M. Yannakakis. Expressing Combinatorial Optimization Problems by Linear Programs. Proc. 20th ACM STOC, 223--228, 1988. 109  Home/Search   Document Not in Database   Summary   Related Articles   Check

....constraints, or a separation oracle. Let dr denote the maximum check (right) degree of the code. As stated, LCLP has O(n m2 dr ) variables and constraints. For turbo and LDPC codes, this complexity is linear in n, since dr is constant. For arbitrary linear codes, we use results of Yannakakis [Yan91] to obtain a characterization of LCLP with O(n md r ) O(n 3 ) variables and constraints. We leave these details for the complete version of the paper. V. Fractional Distance A classical quantity associated with a code is its distance, which for a linear code is equal to the minimum ....

....solver. For our problem, we are interested in the polytope Q, and the special vertex 0 Q. In order to run the above procedure, we must provide a small representation of Q = P . The following definition of Q in terms of constraints on f was derived from the parity polytope of Yannakakis [Yan91] We first enforce 0 # f i # 1 for all and then for all j # J , T N(j) T odd, we require: i#T f i i#(N(j) T ) 1 f i ) # N(j) 1. 7) Thus, the number of facets in Q has an exponential dependence on the check degree of the code. For an LDPC code, the number of ....

Mihalis Yannakakis. Expressing combinatorial optimization problems by linear programs. Journal of Computer and System Sciences, 43(3):441--466, 1991.

....work with subfamilies of linear relaxations. An integrality gap result for a large subfamily of relaxations may then be viewed as a lowerbound for a restricted computational model, analogous say to lowerbounds for monotone circuits [17] and for proof systems [4] An example is Yannakakis s result [21] that representing TSP (the exact version) using a symmetric linear program requires exponentially many constraints. In this paper we prove nonexistence of tighter relaxations for Vertex Cover among three fairly general families of LPs. For all families we prove an integrality gap 2 o(1) ....

M. Yannakakis. Expressing combinatorial optimization problems by linear programs. J. Comput. Syst. Sci., 43 (1991), pp. 441--466. 10

....size of a set of submatrices covering the ones of a Boolean matrix M so that each is covered at most r times. Then 14 (1 r (M) Rank(M) For the other claim again simulate A by a one way k ambiguous nondeterministic protocol with size A messages. Results of [KNSW94] see also [L90] [Y91]) imply that a k ambiguous nondeterministic one way protocol with m messages can be simulated by a deterministic two way protocol with communication log(m 1) Delta k Delta log(m 2) Thus cc(L) log(size A 1) Delta k Delta log(size A 2) log ( size A 2) and c) follow. ut ....

Yannakakis, M.: Expressing combinatorial optimization problems by linear programs. Journal of Computer and System Sciences 43 (1991), pp. 223-- 228.

....the k ABP complexity of f , B k (f) is the minimum, over all ABPs that compute f , of the size of the k th level of the ABP. The k monotone ABP complexity of f , B k (f) is the minimum, over all monotone ABPs that compute f , of the size of the k th level of the ABP. Definition 6 (Yannakakis [16]) For a real matrix M , the positive rank of M , rank (M ) is the minimum integer t such that there exist nonnegative matrices A of dimension n by t and B of dimension t by m such that M = AB. Lemma 5 For every homogeneous function f of degree d and all 0 k d, B k (f) rank (M k ....

....decomposition of M k (f) giving B k (f) 2 For a boolean function b(x; y) denote by C(b) the communication complexity of b and by rank(b) the rank of the real matrix associated with it. Lovasz and Saks (see [7] asked whether C(b) log rank(b) for every boolean function b. It is known ([16], see also [7] that the answer to this question is positive if and only if log(rank (M ) log(rank(M ) for every real matrix M with 0 1 entries. Define an algebraic function to be simple if all of its coefficients are zero or one. Theorem 5 The following are equivalent: 1. For every ....

M. Yannakakis, Expressing combinatorial optimization problems by linear programs, 1988.

....problems (Edmonds [11] are easily shown to be of this type. Note that, if the blossom inequalities were natural split cuts, as an immediate corollary one would get a compact (polynomial size) LP formulation of the matching problem thus solving a long standing open problem (see Yannakakis [31]) Thus, the natural split cuts should be placed immediately above the lift and project cuts in Figure 1. In the next section we examine the situation with the Symmetric Travelling Salesman Problem. 4 The Symmetric Travelling Salesman Problem Given a complete undirected graph G = V; E) with ....

....inequalities with that handle. These separation results are new. At rst sight it might be thought that these corollaries rely on the use of the ellipsoid method (which, while polynomial, is inecient in practice) However, by considering a compact LP formulation for the subtour polytope (see e.g. [31]) one can derive the same results without requiring the ellipsoid method. Moreover, the standard simplex method provides a practically useable, though theoretically non polynomial, method for separation. To use the separation result in a practical cutting plane algorithm for the STSP, a heuristic ....

