A. Karzanov and L. Khachiyan. On the conductance of order markov chains. Order, 8(1):7--15, 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the other, provided the combinatorial sets have a certain structural property called self reducibility. The set of linear extensions of a poset has this property and, not surprisingly, a number of algorithms for generating (almost) uniformly randomly linear extensions of posets have been developed [18, 4]. These algorithms are all randomized algorithms based on the Markov chain Monte Carlo technique . In Appendix This is a fundamental problem in the theory of ordered sets with applications in computer science (sorting) and social sciences. The complexity class #P, introduced by Valiant ....

A. Karzanov and L. Kachiyan. On the conductance of order markov chains. Order, 8(1):7--15, 1991.

....for the application of algorithms that work on polyhedra or more general convex bodies (e.g. see Lovz [Lov92] In their paper about volume approximation for polyhedra Dyer et al. DFK89] already mention as consequence the approximation of the number of linear extensions. Karzanov and Khachiyan [KK91] study a natural Markov chain on the set of linear extensions of an order. Their algorithm returns an approximately uniformly distributed linear extension of an n element order after O(n 6 log n) steps. Here is the process analyzed in [KK91] for generating a random linear extension of order P: ....

....the number of linear extensions. Karzanov and Khachiyan [KK91] study a natural Markov chain on the set of linear extensions of an order. Their algorithm returns an approximately uniformly distributed linear extension of an n element order after O(n 6 log n) steps. Here is the process analyzed in [KK91] for generating a random linear extension of order P: start from an arbitrary linear extension L = Xl, Xn) of P. Pick a position i G 1, n l uniformly at random. If elements xi and xi l are incomparable in P interchange them with probability 1 2 otherwise leave L unchanged. Repeat this ....

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A. Karzanov and L. Khachiyan. On the conductance of order markov chains. Order, 8:7-15, 1991.

....initial state is fixed, the process is reversible (representable as a random walk on a graph) and some bound is obtained for the mixing time m. The payo# has been polynomial time randomized approximation algorithms for counting combinatorial objects such as matchings [17, 10] linear extensions [18], and Eulerian orientations [20] estimating the volume of a convex body [16, 19] and for Monte Carlo integration [6] There is no apriorireason why a state must be sampled at a fixed number of steps. If the transition probabilities are known, a stopping rule which looks where it is going is ....

A. Karzanov and L. Khachiyan, On the conductance of order Markov chains, Technical Report DCS TR 268, Rutgers University, June 1990.

....adapted this technique to give the rst provably polynomial time algorithm to approximate the volume of a convex body. Developments related to the volume problem were sometimes applied back to the original problem of rapid mixing on combinatorial Markov chains. In one case Karzanov and Khachiyan [12] used geometric properties of the underlying graphs through isoperimetric inequalities to show rapid mixing on a Markov chain related to counting linear extensions. In all these early techniques the key to bounding conductance was nding a lower bound on the cutset expansion, the in num inf ....

....Edge Isoperimetry on Linear Extensions The rst case in which we use edge isoperimetry to show faster mixing are Markov chains whose underlying graphs (G; V ) have a natural geometric structure. The key to exploiting this geometric structure will be a type of isoperimetric inequality rst used in [12], strengthened in [5] and to be further strengthened in this paper. 3.1 Isoperimetry The key to bounding the edge isoperimetry is an isoperimetric inequality relating the surface area of a cut to the volume it encloses. First a few de nitions. De nition 3.1 Let F be a real valued function on a ....

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A. Karzanov and L. Khachiyan, On the conductance of order Markov chains, Order, 8(1):7{ 15, 1991.

.... x = Ay for some y 2 [0; 1] n g: The mixing time of Markov chains has attracted much attention lately in the Computer Science community, since the eciency of various algorithms depends on this, e.g. Broder [5] Jerrum and Sinclair [22, 13, 14] Dyer, Frieze and Kannan [11] Karzanov and Khachyan [16], Lov asz and Simonovits [17] Applegate and Kannan [3] Dyer and Frieze [10] Mihail and Winkler [19] Theorem 1 could be used, in particular, to show how to estimate jBj although as we shall see, this can be done more eciently using the Binet Cauchy formula for the determinant of the product of ....

