Integer Solution to a Graph-based Linear Programming Problem

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1 Integer Solution to a Graph-based Linear Programming Problem E. Bozorgzadeh S. Ghiasi A. Takahashi M. Sarrafzadeh Computer Science Department University of California, Los Angeles (UCLA) Los Angeles, CA 90095, USA felib,soheil,majidg@cs.ucla.edu Department of Communications and Integrated Systems Tokyo Institute of Technology Tokyo , Japan atsushi@lab.ss.titech.ac.jp Abstract Integer linear programming problems are in general NPhard. However, some integer linear programming problems have efficient optimization properties by which ILP is solved in polynomial time. In this paper, we study the ILP problem formulated as maxf n j=1 x jja mn x b x = [x 1 x 2 ::: x n ] 2 Z+g. n We propose graph-based sufficient conditions to solve the integer program. We show that the ILP corresponding to a directed acyclic graph is equivalent to formulation of integer budgeting in a graph [7, 6], a well-known problem in area of VLSI CAD optimization. Given an optimal solution to LP relaxation problem of integer budgeting on a given graph, we transform the fractional solution to optimal integer solution. The budgeting problem with many applications has been solved in practice and research using sub-optimal heuristic algorithms. In this work, it is proven that integer budgeting on a given directed acyclic graph is optimally solvable by using the LP solution followed by our efficient algorithm to produce integer solution. We prove that during this transformation, the objective value remains optimal. The complexity of transformation is O(jVj 2 ), where V is set of the nodes in graph G =(V E). Moreover, the LP relaxation problem of integer programs as formulated above which have special graph-based structure can also solve the ILP efficiently in O(n 2 ) for a given constraint matrix A mn. 1 Introduction There are many practical combinatorial optimization problems, which can be formulated and solved using integer linear programming. The general formulation of an integer linear programming (ILP) is maxfc T xja mn x b x b c 2 Z n g where A + mn is an integer matrix. General integer linear programming is NP-hard [1]. There have been remarkable research effort to propose good heuristics and techniques to tackle these problems. However, there are classes of integer programming problems with efficient optimization properties such as network flow problem. Although some of those problems might not be as general and practical as other more difficult integer programs, there are advantages in studying such problems. The efficient optimization property and algorithmic ideas derived from theoretical study of such problems can often be adjusted so as to provide near optimal feasible solutions for difficult problems. Furthermore, many of these well-solved and rather easy problems are used repeatedly as subproblems in algorithms for more difficult problems. A well-known technique to discover the efficient optimization property in a given ILP is to examine whether the linear programming relaxation problem of the integer program can give an integer solution. Linear relaxation problem of an ILP is formulated as maxfc T x : A mn x b x 2 R n +g. Solving the LP will give an optimal solution x which is fractional in general [3]. Based on the classical theorem of Hoffman and Kruscal, if constraint matrix A mn has special structure, the polytope A mn x b will have integer points [2]. In such problems, the extreme points of LP are all integer, hence the solution obtained by simplex method gives an integer solution [3, 4]. The theorem is general, independent of expression of the objective function and vector b. There are set of problems which might not have such special property in their corresponding constraint matrix. However, based on the given objective and vector b, other methods such as rounding might lead to optimal integer solution. In such problems, the extreme points might not necessarily be integer. However, the argument and explanation might be only applicable to particular objective functions or input with special prop- 1

2 erty. In this paper, we propose graph-based sufficient conditions for integer program formulated as maxf n j=1 x jja mn x b x 2 Z n + g such that the optimal integer solution can be obtained efficiently. In the ILP formulation, constraint matrix A mn has to hold special properties in order to represent a directed acyclic graph. One condition is that there has to exist one-to-one correspondence between each path in the graph and each constraint expression in ILP formulation. Then, we show that the problem formulation is exactly equivalent to ILP formulation of integer budgeting problem in a directed acyclic graph, a well-known problem with several applications in VLSI CAD optimization area [8, 9, 6, 7]. Each circuit is represented by a directed acyclic graph. In the directed acyclic graph, there is a latency associated with each node. Timing, a primary objective, is related to total latency of the nodes along the longest paths in graph. Assigning budget to each node allows the extra latency each node can take under the constraints of edge dependency and timing constraint. Increasing the feasible potential latency in each node of graph enables designers to focus on optimization of secondary objectives rather than timing while timing constraint is met. Examples of secondary objectives are power minimization, area minimization, etc. In general, the variables in budget management problems are some characteristics of components in a given library which inherently is discrete rather than continuous. The budgeting problem in a graph is well studied in theory and practice and is widely used in today industry and research [6, 7]. There are heuristic algorithms in literature and industry to solve this problem such as MISA [6] and ZSA [7] algorithms. In different applications the goal is to maximize the value of an expression, which is a function of budgets associated with nodes in the graph. None of the proposed algorithms can solve the budgeting problem optimally. In this paper