M. Yannakakis, \Expressing combinatorial optimization problems by linear programs", J. Compt. Syst. Sci., vol. 43, pp. 441-466, 1991. 17

....or those with linear objective functions) can be written as an integer program, provided that it has a finite solution space [Iba76] though exponentially many constraints are needed. Yannakakis has explored finding small linear programming formulations for many NP hard optimization problems [Yan91] There is also some previous work regarding the approximation of integer programs subject to syntactic restrictions. Hochbaum et al. have shown that integer programs with only two variables per inequality are two approximable[HMNT93] In Section 7, we show that extending this to three ....

M. Yannakakis. Expressing combinatorial optimization problems by integer programs. Journal of Computer and System Sciences, 91:441 -- 466, 1991.

....size of a set of submatrices covering the ones of a Boolean matrix M so that each is covered at most r times. Then (1 r (M) r Rank(M) For the other claim again simulate A by a one way k ambiguous nondeterministic protocol with size A messages. The results of [KNSW94] see also [L90] [Y91]) imply that a k ambigu ous nondeterministic one way protocol with m messages can be simulated 14 by a deterministic two way protocol with communication log(m k 1) k log(m 2) Thus cc(L) log(size k A 1) k log(size A 2) log 2 ( size A 2) k ) and c) follow. ut ....

Yannakakis, M.: Expressing combinatorial optimization problems by linear programs. Journal of Computer and System Sciences 43 (1991), pp. 223-228.

....both practical and theoretical terms. In addition, communication complexity has found surprising applications in the complexity of Boolean circuits (Karchmer and Wigderson [14] Raz and Wigderson [19] Boolean decision trees (Hajnal, Maass, and Tur an [13] combinatorial optimization (Yannakakis [23]) VLSI (Aho, Ullman and Yannakakis [3] Lipton and Sedgewick [16] Mehlhorn and Schmidt [18] Yao [25] and pseudorandom number generators (Babai, Nisan, and Szegedy [5] Nearly all previous work on the communication complexity of various problems has focused on their communication ....

M. Yannakakis, Expressing combinatorial optimization problems by linear programs, in Proceedings of the Twentieth Annual ACM Symposium on Theory of Computing, Chicago, IL, May 1988, pp. 223--228.

....connections. So y ij =1if cluster V i is connected to cluster V j and y ij = 0 otherwise. The convex hull of all these y vectors is generally known as the spanning tree polytope (on the contracted graph with vertex set V 1 , Vm which we assume to be complete) Following Yannakakis [9] this polytope can be represented by the following polynomial number of constraints: # i,j y ij = m 1 y ij = # kij # kji , for 1 # k, i, j # m and i #= j (6) # j # kij =1, for 1 # k, i, j # m and i #= k (7) # kkj =0, for 1 # k, j # m (8) y ij ,# kij # 0, for 1 # ....

M. Yannakakis, Expressing combinatorial optimization problems by linear programs. Journal of Computer and System Sciences 43, 441-466 (1991). 10

....small (exactly one for UP and polynomially bounded for FewP) The class UP was introduced by Valiant [16] in 1976 and was intensively studied. It is not known whether UP is strictly larger than P, although this is believed to be the case, for related results see [18] In contrast, Yannakakis [19] proved that in communication complexity this 4 restriction is as severe as can be, namely, that P cc = UP cc . The somewhat less restrictive class FewP was introduced by Allender [1] and studied in [5, 6] As it contains UP , it is at least as hard. A corollary to our main result (Corollary ....

....c(f) rk(M f ) has been a major tool for proving explicit lower bounds for specific functions, but it is not known how good it is in general. The largest known gap between these two quantities is a constant factor [15] but it may well be exponential (see [11] On the other hand, Yannakakis [19] showed that the gap between c(f) and n 1 (f) which he called the unambiguous non deterministic complexity) can not be more than polynomial. Proposition 7 [19] c f n 1 (f) 2 . In particular, P cc = UP cc . It is not known if this bound is best possible. 3 Main Results The main results ....

[Article contains additional citation context not shown here]

M. Yannakakis, Expressing Combinatorial Optimization Problems by Linear Programs, Proc. 29th FOCS, (1988), 223--228.

....Allender and Reinhardt [6] have shown that unambiguous nondeterminism is indeed as powerful as the unrestricted version: the nondeterministic analogs of the classes UL and NL coincide, i.e. we have UL Poly = NL Poly. A contrary result holds for two party communication protocols. Yannakakis [58] has proven that deterministic communication complexity is at most quadratically larger than unambiguous communication complexity. Furthermore, the results of Mehlhorn and Schmidt [36] can be exploited to obtain a function of input size n which has nondeterministic communication complexity O(log ....

