CNRS 618 Université de Clermont-err II Complexe Scientifique des Cézeaux Aubière Cedex rance Abstract for each edge a partition in- Given a graph with nonnegative weights equality is of the form "!$#&% Here '()* defined by '()* problems related to + -connectivity In this paper we will show that if - denotes the multicut of Partition inequalities arise as valid inequalities for optimization decomposes into by 1 -node cutsets then the separation problem for the partition inequalities on can be solved by mean of a polynomial time combinatorial algorithm provided that such an algorithm exists for / where ' are graphs related to / We will also discuss some applications Keywords: separation problem partition inequalities graph connectivity 1 Introduction 0 1 Given a graph a subset of nodes whose cardinality is is called a + -node cutset if the removal of increases the numbers of connected components in If /'*!687 is a partition of we will denote by 9'()* the set of edges with endnodes in different sets of the partition We will also write for :/* or ;=< we will use ; to denote?a@bcd % Given an inequality of the form "!#%G (1) is called a partition inequality Partition inequalities arise as valid inequalities for network design + -connectivity problems Given a vector the separation problem for inequalities (1) is to find an inequality of type (1) violated by if there is any In [Baïou et al (000)] Baïou et al show that the separation problem for inequalities (1) reduces to the minimization of a submodular function thus it is solvable in polynomial time

2 G In this paper we consider the separation problem for inequalities (1) in the graphs that decompose by 1 -node cutsets We will show that the problem can be solved by mean of a polynomial combinatorial algorithm if such an algorithm exists for graphs related to the pieces If the separation problem for inequalities (1) reduces to a minimum cut problem Indeed if a partition 9/'* with! 7 induces a violated inequality then any partition obtained by collapsing two elements also induces a violated inequality So in the sequel we will suppose In [Cunningham (198)] Cunningham shows that if % the separation problem can be reduced to minimum cut problems In [Barahona (199)] Barahona proposes a reduction of the problem in this case to minimum cut problems In this paper we then consider the case % /' If between two given nodes there are multiple edges then by replacing the edges by only one edge associated to this edge the sum of the weights of ' then the separation problem for inequalities (1) in is equivalent to the separation problem in the new graph or this we will consider in the paper graphs without multiple edges Let be a graph Given a partition of we let denote the graph obtained from by contracting the elements 3*'! '( We denote by the nodes of A partition of is said to be a most violated partition if for all violated partition of we have "! &!! where! are the number of elements of respectively Given two sets /( with! we let " Decomposition separation denote the set of edges between In this section we give some properties of the violated partitions We first give a basic lemma Lemma 1 Let / be a most violated partition Suppose that is a node cutset in ' 7 $#%! Suppose that are such that there is no edge linking a node to a node '& 3* +#- # # 3'!1 where )($* /(0* Let 8 3 4:76 ' ( 96 '* Then are violated Proof Note that #!:;##=3 are the cardinalities of Suppose first that both partitions are not violated Hence -#*#-% &!<=#*# 3"(#-% By summing these inequalities we obtain #1#!#1#3G # 7% 8!# 3" # 7*% Since "!#%3 & is violated it follows that!# 3" #7% Hence?% % a contradiction Now suppose that exactly one of the partitions say is violated Thus &!@# # 3G"# % Since is a most violated partition we also have "A-#B% C"!D % By summing these two latter inequalities making some simplifications we obtain?% % yielding again a contradiction Lemma If /()* is a most violated partitions of then 3'! is connected for Proof Suppose for instance that 9 is not connected let ( be a partition of such that Let 1 '( 6 be the partition given by 3*7 for IH KJ '!#83 for Note that oreover! # 3" # %L : "! #=%L which contradicts the fact that is a most violated partitions Lemma 3 Let 1 G be a graph Suppose that N =" P O be such that contains a node cutset " Let * " - A ' = A Q = $ If there is a violated partition in then at least one of the graphs contains a violated partition

3 G G + Proof Let ( be a most violated partitions in By Lemma 3*'! is connected for Wlog we may suppose that " 7! ( Then for is contained either in or in Suppose that there are two elements among '( such that one is in the other in Then we may suppose that ( < 96 G < 7 for some # %=! As is a node cutset in the graph by Lemma 1 the partition obtained from :/(( G by collapsing ( 96 /' ) is violated This implies that the partitions * " *(( of * " *( 96 /'* of are violated < Now suppose that all the elements are for instance in This means that Consider the partition "P * " * of As is violated is violated 3 -node cutsets Consider a graph $ G that decomposes into O =G N by a -node cutset * " / such that * " / " = D = Let N be the graph obtained from by contracting " Let be the node that arises from the contraction The following is easily seen to be true Lemma 4 Let ' be a partition of suppose that $( G' * If ' is violated then / *' In what follows we suppose that there is no violated partition in in this case determining a violated partition in A Let is a violated partition in We will show that if there is any reduces to determining a violated partition in the graph obtained from by adding an edge between " Let N be the graph obtained from by adding an edge between " Consider the problem ' minimize * ' where is a partition of We will denote by solution of () Let! 7 We have the following lemmas Lemma If 3*'! '( Proof Suppose for instance that 7 given by We have!? 3" #% if ( if * ( be defined as & is a most violated partitions of then + A A 86! 3 for Let 1 A7'!3 ( H 1! 3"# %D #! +! 3"# %D # "!#% () an optimal for be the partition contradicting the fact that is a most violated partition Lemma 6 If there exists a violated partition in with at least three elements then there exists a violated partition in with respect to Proof Let ()! be a most violated partitions in with J Let us consider first the case where " are in the same element of 9 say As does not contain violated partition at least one of the elements of intersect Since is a most violated

4 ( ( ( partition hence by Lemma 3'! 7! is connected for it follows that for 7 either or! Hence we may suppose that for some are the elements of in! We claim that In fact if not then would be a node cutset in by Lemma 1 it follows that the partition 84 6 ( of is violated As " "! this implies that LA contains a violated partition A contradiction Consequently that is Let (( where =Q Note that is a partition of Since this partition has the same number of elements same weight as we have that is violated Now suppose that the nodes " are in two different elements Assume for instance that " Let be the elements of in +6 /' those in Consider the partition with!! #7 of given by '1 '1 61 H * " * " for KJ! We have 1 Q ':# %&"!# %D ( ( :# Q! 7 A! #L 7* # %D ( '( :#! 7' Since is the partition of which minimizes () we have 9/(*( "! 7*:#% # "! #! 7 "! #%G Therefore the partition inequality associated with Lemma 7 If is violated with respect to is a most violated partitions in ( 1 such that " is violated oreover there are two different elements of elements to the other Proof Suppose that is not violated Then then belongs to one of the "! #% (3) Since 1 is violated in by Lemma it follows that 1! 7 Thus L! &3" By summing this inequality (3) we obtain % a contradiction So is violated oreover as does not contain violated partition we have that " are in different elements By Lemmas 6 7 there exists a violated partition in violated partition in with respect to The following lemma is easy to prove Lemma 8 Let 1 if only if there exists a Sup- be a violated partition in with respect to pose that " are in the same element of say Then the partition is a violated partition in Now suppose that contains a violated partition let '( violated one Suppose that ( 1 let us assume for instance that is between Then by Lemma 7 is violated belong to different elements of say respectively Let 9/'*!! with #!= 7 such that be a most be the partition of

5 A A H for for J '!! #;J '! #! Lemma 9 is a most violated partition in Proof We first show that is violated Note that :# 1 "! A! #! 7< A"! A"! Since 1 is violated is so Let /!9 be a partition of Let respectively Let! be the number of elements of We have on be the restrictions of respectivelywe will discuss three cases 3!B < Case 1 There is D( * such that Thus! A! " &! A! The second inequality comes from the fact that is a most violated partition in 3!1 < Case There is no =()* such that of As does not contain a violated partition it follows that! #8% implies that!? 8!1#! 3" /!1#! 3"!D %D / 8! %D /! G 8"! ' are in the same element This The third inequality comes from the fact that %=%& Case 3 " are in two different elements We have! 1A!9 #! 7*< " G &! #! 7< " A! As 1 is the most violated partition in it follows that! &! A!! In all cases we obtain that "! than As is an arbitrary partition this implies that is a most violated one By Lemma 9 if we know a most violated partition in that is to say is more violated such that belong to different sets of the partition then we can extend this partition to a most violated one in " 4 A polynomial combinatorial algorithm with respect to The previous lemmas lead to a technic for separating the partition inequalities in the graphs that decompose by 1 -node cutsets If has a node cutset decomposes into then by Lemma 3 looking for a violated partition in reduces to looking for a violated partition in each of the graphs We may thus suppose that decomposes only by -node cutsets or separating the inequalities in we recursively apply the procedure below

6 If has a -node cutset " " decomposes into then apply the following Solve the separation problem for inequalities (1) in If a violated partition is found then extend the found partition to a violated partition in according to Lemma 4 stop If not then solve problem () for Consider the graph obtained from by adding an edge between " the weight vector Solve the separation problem in with respect to partition if there is any to a violated partition in according to either Lemma 9 or Lemma 8 depending on whether belongs to the partition or not extend the found violated If decomposes into ' by 1 -node cutset /' are the graphs obtained from by adding an edge between every pair of nodes consisting of a -node cutset the above procedure yields a polynomial time combinatorial algorithm for separating inequalities (1) in if such an algorithm exists for - oreover by Lemma 9 the algorithm may give a most violated inequality Applications 4 A graph is called + -edge connected (where + is a positive integer) if between any pair of nodes ( there are at least + edge-disjoint paths Given a weight function the + -connected spanning subgraph problem (+ CSP) is to find a + -edge connected subgraph D of spanning all the nodes in such is minimum A homeomorph of (the complete graph on 4 nodes) is a graph obtained from where its edges are subdivised into paths by inserting new nodes of degree two A graph is called series-parallel if it does not contain a homeomorph of as a subgraph In [Didi Biha ahjoub (1996)] Didi Biha ahjoub show that the following inequalities are valid for the polytope associated with the + CSP when + is odd '* + 7! 3 for all partition /'* such that is series-parallel These inequalities are called SP-partition inequalities Series-parallel graphs decomposable by node -node cutsets into graphs consisting of paths Since the separation problem for inequality (4) can be easily solved into these graphs we have that this problem can be solved in polynomial time using a combinatorial algorithm in series-parallel graphs Graphs with no (the wheel on nodes) as a minor decompose by 1 -node cutsets ach piece is either an edge or a diamant (the graph ) [Halin (1981)] It is clear that the separation problem of (1) can be solved in polynomial time (by enumeration) in the pieces Then the problem can be solved using a polynomial combinatorial algorithm on the -free graphs References - Baïou Barahona AR ahjoub: Separating Partition Inequalities athematics of Operations Research Vol No pp Barahona: Separating from the dominant of the spanning tree polytope Operations Research Letters Vol 1pp WH Cunningham: Optimal attack reinforcement of a network AC Vol 3 pp (198) - Didi Biha A R ahjoub: + -edge connected polyhedra on series-parallel graphs Operations Research Letters Vol 19 pp R Halin Graphentheorie II Wissenschaftliche Buchgesellschaft Darmstad 1981 (4)