we focus on linear objective of budgeting with vector c = I. To best of our knowledge, there is no optimal algorithm proposed for budgeting problem with linear objective even when c = I. Hence, up to now, the complexity of this widely-used problem has been an open question. In this paper, we propose our novel efficient graph-based transformation technique to produce optimal integer solution from optimal LP solution. we prove that this version of budgeting problem can be solved optimally in polynomial time by transformation of LP relaxation solution to integer solution while objective value is still optimal. The rest of the paper is organized as follows: In Section 2, the sufficient conditions and special property in constraint matrix of the targeted ILP are proposed. In Section 3, ILP formulation of integer budgeting problem on a graph is formally defined and properties of constraint matrix is studied. In Section 4, we propose a set of sufficient conditions and our method to have a feasible reassignment of budgets among the nodes in a given graph. In Section 5, we show that by budget re-assignment on the solution of LP relaxation of integer budgeting problem, integer solution with optimal objective value can be obtained. Finally, conclusion and outline of some future directions are presented in Section 6. 2 Integer Solution to LP Relaxation In area of linear programming theory, there has been a deep study on the linear programs that automatically have optimal integer solutions. In particular, it is the case for network flow problems. The following claims and propositions provide the sufficient conditions for a linear programming problem in order to yield the optimal integer solution (ILP). Totally Unimodular Matrix (TU): A matrix A is totally unimodular (TU) if every square sub-matrix of A has determinant +1, ;1, or 0 [4, 5]. Lemma 1 If matrix A is TU, the linear programming relaxation can solve the ILP [2, 4]. A sufficient condition for matrix A =(a ij ) mn to be totally unimodular, proposed by Heller and Tompkins [4], is as follows: Lemma 2 A matrix A mn =(a ij ) mn is TU if a ij 2f;1 +1 0g, 8a ij 1 i m 1 j n. Each column contains at most two non-zero coefficients, i.e. m i=1 ja ijj2). There exists a partition (M 1 M 2 ) of the set M of rows such that each column j containing two nonzero coefficients satisfies i2m1 a ij ; i2m2 a ij = 0 [4]. By total unimodularity (TU) of coefficient matrix every extreme point of LP relaxation is integral regardless of objective function and integer vector b. In this paper, we propose a new graph-based sufficient condition to show that the LP relaxation of integer programming as formulated below can give optimal integer solution. The ILP formulated as: maxf n j=1 x j jax b x 2 Z n +g (1) can be solved if there is a one-to-one correspondence between each row of A and a path in a directed acyclic graph G =(V E). That is a ij = 1 if node v j 2 V is along the i th path of the graph otherwise a ij = 0. Our proposed 2

3 sufficient condition on existence of optimal integral solution to LP relaxation is based on a directed acyclic graph G =(V E). Matrix A and variables x are representation of a directed acyclic graph. The following theorem outlines a set of sufficient constraints such that the ILP formulation represents a directed acyclic graph. Theorem 1 ILP max f n j=1 x jja mn x b x 2 Z n + g with A mn =(a ij ) mn can be solved optimally in polynomial time, if there exists a directed acyclic graph G=(V,E) such that, a ij 2f0 1g, 8a ij 1 i m 1 j n, there is a one to one correspondence between each variable x j and a node v i 2 V, There is a one to one correspondence between each constraint n j=1 a ijx j b i and a directed path P i in the graph. That is there is a one to one correspondence between variables x j with non-zero coefficients and the nodes on the directed path P i in graph G, and n j=1 (1 ; a ij)=b i for each i = 1 ::: m and ε i 2 Z +. In the next section we will discuss the relation between the last condition outlined in Theorem 1 and graph property. Matrix A corresponds to directed acyclic graph G. The number of nodes in graph G, jvj is n, where n is the number of variables in ILP formulation. m, the number of rows in matrix A corresponds to the number of paths in the graph. The number of paths in a graph can be O(2 n ) in the worst case. Since we are given a matrix A mn (fixed m and n), the existence of exponential relation between m and n does not lead to exponential growth of the analysis. In a directed graph G =(V E), let P(G) is a directed path in the graph. Assume that x j in ILP formulation represents the budget associated with node v j in graph G =(V E) and T is the upper bound on total budget along the path. This is a well-known problem on directed acyclic graph called integer budgeting problem. For a given graph G =(V E), LP formulation of budgeting problem based on the paths of the graph is maxf j x j g, subject to: vi 2P i x j T for each path P i in graph G (2) For a given directed acyclic graph G =(V E), there are exponential number of paths. Hence this formulation of linear programming for budgeting problem is not efficient. In the next section we formally define the integer budgeting problem in the widely-used efficient formulation. However, the aforementioned formulation can be used to examine whether a given constraint matrix and variables of a given ILP can be a representation of paths and budgets of the nodes in a directed acyclic graph. We propose sufficient conditions to prove that the linear programming relaxation problem of integer budgeting problem can give optimal integer solution. According to Theorem 1 our arguments can be applicable to more general ILP problems rather than only integer budgeting problem. 3 Integer Budgeting in a DAG In a given directed acyclic graph G =(V E), associated with each node v i, there is a delay variable d i > 0 and budget variable b i. 1 d i is the latency associated with node v i. If inputs to node v i are ready at time t, the output of node v j is ready at time d i +t. b i is the extra potential delay the node can accept. Budget is given to each node to improve other properties and objectives rather than delay. Improve of such properties are realized with the cost of increase in latency of the nodes. In a directed graph G, the edge e ij is incident to node v j and incident from node v i. Edge e ij is called an outgoing edge with respect to node v i and an incoming edge with respect to node v j. V I (i) is the set of incoming edges to node v i. V O (i) is the set of outgoing edges from node v i. Primary inputs (PIs) are the nodes with no incoming edges. Primary outputs (POs) are the node with no outgoing edges.the following are some basic definitions corresponding to a directed acyclic graph G. arrival time of v i : Arrival time at node v i is defined as the maximum of total delay and budget assigned to the nodes along a path from PI to node v i among all the paths from PIs to node v i. If input to primary input of graph is ready at time 0, the output of node v i is ready at a i which can be calculated as a i = max v j 2V I (i) a j + (d i + b i ), a i = 0 for v i 2 PI. Arrival time at a primary output is maximum summation of budget and delay associated with each node along the path from primary input up to primary output. Arrival time at each primary output cannot exceed a fixed value, say T. This is referred as required time at primary outputs. Hence this constrains the amount of budget which can be assigned to each node. Budgeting problem can be formally defined as follows: Budgeting formulation: Budgeting problem on a directed acyclic graph G =(V E) with given d i associated with each node v i and given required time T can be formulated as follows: Objective: Maximize vi 2V b i 1 b i represents budget associated with node v i in graph G and is not related to constant b i in general ILP formulation. To avoid confusion, from now on in this paper, b i is budget of the node. 3

4 Subject to: a j a i + b j + d j 8e ij 2 E (3) a i T 8v i 2 PO (4) a i = 0 8v i 2 PI (5) Although requited time at primary outputs and arrival time at primary inputs can be different, for simplicity, we assume that arrival time at each primary input is zero and required time at primary outputs is T. The budgeting problem is assignment of budget b i to each node in the graph such that the total budget is maximized. In integer budgeting problem, there is additional constraint of: d i b i T 2 Z + 8v i 2 V : (6) The last sufficient condition in Theorem 1 implies a directed acyclic graph with d i = 1 for each node v i in graph G. The required time at primary output,t, nodes is set to jv j. In a general graph, this is equivalent to existence of integer positive solution to equation set T I m ;A mn d m + ε m = b. d i is a constant value associated with each node. T is a scalar variable and ε is the slack variable associated with conversion of inequalities in constraint matrix to equalities. Constraint matrix A corresponding to abovementioned LP formulation of budgeting problem is as follows. Variable x j corresponds to a j if 1 j n and corresponds to b j if n < j 2n. Each row index corresponds to an edge in graph G. Assume A =(a ij ) mn m. For 1 j n, a ij = 1 if edge e i is incident from node j and a ij = ;1 if edge i is incident to node j (e ij is incoming edge to node v i ). Otherwise it is set to zero. For n < j 2n, a ij = 1 is edge i is incident to node j ;n. We observe that the linear programming relaxation of integral budgeting for a given directed path holds the sufficient condition to give optimal integer solution. Theorem 2 The linear programming relaxation of integer budgeting problem gives optimal integer solution if the input graph is a directed path. Proof: matrix A corresponding to budgeting problem formulation holds the sufficient conditions outlined in Lemma 2 graph G is a directed path. Hence the optimal integral solution can be obtained from LP relaxation solution. The aforementioned sufficient condition does not necessarily hold for any directed acyclic graph rather than a directed path. In the next section, we prove that the integral budgeting problem can be solved optimally in polynomial time, using the solution of the linear programming relaxation problem. 4 Budgeting Assignment in a DAG In this section, we first define the maximal budgeting on a given directed graph G =(V E) with required time T at primary outputs. T is an important parameter. Arrival time of any node cannot exceed T. Otherwise the dependency constraints in Equation 3 are not satisfied. Most of the definitions, claims and discussions in this paper are based on a given fixed T on graph G. Maximal budgeting is defined as follows: Maximal Budgeting Graph (G B m ): B m is a feasible solution to budgeting problem on a directed acyclic graph G. B m with associated objective value, jb m j, is called maximal budgeting if no more budget can be given to any node while the budget of any other node does not decrease. The maximum solution B is also a maximal solution. Maximal budgeting solution B m can be obtained by applying different algorithms such as MISA algorithm [6], ZSA algorithm [7], etc. We then propose a budget re-assignment method on a given maximal budgeting. A solution B LP of LP relaxation problem of budgeting problem is a maximal budgeting. The solution B LP is fractional. We show that by applying budget re-assignment on graph (G B LP ) an integer maximal budgeting can be obtained in O(jV j 2 ). Note that during this transformation, the total objective value can change. However, we prove that if the LP solution B LP is the optimal LP solution, i.e. B, integer solution resulted from budget re-assignment holds the optimal objective value. Hence an integer optimal solution for budgeting problem is obtained. The followings are some basic definitions used in this section: required time of v i : r i = min v j 2V O (i) (r j ; (d j + b j )). r i = T for v i 2 PO. T is required time at primary outputs in graph G. slack of v i : s i = r i ; a i a-slack of e ij : ε aij =(a j ; (d j + b j )) ; a i, e ij 2 E. r-slack of e ij : ε rij =(r j ; (d j + b j )) ; r i, e ij 2 E. Critical Edge: Edge e ij is said to be critical if the a-slack value and r-slack value associated with edge e ij are zero. A path in a graph which includes only critical edges is called critical path. The following two lemmas can be easily derived from the abovementioned definitions: Lemma 3 In a directed graph G, if e ij 2 E and s i = s j, then ε aij = ε rij = ε ij. 4

5 Lemma 4 If in (G B m ), the slack of each node is zero, B m is a maximal budgeting. Non-critical edges are referred to as ε-edges. According to Lemma 3 the a-slack and r-slack of a ε-edge in (G B m ) are equal, that is ε ij = ε aij = ε rij, 8e ij 2 (G B m ). Lemma 5 In a maximal budgeting (G B m ), each node (except PIs and POs) has at least one critical incoming edge and at least one critical outgoing edge. Proof: By the way of contradiction, we assume that there is no critical outgoing edge from v i. v i is not a primary output. There has to exist at least one outgoing edge from v i. If there are k non-critical outgoing edges from node v i then ε ij, j = 1 ::: k is not zero. The slack of each node is zero. Therefore min(ε ij ), j = 1 :: k can be added to the budget of node v i while all the arrival time and required time constraints for the whole graph are met. Hence we get more budget and this contradicts the definition of maximal budgeting. Similar argument can be applied to prove that at least one incoming edge to each node must be critical. Critical Graph (G T B m ): Associated with solution B m, critical graph G T G =(V E) is the graph obtained from the graph G by deleting all non-critical edges in G. G T = (V E T ), E T = E ;fe ij jε ij 6= 0g. In any budgeting on graph G, slack of each node and edge must be non-negative or in other words a i T. This is referred to as feasibility in graph. A graph is budgeting B is not feasible if slack of a node or an edge is negative. Feasible Budget Re-assignment on (G B m ): In a graph G with maximal budgeting solution B m, the budgets of the nodes are changed such that the new budgeting B 0 m is still a maximal budgeting (G B0 m ). Budget re-assignment on graph G transforms the budgeting from solution B m to B 0 m. Feasible β-budget Re-assignment on (G B m ): β-budget re-assignment on (G B m ) is a feasible budget reassignment in which the change of budget in each node is either ;β, +β,or0. After feasible budget re-assignment, the budgeting is maximal and feasible. Assume that in a re-assignment of budget of fβ 0g at each node in graph G, the total amount of change in the budget of the nodes along each critical path is zero. In this case, arrival time at each node v i is changed by k i β. Since budget of each node changes either β or ;β, the change of budget along each critical path from PI to node v i is multiple of β, called k i, k i 2 Z. Theorem 3 presents two sufficient conditions for feasible β-budget re-assignment. Theorem 3 The re-assignment of budget of f0 βg at each node in graph (G B m ) is a feasible β-budget reassignment if the total amount of change in the budget of the nodes along each critical path from PI to PO is zero, and for each ε-edge e il, ε il (k i ; k j ) β, where edge e jl is critical. k i β and k j β are the amount of change in total budget along any critical path from PI to node v i and v j, respectively. i (a) j l ε edge Figure 1: Two Required Conditions for β-budget Reassignment. Proof: We first prove that after the budget re-assignment of fβ 0g, budgeting on graph G, i.e., (G B 0 m ), is feasible and maximal. We need to observe the effect of budget reassignment on arrival time at each node. Assume (G T B m ) is critical graph before budget re-assignment. a j is arrival time at node v j before budget re-assignment. By induction we prove that arrival time a j is a j + k j β after budget reassignment. k j β is total budget re-assignment along the critical paths from PI to node v j.in(g B 0 m ), all the nodes are sorted in topological order. Arrival time of each node is calculated. Arrival time at primary inputs are a PI β or a PI. See Figure 1(a). Assume that up to (including) node v j in graph G (level p), arrival time is changed as claimed, that is by k j β. According to Lemma 5, each node has a critical outgoing edge. Hence, the child of node v j in G T, say node v l is in the next level p + 1. Assume that there is an ε-edge e il incident to node v l as well. Since ε il (k i ;k j )β, arrival time at node v j, a j +k j β is at least as large as arrival time at node v i. Hence Arrival time at node v l is a j +k j β +d l +b l β which can be reformulated as a l + k l β. In the next step, we need to show that the change in arrival time at node v j as formulated above applies for each critical path from PIs to node v j. Look at Figure 1(a). Edges e ij and e lj are both critical in graph (G B m ). After budget re-assignment, arrival time at node v i is a i +k i β. Arrival time at node v l is a l + k l β. Since along the critical path from PI to PO through edge e lj total change in budget is zero, the amount of change in budget from node v j until i (b) l j 5

6 PO is ;k l β β if budget of β changes at node v j. Since the critical path from PI until node v j through node v i to PO is critical, total budget of k i β β ; k l β β = 0. Hence k l = k i. Now assume node v i is a primary output. Arrival time at node v i is a i + k i β. k i β is the total change of budget along the critical paths from PIs to node PO. Due to first condition k i is zero. Hence, arrival time at node v i does not change after budget re-assignment, i.e. feasibility is satisfied (a i T). The arrival time at node v j is a j + k j β. Similar argument can be applied to show that the amount of change in the required time at node v j is r j ; k 0 j β, where k0 j β is the amount of change in the total budget along the critical paths from primary outputs to node j. Slack of node v i after budget change is r i ; a i ; ki 0 ; k i. According to Lemma 4, s i = r i ;a i = 0 since B m is a maximal budgeting. k i β + ki 0 β is equivalent to total change of budget along the critical path through node v j, which is zero. Hence slack of node v j after budget change is still zero. Therefore we have a feasible maximal budgeting. According to Lemma 3, if the budget of β is re-assigned among the nodes under the aforementioned conditions, another maximal solution on graph G is obtained. We show that the budget exchange between two sub-graphs under child-parent relation satisfies the conditions, hence it is a feasible β-budget re-assignment in graph (G B m ). Child (parent): In a directed graph G, edge e ij 2 E and e ij is critical. Node v j is child of node v i. c(v i ) is used to refer to a child of node v i. Node v i is said to be the parent of node v j. p(v j ) is used to refer to a parent of node v j. Parent Relation: If v i and v j have common child, v i p v j.ifv 1 p v 2 ::: p v n, then v 1 p v n. p is an equivalent relation, called parent relation. Child Relation: If v i and v j have common parent, v i c v ; j. Ifv 1 c v 2 ::: c v n, then v 1 c v n. Similar to parent relation, c, called child relation, is an equivalent relation. Lemma 6 v i c v j,iffp(v i ) p p(v j). Proof: If v i c v j, there exists node v l such that v i c v l and v j c v l. Hence, nodes v i and v l share a parent, i.e., p(v i ) p p(v l ). According to transitive property in child and parent relation, it can be shown that p(v i ) p p(v j ). Lemma 7 In (G B m ),ifv i p v j, arrival time at nodes v i and v j are equal; a i = a j. Proof: Nodes v i p v j. Let v k be the child node of nodes v i and v j. Since v i is a parent of node v k, a k = a i + b k + d k. Similarly a k is equal to a j + b k + d k. Hence, a i = a j. If v i and v j do not share a common child, due to transitive property in equivalent parent relation, v i p v k p :::v l p v j, arrival time at v i is equal to arrival time at v j. According to Lemma 5, each node is incident to/from a critical edge. Consider node v i in graph G =(V E). Let S p (v i )=fv j jv i p v j g be a parent set. Each node shares at least one child node with another node in S p. Let v l be a child node of v i. S c (v l )=fv j jv j c v l g is the set of nodes in which each node in the set has a common parent at least with one other node in the set. According to Lemma 6, sets S p (v j ) and S c (v l ) are a pair of sets such that all the child nodes of the nodes in S p are in S c.similarly, all the parent nodes of the nodes in set S c are in S p The sets S p (v i ) and S c (v l ) are called parent-child set (S p S c ) associated with node v i. Parent-child set (S p S c ) is shown in Figure 2. The followings are the propositions regarding the parent-child set in (G B m ). Lemma 8 If nodes v i p v j, there is no directed critical path between v i and v j if 8v i 2 V d i > 0. Proof: Since v i p v j, by Lemma 7, a i = a j. If there is a critical path between v i and v j, then a i cannot be equal to a j according to definition of arrival time and critical edges with assumption of d i > 0. Hence there cannot exist any critical path between any two nodes in the parent set. Lemma 9 If nodes v i c v j, there is no directed critical path between v i and v j if 8v i 2 V d i > 0. Proof: Assume that there is a critical path P ij = fv i :::v k v j g connecting nodes v i and v j. Since v k is p(v j ), v k belongs to the parent set according to Lemma 8. Hence p(v i ) p v k. That is a k = a p(vi ). However, since there is path between v i and v k and delay of each node is non-zero, a p(vi ) < a v i < a vk. Therefore, by the way of contradiction, the path P ij cannot be critical. Lemma 10 In a parent-child set (S p S c ),S p and S c do not intersect if 8v i 2 V d i > 0. Proof: Assume that the two sets intersect at node v k. Since node v k is in set S p, node v k has at least a child in set S c, say node v l. Therefore, there is an edge from node v k to node v l both belonging to S c. This contradicts Lemma 9. Let β-budget exchange in parent-child set (S p S c ) be decreasing the budget of the nodes in S p by β and increasing budget of nodes in S c by β. 6

7 S p (S p,s c ) 1 3 S c Similarly, the budget can be increased by β in parent set and reduced by β in child set. This is called (;β)- budget exchange in (S p S c ). Lemma 11 can be adjusted to be applied for (;β)-budget exchange on parent-child set as well. In this paper, we apply β-budget exchange on a given parent-child set. In the next section, we apply β- budget re-assignment on LP solution which is a maximal budgeting on G in order to obtain integer solution. 2 4 ε-edge critical edge Figure 2: In graph G f, nodes v i and v j share a child (Node v k ) while there is a directed path from v i to v j. Lemma 11 In a given (S p S c ) in (G B m ),ifβ min(ε ij ), where e ij is an ε-edge with v j 2 S c and v i =2 (S p S p ) (incoming ε-edges to S c ), the β-budget exchange is a feasible β-budget re-assignment in (G B m ). Proof: In order to prove that β-budget re-assignment can be applied to a parent-child set, we show that the sufficient conditions mentioned in Theorem 3 are satisfied during budget exchange between a parent-child set. Since there is no critical path between any two nodes in S c or S p, the critical paths in S c [ S p consist of two nodes one in parent-set and the other in child set. Budget of the nodes in S p are reduced by β and budget of the nodes in S c are increased by β. Along each critical path in this subgraph, the total amount of change in budget is zero. There is no change in budget or arrival time at any other nodes outside the parent-child set in graph G. Therefore the first sufficient condition in Theorem 3 is satisfied. The ε-edges can be categorized based on where the two ends of the edges are located. Figure 2 shows all different possible such edges with respect to a given parent-child set in (G B m ). At each parent node v i, the amount of change in arrival time is ;β. At each child node v j, the amount of change in arrival time is zero. Therefore, arrival time at a child node does not change. Hence the criticality of the edges connecting the child nodes to the rest of the graph remain unchanged. Similarly, the criticality of incoming edges to parent nodes are unchanged after budget exchange. That is ε-edges 3 and 2 remains unchanged, hence they remain non-critical. The inequality ε 1 β is satisfied as well. For ε-edge 4, the inequality ε 4 β is held since β ε for incoming epsilon-edges to child set. There also cannot exist any ε-edges between two parent nodes, two child nodes, or between a child and a parent node as well. Hence the second sufficient condition in Theorem 3 is satisfied as well. This ends the proof. 5 Integer Solution of LP Budgeting Problem (G B ) is the solution to linear programming relaxation of integer budgeting problem. B is also a maximal budgeting. Hence, budget re-assignment is applicable to (G B ). In addition, since B is the optimal solution, B m B for any maximal budgeting B m. We define β in β-budget reassignment on graph (G B ) such that the budget of all the nodes become integer. We show that during this transformation from optimal solution to integer solution (B ) 0, the objective value of new solution is equal to jb j. Integral sequence: A sequence of nodes IS n =< v 1 v 2 ::: v n > along a critical path in (G B ) is called Integral Sequence if a 1 a n 2 Z + and a 2 ::: a n;1 =2 Z. Lemma 12 The total budget of the nodes along any integral sequence in (G B m ) is integer. Proof: Since the arrival time of the nodes at the two ends of an integral sequence IS n is integer, j2isn (d j + b j ) 2 Z +. Since each d j is integer, j2isn b j 2 Z +. Corollary 1 The total budgeting on any critical path from PI (Primary Input) to PO (Primary Output) is integral. Assume there are a set of nodes in an integral sequence of (G B ) whose corresponding budgets are not integer. Since the total fractional value of those budgets are integer, budget of the nodes can become integer by re-assigning the fractional budget in the integral sequence. Assume there is a critical path from node v n, at the end of an integral sequence IS n, to some other node v k in graph with fractional budget. Since the total fractional budget on the integral sequence IS n up to node v n is integer, there is no need to apply budget re-assignment between node v n and v k. On the other hand, based on Lemma 12, each node with fractional budget belongs to an integral sequence. Hence, node v k belongs to an integral sequence of nodes with fractional budget as well. Hence, within an integral sequence, it is sufficient enough to re-assign the fractional budget only on the nodes in an integral sequence. On the other hand, in graph G, there are several integral sequences connected to each other. Therefore in re-assigning the budget between 7

8 the nodes, the required conditions in Theorem 3 have to be satisfied in all those sequences. Hence, the goal is to apply budget re-assignment of the fractional budgets on the nodes in graph in (G B ) to obtain integer solution. Since the budget re-assignment needs to be applied between the nodes with fractional value, we reduce the graph (G B ) to graph G f, the fractional adjacency graph defined as follows: Fractional Adjacency Graph : Graph G f is the fractional adjacency graph corresponding to given graph (G B ). The nodes in graph G f are a subset of nodes in graph G that have non-integer (fractional) budget. The edge between two nodes in graph G f represents the existence of a directed critical path between two nodes in graph G such that there is no fractional budget along the path and arrival time of each node along the path is not integer. There is a ε-edge between two nodes v i and v j, if there is no critical path between the two nodes but at least a path with ε-edges along the path. Among all different paths between the two nodes, the minimum of total ε value of the ε-edges along each path is the ε value of the ε-edge in graph G f. i j k i l j k Figure 3: In graph G f, nodes v i and v j share a child (Node v k ) while there is a directed path from v i to v j. Two adjacent nodes v i and v j in graph G f represents the two immediate nodes on a directed critical path in graph G with fractional budget, both belonging to same integral sequence. Although the nodes along the critical path between nodes v i and v j in graph G have integer budget, the arrival time at each of those nodes are fractional. The fractional value at arrival time of the nodes along the critical path are equal to arrival time at node v i. β-budget re-assignment is applied on graph G f such that the budget of all the nodes become integer. Only fractional value of budgets need to be re-assigned in order to obtain integer solution. Hence β is a fractional value less than unit. As described in previous section, feasible budgetreassignment can be applied on a parent-child set on graph G. Similar argument can be applied to graph G f as follows: Lemma 13 In graph G f, if node v i p v j, the fractional values of arrival time at both nodes are equal, i.e., a i ; [a i ]=a j ; [a j ]. Proof: Assume v i p v j. Let v k be the child node of both nodes v i and v j. Arrival time at node v k is equal to fractional value of summation of fractional value of arrival time at node v i and fractional value of budget at node v k. Budget of the nodes along the critical path from v i to v k in graph G are all integer. Similarly, arrival time at node v k is equal to fractional value of summation of fractional value of arrival time at node v j and fractional value of budget at node v k. Hence, a i ; [a i ]=a j ; [a j ]. If v i and v j do not share a child node, due to transitivity in parent relation, we still have a i ; [a i ]=a j ; [a j ]. Lemma 14 If nodes v i p v j in graph G f and there is a directed critical path between nodes v i and v j in graph G, there has to exist at least one node on the path between the nodes v i and v j in graph G. t i j k l Figure 4: In graph G f, v i p v j while there is a directed path from v i to v j. Proof: Assume there is a path between node v i and v j in graph G. Let node v k be the child node of nodes v i and v j. There are two paths from node v i to v k, one is the direct edge e ik and the other is the path (v i v j )(v j v k ). See Figure 3. The fractional value at the node v k from the first path is a i ; [a i ] and from the other path is a i ; [a i ] ; a j +[a j ]. According to Lemma 13, these two values need to be equal. This is possible iff a j ; [a j ]=0 which contradicts that e jk 2 E(G f ). Therefore there has to exit at least one node say v l on the path from v i to v j such that a i ; [a i ]+a l ; [a l ]=0. Similarly if the two nodes v i and v j do not have a same child, we can prove that the total fractional value on the path from v i to v j including v j needs to be integral, i.e. there has to exist at least one node on the path between v i and v j. Look at Figure 4. The set S p (v i )=f jjv j p v i g is the set of nodes in graph G f such that each node shares at least a common child with another node in S c (v i ). The set S c (v i )=f jjv j c v i g is the set of nodes in which each node in the set shares a parent at least with one another node in the set. Lemma 15 Set S p (v i ) and S c (v j ) do not intersect (e ij 2 E(G f )). Proof: Assume that the two sets intersect. Then there is a node v k belonging to both sets. Since node v k is in set t i P m j l k C 8

9 Parent Set S p 1 S p 5 7 S c Child Set S c (S p, S c ) 4 3 ε-edge Figure 5: Parent-Child Set (S p,s c ) in graph G f of graph G. S c (v j ), it has at least one parent, say v l 2 S p (v i ). Therefore there is an edge from node v l to node v k ). On the other hand, v k 2 S p (v i ). That is there is a direct edge between two nodes v k v l 2 S p (v i ). This contradicts Lemma 14. On a given parent-child set in graph G f, we apply β- budget exchange. That is budget of parent nodes are decreased by β and budget of child nodes are increased by β. If fractional budget in graph G f are re-assigned by budget re-assignment on parent-child set, the fractional budget is removed from each parent node and re-assigned to one of its successor in the graph. Hence, the fractional budgets are re-assigned from PIs to POs, in one direction within an integral sequences. There are ε-edges in a given graph G f. In order to have a feasible budget re-assignment on parentchild set, we show that the sufficient conditions outlined in Theorem 3 are satisfied in a given graph G f as well. Lemma 16 β-budget exchange on a parent-child set in graph G f is a feasible β-budget re-assignment if β min(ε ij α), where ε ij is ε-edge. ε ij is an incoming edge to child set. α i is the fractional value at parent nodes. i j S p (α) α j ε-edge p α= α i S c (α) β-budget re-assignment Figure 6: ε-edge incident to a child node in (S p S c ) α. critical edge Figure 7: ε-edges in graph G f with respect to parent-child set (S p S c ). Proof: In Figure 5, a set of parent-child set is shown in a given graph G f. In a budget exchange on the set, there is an alternative β budget exchange along each critical edge in graph G. k i β corresponds to total change of budget along the critical paths from PI to node v i. At each child node v i, the corresponding k i is zero. At each parent node v i, the corresponding k i = ;1 when budget in parent set is decreased by β. Hence the first sufficient condition in Theorem 3 is satisfied. We prove that as long as β ε, the budget exchange is a feasible β-budget re-assignment on a given graph G. There are 8 possible type of ε-edges with respect to (S p S c ). At each edge, we check if the inequality defined in Theorem 3 is satisfied after budget exchange or not: ε-edges 2 and 4 will not change since the arrival time at the child set and incoming edges to parent set are not affected by budget exchange. At ε-edge 1, the inequality ε (;1 ; 0)β is True for any β > 0. At ε-edge 5, the inequality ε (;1 ; (;1))β = 0is True for any β > 0. At ε-edge 6, the slack will not change since the arrival time at child node does not change. At ε-edge 8, the inequality ε (;1 ; (0))β is True for any β > 0. At ε-edges 7 and 3, the arrival time at the ε-edges do not change. Therefore ε (0 ; (;1))β. Since β < 1, for ε 1, ε > β is True. Assume ε < 1. In Figure 6, α i and α j are the fractional value at nodes v i and v j, respectively. The value of ε is: 9

10 ε jp = αi ; α j if α i > α j 1 + α i ; α j if α i < α j : (7) When α i < α j, ε > α i. Since β α i, ε > β. Hence the inequality is held. On the other hand, if α i > α j, the inequality is held since β ε. Thereofore both sufficient conditions in Theorem 3 are staisfied. If β is less than the fractional value of budget in parent nodes, after budget re-assignment, arrival time at parent node is reduced by β. Hence, if β is equal to fractional value of the arrival time, arrival time at all parent nodes become integer. On the other hand, β need to be at most as large as the minimum available budget in parent nodes. Lemma 17 Let (S p S c ) be a parent-child set with α p, the fractional value at the arrival time at the parent nodes. Assume that α p is the smallest fractional value of arrival time at all the nodes in graph G f. β-budget exchange of β = α p from parent nodes to child nodes is a feasible budget re-assignment. Proof: In order to be able to re-assign budget of β from parent nodes, each parent node must have at least budget of β, i.e. 8v i 2 S p b i β. Assume that there is a node v j 2 S p such that b j β, hence b j α p. In this case, arrival time at parent of node v j is α p ;b j < α p and this contradicts the fact that fractional value of arrival time at no other nodes other than parent nodes can be as small as α p. Hence each b i β. Next, consider ε-edges connected to (S p S c ). According to Lemma 16, only two types of ε-edges, ε-edges 3 and 7 as shown in Figure 2 are under the condition that ε value of such edges have to be larger than β. Since β < 1, all ε 1 are safe edges. Assume ε < 1. Since α p is the smallest fractional value of arrival times, the arrival time at the node incident from ε-edge, say α x, is greater than α p. Hence, ε = 1 + α p ; α x ) ε > β. Therefore ε-edges with slack, less than unit are safe as well. This ends the proof that the budget re-assignment is feasible. After budget re-assignment on parent-child set (S p S c ), arrival time at each parent node becomes integer with β = α p. If budget of any node in parent set or child set becomes integer, the node is removed from G f. In this budget re-assignment, an integer budget of any node in graph G never becomes fractional. Hence no node is added to graph G f after budget re-assignment. Since arrival time at parent node becomes integer, all the edges connecting the parent nodes to the child nodes are removed from graph G f. Similarly no edge is added to graph G f after budget re-assignment. Assume that generating the parent-child sets and applying budget re-assignment on the parent-child sets in graph G f continues. A more important fact is that after budget re-assignment, the parent nodes do not have any outgoing edges in graph G f. Hence, the corresponding nodes cannot become parent nodes anymore. Therefore we have the Lemma as stated below: Lemma 18 Each node in graph G f can only be once in a parent set during sequential parent-child budget reassignment. Note that after each β-budget exchange, the outgoing edges of parent nodes are removed. No more outgoing edges are added to parent nodes in G f since arrival time at parent nodes are integer. On the other hand, integer budget of a node never becomes fractional after any β-budget exchange. Since each node can only once appear in a parent set, the number of parent-child which can be generated followed by budget re-assignment on each set is O(jV j), where V is set of nodes in graph G. Theorem 4 Sequentially generating parent-child set followed by β-budget re-assignment in the order of increasing fractional value of arrival time at parent nodes of the parent-sets with β = α p,g f = /0 in O(jV j). If graph G f = /0, the budget of all the nodes in graph G are integer. Hence, Theorem 4 shows that a maximal integer solution can be obtained from LP solution using β-budget exchange on graph G f. Lemma 19 In graph G f corresponding to (G B ), js p (v i )j = js c (v j )j if e ij 2 G f. Proof: Assume that there are more number of nodes in one of the sets, say S c (u). In that case, once the minimum budget, say f min, is removed from all the nodes in set S c (u) and added to the budget of the nodes in S p (v), the total budget obtained after this move is js p (v)j f min > js c (u)j f min. This contradicts the optimality of budget in (G B ). Similar discussion applies when the other set is larger. Therefore js c (u)j = js p (v)j. Theorem 5 In any feasible β-budget re-assignment on parent-child set (S p S c ) in graph (G B ), the total budget does not change. According to Theorem 5, during each budget reassignment the amount of total budget will not change. Hence the solution is still optimum after applying the budget re-assignment on (G B ). Each parent-child set construction takes O(jEj), budget re-assignment takes O(jEj). Updating graph G f takes O(jEj). This repeats O(jV j) times. However, by amortized analysis we see that the complexity of O(jEj) during the process applies to a set of edges during the current iteration and then those 10

11 edges are removed from graph G f before the next budget re-assignment. Hence the total complexity is O(jEj) = O(jV 2 j). The result is transformation from solution B to a new solution (G (B ) 0 ) in which integer budget is assigned to each node while objective value does not change, i.e., jb j. Theorem 6 The solution to linear programming relaxation problem of integer budgeting problem on graph G = (V E) can be transformed to equivalent integer solution in polynomial time (O(jV j 2 ). Corollary 2 The linear programming relaxation of ILP maxf n i=1 x jja mn x b x 2 Z n + g such that gives integer solution if matrix A mn and variable vector x hold the sufficient conditions in Theorem 1. 6 Conclusion and Future Work In this paper we studied a class of ILP problems formulated as maxf n i=1 x ijax b x 2 Z+g. n Although ILP problems are in general NP-hard, there are efficient optimization properties in some integer programs. In this paper we study the optimization property of aforementioned ILP problem on a directed acyclic graph. It is observed that the ILP formulation on a graph is equivalent to integer budgeting problem, widely used in VLSI CAD optimization area. Using optimal solution to LP relaxation of budgeting problem, we transform the solution to optimal integer solution. For this purpose, we introduce budget re-assignment in a directed acyclic graph. Particularly we define parent-child set in graph on which budget re-assignment can be applied. Using this technique on the graph with budget obtained from optimal LP solution, we re-assign the fractional value of budget associated with the nodes in the graph such that budget of each node becomes integer. We prove that during this transformation, objective value from optimal LP solution does not change. Hence an optimal integer solution is obtained. This transformation takes O(jV j 2 ) where V is set of the nodes in directed acyclic graph G. As a future work, we are planning to develop an efficient polynomial algorithm to solve this class of graph-based ILP problems rather than using the LP solution. Also study on the structure and property of constraint matrix A and relation between A and graph topology are other ongoing research related to this work. [2] A.J. Hoffman and J.B. Kruskal. Integral boundary points of convex polyhedra. H.W. Kuhn and A.W. Tucker, eds. Linear Inequalities and Systems, Princeton University Press, Princeton, N.J., pp , [3] V. Chvatal. Linear Programming. Freeman Publisher, New York, [4] L. A. Wolsey. Integer Programming. In Wiley- Interscience Publisher, pg , [5] P. D. Seymour. Decomposition of Regular Matroids. In Journal of Combinatorial Theory, B28, pp , [6] C. Chen, E. Bozorgzadeh, A. Srivastava, and Majid Sarrafzadeh. Budget Management with Applications. In Algorithmica, vol 34, No. 3, pp , July [7] R. Nair, C. L. Berman, P. S. Hauge, E. J. Yoffa. Generation of Performance Constraints for Layout In IEEE Transactions on Computer-Aided Design, Vol. 8, No. 8, pp , August [8] M.Sarrafzadeh, D. A. Knol, G.E. Tellez. A Delay Budgeting Algorithm Ensuring Maximum Flexibility in Placement In IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems,, Vol. 16, No. 11, pp , Nov [9] C. Kuo and A. C.-H Wu. Delay Budgeting for a Timing- Closure-Design Method, In International Conference on Computer-Aided Design, pp , References [1] M. R. Garey, and D. S. Johanson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman and Company,