M. Yannakakis. Expressing combinatorial optimization problems by linear programs. Journal of Computer and System Sciences, 43(3):441--466, 1991. 62

.... It was originally motivated by applications to distributed computing and VLSI, where it captures essential features in an natural way (see [2] and the references within) Recently, unexpected connections were found between this model and seemingly unrelated areas of combinatorial optimization [21] and circuit complexity [15] A very natural question to ask is the direct sum question: Is it easier to solve two problems together than separately This question is related, in its essence, to similar questions in algebraic complexity [3] and other models [7] For the original model of Yao ....

M. Yannakakis, Expressing Combinatorial Optimization Problems by Linear Programs. In Proc. Twentieth ACM Symp. Theor. Comput., 1988, 223--228.

....material the reader is referred to [4, 3] Let f : f0; 1g n Theta f0; 1g n f0; 1g, and let F be the 2 n Theta 2 n associated matrix. A lower on the deterministic communication complexity for multi rounds game is log(rank(F ) A (possibly) better bound that was defined by Yannakakis, [6], is the positive rank . For a positive n Theta n matrix F , the positive rank of F , denoted by prank(F ) is the smallest integer r for which F can be expressed as F = A Theta B T where A; B are positive m Theta r matrices. One can relate these measures to the randomized complexity too. ....

M. Yannakakis, Expressing combinatorial optimization problems by linear programs, STOC, 1988 pp.223-228.

....cannot be done in PTIME even for sets of linear inequalities. If we have a set C of linear inequalities, it might not be possible to describe the result of a variable elimination operation on C by a set of linear inequalities with size less than exponential in the number of eliminated variables [50]. The following is a weaker result which considers Horn disjunctive linear constraints and a fixed number of variables. Theorem 6 ( 29] We can eliminate a fixed number of variables from a set of Horn disjunctive linear constraints in PTIME. Luckily there are subclasses of linear constraints ....

M. Yannakakis. Expressing Combinatorial Optimization Problems by Linear Programs. In Proc. of ACM Symposium on the Theory of Computing, pages 223--288, 1988.

....fact variable elimination cannot be done in PTIME even for LIN. If we have a set C of linear inequalities, it might not be possible to describe the result of a variable elimination operation on C by a set of linear inequalities with size less than exponential in the number of eliminated variables (Yannakakis 1988). The following is a recent result of the authors which extends (Koubarakis 1997a) Theorem 0.2 Let C be a set of UTVPI 6= constraints. We can eliminate any number of variables from C in O(dn 4 ) time where d is the number of disequations and n is the number of variables in C. The Scheme of ....

Yannakakis, M. 1988. Expressing Combinatorial Optimization Problems by Linear Programs. In Proc. of ACM Symposium on the Theory of Computing, 223-- 288.

....: p (f j ) T w j Gamma h j x k j j = 1; 2; p x 0 w j 0 j = 1; 2; p Note that this approach to obtaining a compact formulation is predicated on being able to formulate the separation problem as a compact linear program. This may not always be possible. In fact, Yannakakis [82] shows that for a b matching problem under a symmetry assumption, no compact formulation is possible. This despite the fact that b matching can be solved in polynomial time using a polynomial time separation oracle. An Application: Neural Net Loading The decision version of the Hopfield neural ....

M.Yannakakis, Expressing Combinatorial optimization problems by linear programs, in Proceedings of ACM Symposium of Theory of Computing, (1988) 223-228.

....this separation problem is known. Martin [92] has shown that if the separation problem can be expressed as a compact linear program then so can the optimization problem. Hence an unresolved issue is whether there exists a polynomial size (compact) formulation for the b matching problem. Yannakakis [123] has shown that, under a symmetry assumption, such a formulation is impossible. 28 4.3.3 Other Combinatorial Problems Besides the matching problem several other combinatorial problems and their associated polytopes have been well studied and some families of facet defining inequalities have ....

M.Yannakakis, Expressing Combinatorial optimization problems by linear programs, in Proceedings of ACM Symposium of Theory of Computing, (1988) 223-228.

....which can also be viewed as the stable set polytope of the line graph) Consider the Edmonds constraints: P i2S x i (jSj Gamma 1) 2 for jSj odd. Their N index is unbounded (as a function of jSj) as was shown by Lov asz and Schrijver [45] and an indirect consequence of a result of Yannakakis [66]) However, their N index is unknown and could possibly be bounded. 5 The Maximum Cut Problem Given a graph G = V; E) the cut ffi(S) induced by vertex set S consists of the set of edges with exactly one endpoint in S. In the NP hard maximum cut problem (MAX CUT) we would like to find a cut ....