A. Karzanov and L. G. Khachyan, On the conductance of order Markov chains, Technical Report DCS TR 268, Rutgers University, 1990.

....observing, as we did earlier, that the volume of the 21 order polytope of the partial order P with ground set f1; ng is exactly e(P ) n . An algorithm more tailored to the special case of approximating e(P ) for an n element partial order P , was investigated by Karzanov and Khachiyan [20]. We briefly sketch their approach. As we saw above (essentially) to approximate e(P ) it is enough to be able to approximate IP(x OE y) for elements x; y. To do this, it is enough to be able to sample approximately uniformly from the set of all linear extensions of P . Karzanov and Khachiyan ....

A. Karzanov and L. Khachiyan, On the conductance of order Markov chains, Order 8 (1991) 7--15.

....in positive results, lower bounds on Phi are generally of greater interest and we focus on them for most of the rest of this paper. We shall consider negative results in Section 4. In some cases such a bound can be obtained directly, using elementary arguments [16, 24] or geometric ideas [9, 14]. However, in many important applications the only known handle on Phi is via the canonical path approach sketched in the previous section. Thus we attempt to construct a family Gamma = ffl xy g of simple paths in G, one between each ordered pair of distinct states x and y , such that no edge is ....

Karzanov, A. and Khachiyan, L. On the conductance of order Markov chains. Technical Report DCS 268, Rutgers University, June 1990.

....the other, provided the combinatorial sets have a certain structural property called self reducibility. The set of linear extensions of a poset has this property and, not suprisingly, a number of algorithms for generating (almost) uniformly randomly linear extensions of posets have been developed [20, 6]. These algorithms are all randomized algorithms based on the Markov chain Monte Carlo technique 3 . In appendix B we describe the best known algorithm, due to Bubley and Dyer [6] that has a running time of O(n 3 log n# 1 ) where n is the poset s cardinality, and # is the desired accuracy. ....

A. Karzanov and L. Kachiyan. On the conductance of order markov chains. Order, 8(1):7--15, 1991.

.... in time polynomial in the size of the input, Gamma1 and log(ffi Gamma1 ) Examples of problems amenable to this type of approximation are dense 0 1 permanent [4, 12] matchings [12] volume computation [7] counting Eulerian orientations [17] counting linear extensions of a partial order [16] and computing the partition function for the ferromagnetic Ising model [13] The algorithms in the papers cited use a random walk to generate an almost uniform random solution to the problem (e.g. a random matching) and then apply multistage statistical sampling methods to obtain the desired ....

A. Karzanov and L. G. Khachiyan, On the conductance of order Markov chains, Technical Report DCS TR 268, Rutgers University, 1990.

.... body in high dimensional Euclidean space, which has been variously refined by the original authors and by Lov asz and Simonovits [14] and extended to the integration of log concave functions over convex regions by Applegate and Kannan [1] The approach was also used by Karzanov and Khachiyan [11] to validate an fpras for the number of linear extensions of a partial order. It would be interesting to know which #P complete functions can be approximated in the fpras sense, and which cannot. As we have seen, definite progress has been made in the last few years, at least on the positive side ....

A. Karzanov and L. Khachiyan, On the conductance of order Markov chains, Technical Report DCS 268, Rutgers University, June 1990.

....known to be NP Complete, even for r = 1 and zero one variables. The counting problem seems to be even harder. This asks for jKj. This problem is #P complete, again even for r = 1 and zero one variables. Markov chains have been successfully used to approximately solve several #P complete problems [1, 2, 3, 4, 6, 7, 10, 12, 14]. In all of these problems the running time of the algorithm is polynomial in problem size and relative error. On the other hand, the general zero one permanent still resists polynomial time approximation, but Jerrum and Vazirani [9] have reduced the time complexity to O(2 O( p n(log n) 2 ) ....

A. Karzanov and L. Khachiyan, On the conductance of order Markov chains, Technical Report DCS TR 268, Rutgers University (1990).

....time of several important Markov chains with an inherently geometric flavor. These include the work of Dyer, Frieze and Kannan [11] Lov asz and Simonovits [26] and others on computing volumes (see [21] for a survey) and Karzanov and Khachian on counting linear extensions of a partial order [22]. ....

A. Karzanov and L. Khachiyan, On the conductance of order Markov chains, Technical Report DCS 268, Rutgers University, June 1990.

.... [5] was required in [10] Recently, Lov asz and Simonovits [24] generalized the notion of conductance, and gave a sharper proof that this implies rapid mixing (although in a weaker sense than Sinclair and Jerrum [30] They also proved the above conjecture of [10] See also Karzanov and Khachiyan [19]. With these improvements, they improved the analysis of the algorithm and its polynomial time bound. They also simplified the algorithm itself somewhat. In order to obtain rapid mixing, Dyer, Frieze and Kannan were obliged to smooth the boundary of the convex set by inflating it slightly. ....

....(and so it can be approximated by the methods of Section 4) The volume approximation algorithm of Dyer, Frieze and Kannan applied (in the notation of Section 3. 2) to P (OE) gave the first (random) polynomial time approximation algorithm for estimating e(OE) However, Karzanov and Khachiyan [19] have recently given an improvement to the algorithm for this application which is more natural, and which we will now outline. Observe first that it suffices to be able to generate an (almost) random linear extension of OE. For an incomparable pair i; j under OE, let ae ij denote the proportion ....

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A. Karzanov and L. G. Khachiyan, On the conductance of order Markov chains, Technical Report DCS TR 268, Rutgers University, 1990.

....satisfies the detailed balance condition: Q(x, y) #(x)P(x, y) #(y)P(y,x) for all x, y ## ; furthermore, we assume the loop probabilities P(x,x) are at least 1 2 for all x ## . Since 3 For a more direct approach to this problem, using a conductance argument as described below, see [KK90]. 490 CHAPTER 12 THE MARKOV CHAIN MONTE CARLO METHOD the Markov chain M is a constructed one, it is not at all difficult to arrange that these two conditions are met. 12.3.1 CANONICAL PATHS To describe the canonical path argument, we view M as an undirected graph with vertex set# and edge set ....

A. Karzanov and L. Khachiyan. On the conductance of order Markov chains, Technical Report DCS 268, Rutgers University, June 1990.

....a polynomial fraction of the number of near perfect matchings, and hence the expected number of iterations before a perfect matching is obtained is polynomially bounded. Other applications of this method involve counting the number of linear extensions of a partial order (Khachiyan and Karzanov [41]) eulerian orientations of a graph (Mihail and Winkler [59] forests in dense graphs (Annan [7] and certain partition functions in statistical mechanics (Jerrum and Sinclair [37] See Welsh [66] for a detailed account of fully polynomial randomized approximation schemes for enumeration ....

A. Karzanov and L. G. Khachiyan, On the conductance of order Markov chains, Order 8(1991), 7--15. Random Walks on Graphs: A Survey 45

....total orderings are known as the linear extensions of the DAG. Hence, L (s) 1 = log K K(K Gamma 1) 2 Gamma log M: 4) Unfortunately, counting the number of linear extensions of a DAG is known to be an NPhard problem [3] There are efficient means of producing an upper bound to M [12], and we are investigating its use to provide an estimate for the value of M . In the meantime, we calculate M by brute force, with the understanding that this technique will be applicable only to modestly sized causal models. 0.0 0.0 DAGs Length 1 Length 2 2.0 2.0 5.5850 5.0000 4.5850 5.0000 ....

A. Karzanov and L. Khachiyan. On the conductance of order markov chains. Technical Report DCS TR 268, Computer Science, Rutgers University, 1990.

....degree polynomials, has more often than not been the first step towards revealing those structural properties that can be further exploited to obtain practical efficiency. It should also be noted that progress in the area of polynomial time approximations for #P complete functions is very recent [2, 3, 6, 8, 9, 11, 12, 13, 15], etc. For example, the algorithm in [8] to approximate volumes had complexity n 26 ; it was improved in [15] to n 16 ; it is currently at n 10 [2] and for special cases it has complexity n 2:5 [13] which was improved very recently to n 2 [22] In this sense, any polynomial time ....

....time approximations for #P complete functions is very recent [2, 3, 6, 8, 9, 11, 12, 13, 15] etc. For example, the algorithm in [8] to approximate volumes had complexity n 26 ; it was improved in [15] to n 16 ; it is currently at n 10 [2] and for special cases it has complexity n 2:5 [13], which was improved very recently to n 2 [22] In this sense, any polynomial time approximation scheme is of interest even purely for its methodology. On the positive side, from a practical point of view our heuristic for single edge parameters seems very practical for medium size networks. In ....

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A. Karzanov and L. Khachiyan. On the conductance of order Markov chains. Technical report, Rutgers University, 1990.

.... on graphs to obtain the first randomized polynomial time algorithm for approximating the value of the permanent of a matrix; see also Dagum, Luby, Mihail and Vazirani [10] Random walks were used by Dyer, Frieze and Kannan [13] to estimate the volume of a convex body, and by Karzanov and Khachiyan [16] to sort partial orders when comparisons are expensive. Recently, Coppersmith et al. 8] have found an application of random walks to on line algorithms. Aldous [1] gives many other contexts in which random walks on graphs arise, and a valuable bibliography [2] compiled by the same author lists ....

A. Karzanov and L. Khachiyan, On the conductance of order Markov chains, Technical Report DCS TR 268, Rutgers University, June 1990.

....on the rapid mixing of a particular geometric Markov chain. See Jerrum and Sinclair [12] for a recent survey of this approach to approximation problems. Subsequently, Matthews [14] gave a somewhat different geometric approach. Using geometric and conductance arguments [17] Karzanov and Khachiyan [13] showed the rapid mixing of a combinatorial Markov chain on W. Dyer and Frieze [8] improved the conductance estimate, and hence the bound on the mixing time, of this chain. These results all rely on a relationship between W and the geometry of a certain polytope in R n . Felsner and Wernisch ....

Alexander Karzanov and Leonid Khachiyan. On the conductance of order Markov chains. Order, 8(1):7--15, 1991.

.... past ten years many new applications have been found (see, e.g. 3] for sampling via Markov chains; these have resulted in polynomial time randomized approximation schemes for computing the volume of a convex body [12] and counting combinatorial objects such as matchings [13] linear extensions [14], and Eulerian orientations [18] Typically, in these applications, sampling from an approximately correct distribution is accomplished by walking randomly on a graph for a fixed number of steps, after which the distribution of the occupied vertex is nearly stationary. There is no particular ....

A. Karzanov and L. Khachiyan, On the conductance of order Markov chains, Technical Report DCS TR 268, Rutgers University, June 1990.

....initial state is fixed, the process is reversible (representable as a random walk on a graph) and some bound is obtained for the mixing time m. The payoff has been polynomial time randomized approximation algorithms for counting combinatorial objects such as matchings [17, 10] linear extensions [18], and Eulerian orientations [20] estimating the volume of a convex body [16, 19] and for Monte Carlo integration [6] There is no a priori reason why a state must be sampled at a fixed number of steps. If the transition probabilities are known, a stopping rule which looks where it is going is ....

A. Karzanov and L. Khachiyan, On the conductance of order Markov chains, Technical Report DCS TR 268, Rutgers University, June 1990.

....the other, provided the combinatorial sets have a certain structural property called self reducibility. The set of linear extensions of a poset has this property and, not suprisingly, a number of algorithms for generating (almost) uniformly randomly linear extensions of posets have been developed [8, 2] in order to address the fundamental problem of counting linear extensions. These algorithms are all randomized algorithms based on the Markov chain Monte Carlo technique 4 . In the Appendix we describe the best known algorithm, due to Bubley and Dyer [2] that has a run 3 The complexity ....

....let S t 1 be this new permutation. Otherwise, let S t 1 = S t . It is easily seen that Mf is ergodic with uniform stationary distribution. When f is the uniform distribution on f1; 2; n Gamma 1g, M f is the Karzanov Kachiyan chain with mixing time O(n 5 log n n 4 logffl Gamma1 ) [8]. Bubley and Dyer showed that if f is defined as f(i) i(n Gamma i) K, where K = n 3 Gamma n) 6, then M f has mixing time of O(n 3 log nffl Gamma1 ) ....

A. Karzanov and L. Kachiyan. On the conductance of order markov chains. Order, 8(1):7--15, 1991.

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A. Karzanov and L. Khachiyan. On the conductance of order markov chains. Order, 8(1):7--15, 1991.

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