Olivier Coudert. On Solving Covering Problems. In Proceedings of Design Automation Conference (DAC), June 1996.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....and minimization of Boolean relations [12] In recent years, several powerful algorithmic techniques have been proposed for solving BCP, allowing dramatic improvements in the ability to solving large and complex instances of BCP. Examples of these techniques include, among others, partitioning [3], limit lower bound [4] negative thinking [8] for unate covering) and linear programming lower bounds [10] Despite these improvements, and as with other NP hard problems, new effective techniques allow in general very significant gains, both in the amount of search and in the run times. The ....

....We should note that throughout the remainder of this paper some familiarity with backtrack search SAT algorithms is assumed. The interested reader is referred to the bibliography (see for example [1, 14] for additional references) Covering problems are often solved by branch and bound algorithms [3, 8, 15]. In these cases, each node of the search tree corresponds to a selected unassigned variable and the two branches out of the node represent the assignment of 1 and 0 to that variable. These variables are named decision variables. The first node is called the root (or the top node) of the search ....

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O. Coudert. On Solving Covering Problems. In Proceedings of the ACM/IEEE Design Automation Conference, June 1996.

....sets, has proven better [18] Logical reductions can be iteratively applied, until they produce a stable covering matrix, called cyclic core.If this is empty, the essential columns found form a minimal solution. Otherwise, the problem can be solved exactly by branch and bound algorithm [11]. In this scheme, a high lower bound and a low upper bound on the optimal solution are extremely effective in terminating useless searches. Most algorithms in the VLSI literature compute lower bounds by maximal independent sets [23] A theoretically better lower bound is presented in [11] but in ....

....algorithm [11] In this scheme, a high lower bound and a low upper bound on the optimal solution are extremely effective in terminating useless searches. Most algorithms in the VLSI literature compute lower bounds by maximal independent sets [23] A theoretically better lower bound is presented in [11], but in practice this is worse than the classical one, apart from ill conditioned or high density problems. Liao and Devadas [15] show that a tighter lower bound may be obtained by formulating unate covering as an integer linear program and relaxing the integrality constraint. This bound is ....

O. Coudert. On solving covering problems. In Proceedings of the 33rd ACM/IEEE Design Automation Conference, pages 197--202, Las Vegas, NV, June 1996. ACM-SIGDA; IEEE, ACM Press.

....have been mapped onto ILP, graph coloring or graph partitioning tasks. All these optimization tasks are intractable for realistic size examples. Thus different heuristics, which generate solutions quickly without guaranteeing optimality, have widely been used. Pau88] Hem92] GDW93] DMi94] Cou96] Cou97] EKP98] KiPo98] YuKu98] The simplest constructive algorithms were chosen because the fast heuristics showed good performance solutions close to the global optimum are generated in matter of seconds. These simple heuristics were incorporated into the allocation and binding step of ....

O. Coudert, "On Solving Covering Problems", Proc. of 33rd ACM/IEEE Design Automation Conf., Las Vegas, NV, USA, June 1997

....of clauses of each variable all the clauses that are known to be satis ed. Hence, each time a clause becomes satis ed, is hidden from the list of clauses of all the variables with literals in . The technique of hiding satis ed clauses can be traced back to the work of O. Coudert in Scherzo [3] for the Binate Covering Problem. The motivation for hiding clauses is to reduce the amount of work required each time a variable x is assigned, since in this case only the unresolved clauses associated with x need to be analyzed. 5 00 00 00 00 00 11 11 11 11 11 00 00 00 00 00 11 ....

.... 1 2 WLS HS W W TS W HS W TS W W HS W W TS TS HS HS W W TS W W W W W W W W W Figure 1: Operation of lazy data structures 4.1. 4 Satis ed Clause and Assigned Literal Hiding One nal organization of adjacency lists is to utilize the same data structures as the ones used by Scherzo [3]. In this case, unsatis ed literals get removed from literal lists in clauses, and satis ed clauses get hidden from clause lists in variables. The utilization of clause and literal hiding techniques aims reducing the amount of work associated with assigning each variable. As will be shown in ....

O. Coudert. On Solving Covering Problems. In Proceedings of the ACM/IEEE Design Automation Conference, pages 197-202, June 1996.

....eliminated, and as a secondary goal, the literal count is reduced as much as possible. This is essentially a weighted covering problem where the constraint is the r.p.r. faults and the cost is the literal count. One of the many heuristic procedures for solving covering problem can be used (e.g. [11]) Note that some factors that actually increase the 1208 IEEE TRANSACTIONS ON COMPUTER AIDED DESIGN OF INTEGRATED CIRCUITS AND SYSTEMS, VOL. 18, NO. 8, AUGUST 1999 Fig. 4. Example of kernel extraction with an observation point. Fig. 5. Example of cube extraction with a control point (same form ....

O. Coudert, "On solving covering problems," Proc. 33rd Design Automation Conf., 1996, pp. 197--202.

....be noted that throughout the remainder of this paper some familiarity with backtrack search SAT algorithms is assumed. The interested reader is referred to the bibliography (see for example [1, 13] for additional references) Covering problems are often solved by branch and bound algorithms [4, 7, 14]. In these cases, each node of the search tree corresponds to a selected unassigned variable and the two branches out of the node represent the assignment of 1 and 0 to that variable. These variables are named decision variables. The rst node is called the root (or the top node) of the search ....

....search can be pruned whenever the lower bound estimate is higher than or equal to the most recently computed upper bound. In these cases we can guarantee that a better solution cannot be found with the current variable assignments and therefore the search can be pruned. The algorithms described in [4, 7, 14] follow this approach. The most commonly used lower bound estimation procedure for BCP is the approximation of a maximum independent set of clauses [3] However, other procedures can be used, namely the ones based on linear programming relaxations [7] Lagrangian relaxations [11] or the ....

[Article contains additional citation context not shown here]

O. Coudert. On Solving Covering Problems. In Proceedings of the ACM/IEEE Design Automation Conference, pages 197-202, June 1996.

....be noted that throughout the remainder of this paper some familiarity with backtrack search SAT algorithms is assumed. The interested reader is referred to the bibliography (see for example [1, 14] for additional references) Covering problems are often solved by branch and bound algorithms [5, 8, 15]. In these cases, each node of the search tree corresponds to a selected unassigned variable and the two branches out of the node represent the assignment of 1 and 0 to that variable. These variables are named decision variables. The first node is called the root (or the top node) of the search ....

....can be pruned whenever the lower bound estimation is higher than or equal to the most recently computed upper bound. In these cases we can guarantee that a better solution cannot be found with the current variable assignments and therefore the search can be pruned. The algorithms described in [5, 8, 15] follow this approach. Several lower bound estimation procedures can be used, namely the ones based on linear programming relaxations [8] or lagrangian relaxations [11] Nevertheless, and for BCP, the approximation of a maximum independent set of clauses [4] is the most commonly used. The ....

[Article contains additional citation context not shown here]

O. Coudert. On Solving Covering Problems. In Proceedings of the ACM/IEEE Design Automation Conference, June 1996.

....be noted that throughout the remainder of this paper some familiarity with backtrack search SAT algorithms is assumed. The interested reader is referred to the bibliography (see for example [1, 13] for additional references) Covering problems are often solved by branch and bound algorithms [5, 8, 14]. In these cases, each node of the search tree corresponds to a selected unassigned variable and the two branches out of the node represent the assignment of 1 and 0 to that variable. These variables are named decision variables. The first node is called the root (or the top node) of the search ....

....can be pruned whenever the lower bound estimation is higher than or equal to the most recently computed upper bound. In these cases we can guarantee that a better solution cannot be found with the current variable assignments and therefore the search can be pruned. The algorithms described in [5, 8, 14] follow this approach. Several lower bound estimation procedures can be used, namely the ones based on linear programming relaxations [8] or lagrangian relaxations [10] Nevertheless, and for BCP, the approximation of a maximum independent set of clauses [4] is the most commonly used. The ....

[Article contains additional citation context not shown here]

O. Coudert. On Solving Covering Problems. In Proceedings of the ACM/IEEE Design Automation Conference, June 1996.

....and minimization of Boolean relations [12] In recent years, several powerful algorithmic techniques have been proposed for solving BCP, allowing dramatic improvements in the ability to solving large and complex instances of BCP. Examples of these techniques include, among others, partitioning [3], limit lower bound [4] negative thinking [8] for unate covering) and linear programming lower bounds [10] Despite these improvements, and as with other NP hard problems, new effective techniques allow in general very significant gains, both in the amount of search and in the run times. The ....

....We should note that throughout the remainder of this paper some familiarity with backtrack search SAT algorithms is assumed. The interested reader is referred to the bibliography (see for example [1, 14] for additional references) Covering problems are often solved by branch and bound algorithms [3, 8, 15]. In these cases, each node of the search tree corresponds to a selected unassigned variable and the two branches out of the node represent the assignment of 1 and 0 to that variable. These variables are named decision variables. The first node is called the root (or the top node) of the search ....

[Article contains additional citation context not shown here]

O. Coudert. On Solving Covering Problems. In Proceedings of the ACM/IEEE Design Automation Conference, June 1996.

....the amount of search. In particular we show how effective search pruning techniques from the SAT domain can be generalized and extended to the BCP domain. Experimental results are presented in Section 5, and the paper concludes in Section 6. 1 Several examples of these techniques can be found in [4, 5, 10, 12]. 1 2 Preliminaries An instance C of a covering problem is defined as follows, minimize n P j=1 c j x j subject to A x b; x 2 f0; 1g n (1) where c j is a non negative integer cost associated with variable x j ; 1 j n and A x b; x 2 f0; 1g n denote the set of linear ....

....We should note that throughout the remainder of this paper some familiarity with backtrack search SAT algorithms is assumed. The interested reader is referred to the bibliography (see for example [1, 17] for additional references) Covering problems are often solved by branch and bound algorithms [4, 10, 18]. In these cases, each node of the search tree corresponds to a selected unassigned variable and the two branches out of the node represent the assignment of 1 and 0 to that variable. These variables are named decision variables. The first node is called the root (or the top node) of the search ....

[Article contains additional citation context not shown here]

O. Coudert. On Solving Covering Problems. In Proceedings of the ACM/IEEE Design Automation Conference, June 1996.

....to cover all the rows corresponds to the minimal set of nodes for which inserting observation points will make all of the core response vectors map to unique observable OCLB output vectors. Column covering is an NP complete problem, but efficient techniques and heuristics exist for solving it [Coudert 96] The next section describes a systematic procedure for forming this matrix for a specified set of fault free core responses. 5. OBSERVATION POINT INSERTION PROCEDURE Consider all of the fault free core response vectors corresponding to the core test set specified by the core vendor. If all the ....

Coudert, O., "On Solving Covering Problems," Proc. of 33rd Design Automation Conf., pp. 197-202, 1996.

....in the minimum column cover of the conflict matrix. This set of literals is compatible with state p, but conflicts with all L states that come before state p in the sequence. Finding the minimum column cover is an NP complete problem, but efficient heuristics and techniques exist for solving it [Coudert 96] An example of forming the conflict matrix and finding the largest cube d is shown in Fig. 4. The conflict matrix is covered by columns 3 and 4, so the cube d has two literals which correspond to the last two bits in state p. When the circular chain reaches state p, the decoding cube d is ....

Coudert, O., "On Solving Covering Problems," Proc. of 33rd Design Automation Conference, pp. 197-202, 1996.

....in Figure 2. 6. Experimental Results In this section we include experimental results of a tool for computing minimum size prime implicants, minprime, and compare it with other ILP solvers, lp solve [2] opbdp [1] and the commercial optimizer CPLEX. Moreover, the binate covering tool scherzo [3] is also evaluated. For this purpose we use a representative set of the satisfiable instances of the DIMACS benchmarks [4] The experimental results, obtained on a SUN 5 85 machine with 64 MByte of physical memory, are shown in Table 1. For each benchmark and for each tool were allowed 3000 ....

....which show greater promise. Acknowledgments The author would like to thank Arlindo Oliveira for 1. In such a situation, each conflict involves all variables and so backtracking is necessarily chronological, to the most recent decision assignment [7] Benchmark min CPLEX lp solve [2] scherzo [3] opbdp [1] min prime time UB time UB time UB time UB time UB aim 50 1 6 yes1 1 50 116.5 50 3,000 2.33 50 0.09 50 0.05 50 aim 50 2 0 yes1 2 50 109.5 50 3,000 5.65 50 0.64 50 0.02 50 aim 50 3 4 yes1 3 50 62.9 50 377.1 50 0.57 50 0.40 50 0.08 50 aim 50 6 0 yes1 4 50 26.9 50 96.8 50 0.73 ....

O. Coudert, "On Solving Covering Problems," in Proceedings of the Design Automation Conference, June 1996.

....leverage off an existing method for fast implicit generation of prime implicants. Moreover, our novel approach can be nicely incorporated into a very efficient implicit minimizer for hazardfree logic. In particular, the approach makes it possible to use the implicit set covering solver of Scherzo [8, 6, 5, 7], the state of the art minimization method for synchronous two level logic, as a black box. Both Espresso HF and Impymin can solve all currently available examples, which range up to 32 inputs and 33 outputs. These include examples that have never been previously solved. For examples that can be ....

O. Coudert. On solving covering problems. In Proceedings of the 33rd Design Automation Conference. ACM, 1996.

....and early discarding of dominated implicants performs efficiently in practice, the final step of finding a minimum cost unate covering problem is often a bottleneck. Even extending the classical branch and bound based covering algorithm with lower bound computation techniques recently presented in [2] do not provide much improvement. As an alternative, we have adapted the two level heuristic minimization program espresso [1] to perform trace driven two level minimization. This new heuristic program, called elp, is described briefly in the next subsection. 3.2 Trace driven two level heuristic ....

O. Coudert. On Solving Covering Problems. In Proc. of the Design Automation Conf., pages 197--202, June 1996.

....checking, circuit delay computation, logic synthesis and functional vector generation [8] among other applications. See [5, 7, 14, 16, 17, 18] for an overview of applications of SAT to EDA. Moreover, SAT can also play a central role in solving instances of binate covering problems (BCP) [6, 9, 10, 11, 12], in particular for those in which the constraints are hard to satisfy, e.g. in computing minimum size test patterns [10] SAT also plays a key role in other domains, including for example Artificial Intelligence [3, 19] and Operations Research [2] Recent years have seen dramatic improvements in ....

O. Coudert, "On Solving Covering Problems," in Proceedings of the Design Automation Conference, June 1996.

....can be exploited by SAT algorithms, after diagnosing the causes of conflicts, by identifying necessary assignments required for preventing conflicts from being identified during the search. In addition, other pruning techniques, as for example the ones commonly used in covering problems [5], can be straightforwardly applied to SAT algorithms. As we show in the next section, with a suitable organization of the ILP algorithm, each of these search pruning techniques can also be applied in solving ILPs. 4 SAT Based Search Algorithms for Solving 0 1 ILPs One of the first SAT based ....

....lower bounds to the current set of variable assignments are estimated. Different lower bound estimation procedures can be used, including linear programming relaxations and lagrangian relaxations [1, 9] In our current implementation we have used the lower bound estimation procedures described in [5], since these procedures are the most suitable for the target application described in Section 5. 4.2 SAT Based Linear Search Algorithm As described earlier in this section, a different organization for a SAT based ILP algorithm is based on P. Barth s approach [2] Let us consider the cost ....

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O. Coudert, "On Solving Covering Problems," in Proceedings of the Design Automation Conference, June 1996.

....delay fault testing and timing analysis. In addition, SAT algorithms can be used as the back end search engines of algorithms for solving other EDA problems. In recent years there has been extensive research work on algorithms for solving covering problems with very promising experimental results [1, 2, 5, 6]. Similarly, new algorithms for SAT have been proposed [7] also with promising experimental results. The purpose of this paper is twofold. First, we note that some of the pruning techniques used for solving covering problems and for solving satisfiability are equivalent. Next, we show how some of ....

....clauses containing that variable can be removed. Notice, however, that this is equivalent to the pure literal rule of the DavisPutnam SAT algorithm [3] 3. 2 Dynamic Partitioning The identification of partitions in covering matrices can be particularly helpful for solving set covering problems [1, 6]. Similarly, the identification of disjoint sub formulas in a CNF formula can be used for simplifying the computation of the solution of a given instance of SAT, since the solution of each sub formula can then be computed separately. It is straightforward to devise a linear time procedure for ....

O. Coudert, "On Solving Covering Problems," in Proceedings of the Design Automation Conference, June 1996.

....can be exploited by SAT algorithms, after diagnosing the causes of conflicts, by identifying necessary assignments required for preventing conflicts from being identified during the search. In addition, other pruning techniques, as for example the ones commonly used in covering problems [5], can be straightforwardly applied to SAT algorithms. As we show in the next section, with a suitable organization of the ILP algorithm, each of these search pruning techniques can also be applied in solving ILPs. f : 0 1 , n 0 1 , x i x i w l 1 . l k ( l 1 ....

....lower bounds to the current set of variable assignments are estimated. Different lower bound estimation procedures can be used, including linear programming relaxations and lagrangian relaxations [1, 9] In our current implementation we have used the lower bound estimation procedures described in [5], since these procedures are the most suitable for the target application described in Section 5. The two ILP algorithms, ls ilp and bb ilp have significantly different organizations. In general, we believe bb ilp to be a better solution since clauses are not explicitly added to the original set ....

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O. Coudert, "On Solving Covering Problems," in Proceedings of the Design Automation Conference, June 1996.

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Olivier Coudert. On Solving Covering Problems. In Proceedings of Design Automation Conference (DAC), June 1996.

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O. Coudert. On Solving Covering Problems. In Proceedings of the ACM/IEEE Design Automation Conference, pages 197-202, June 1996.

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O. Coudert, "On solving covering problems," in Proc. 33rd Design Automation Conf., 1996, pp. 197--202.

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Coudert, O., "On Solving Covering Problems," Proc. Design Automation Conference, pp. 197-202, 1996.

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