M. Yannakakis. Expressing combinatorial optimization problems by linear programs. In Proc. 29th Symp. on Found. of Comp. Sci., pages 223--228, 1988. 22

.... It was originally motivated by applications to distributed computing and VLSI, where it captures essential features in an obvious way (see e.g. 1] and the references within) Recently, unexpected connections were found between this model and seemingly unrelated areas of combinatorial optimization [16] and circuit complexity [11] A very natural question to ask is: Is it easier to solve Supported by NSF grant CCR 9010533 y Supported by the American Israeli Binational Science Foundation grant 89 00126 two problems together than separately This question is related, in its essence, to ....

M. Yannakakis, "Expressing combinatorial optimization problems by linear programs", Proceedings of 20 th STOC, pp. 223-228 (1988).

....fact variable elimination cannot be done in PTIME even for LIN. If we have a set C of linear inequalities, it might not be possible to describe the result of a variable elimination operation on C by a set of linear inequalities with size less than exponential in the number of eliminated variables (Yannakakis 1988). The following result extends a similar result in (Koubarakis 1997a) Theorem 2 Let C be a set of UTVPI 6= constraints. We can eliminate any number of variables from C in O(dn 4 ) time where d is the number of disequations and n is the number of variables in C. The Scheme of Indefinite ....

Yannakakis, M. 1988. Expressing Combinatorial Optimization Problems by Linear Programs. In Proc. of STOC-88, 223--288.

....and Naor, 1994. c) For a set of m linear constraints with k variables, elimination of some variables, i.e. existential quantifier elimination for any i k variables, has a worst case bound exponential in k. Elimination could be exponential because of the output size. A good example is given by Yannakakis, 1988, consisting of a linear constraint set for representing a parity polytope in k dimensions. This polytope, though simple to describe, has O(2 k ) facets, requiring an exponential number of constraints (if no existentially quantified variables are involved in the representation) However, with ....

....the complexity gap between performing satisfiability checks and performing variable elimination for sets of linear constraints. The former is polynomial whereas the latter is exponential (Schrijver, 1986) What s worse, variable elimination can be exponential just because of the size of the result (Yannakakis, 1988) if a non lazy, or eager , representation is used. Therefore, we propose to use the lazy methodology not just for the intermediate representation during the evaluation of positive algebraic operations, but also for the internal representation when storing generalized relations. Fortunately, ....

M. Yannakakis. Expressing Combinatorial Optimization Problems by Linear Programs. Proc. 20th ACM STOC, 223--228, 1988.

....number of bits that must be communicated is the deterministic communication complexity of f . This complexity measure was introduced by Yao [22] and has been shown to be tightly related to time area tradeoffs in VLSI [1, 7, 8, 11, 12, 19, 21] circuit complexity [9] and combinatorial optimization [20]. It has also been studied for its own sake as an interesting model of computation. Indeed, non deterministic, probabilistic, and alternating variants have been considered and several complexity classes analogous to the more notorius ones in Turing machine complexity have been defined [2, 14, 18] ....

Yannakakis, M. "Expressing combinatorial optimization problems by linear programs ". In Proceedings of 29th IEEE Symposium on Foundations of Computer Science, pages 223--228, 1988.

....of polytopes in higher dimensions but with fewer (polynomially many) facets. For all such polytopes one can apply interior point methods and optimize over them in polynomial time. For a thorough discussion of liftings of polyhedra associated with combinatorial optimization problems consult [65, 42] and the references cited in them. It is an interesting problem to look for easily computable (for instance NC computable or at least polynomial time computable) barriers for combinatorial optimization problems whose linear programming formulation contains exponentially many inequalities. A ....

....where m is the number of edges in the graph. This problem is especially interesting because Yannakakis has shown that under certain symmetry preserving conditions on the lift operator it is impossible to lift the matching polytope to a higher dimensional polytope with polynomially many facets, [65]. Whether the matching polytope can be represented as a projection of a convex set endowed with an O(m) self concordant barrier function remains an interesting open problem. 32 F. Alizadeh 5.2. Maximum cliques in perfect graphs. A particularly nice application of semidefinite programming is to ....

M. Yannakakis, Expressing combinatorial optimization problems by linear programs, J. Comput. Syst. Sci., 43 (1991), pp. 441--466.

No context found.

M. Yannakakis. Expressing Combinatorial Optimization Problems by Linear Programs. Proc. 20th ACM STOC, 223--228, 1988. 109

No context found.

M. Yannakakis, Expressing combinatorial optimization problems by linear programs, Proceedings of the 20 Annual ACM Symposium on Theory of Computing, Chicago, Ill. ACM Press (1988), 223-228. 15

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC