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Outline Markov Chains are . . . Markov Chains under . . . From the . . . Symmetric Markov Chains In General, Symmetry . . . Proof of NP-Hardness Proof (cont-d) Acknowledgments Title Page ◭◭ ◮◮ ◭ ◮ Page 1 of 10 Go Back Full Screen Close Quit

Under Interval and Fuzzy Uncertainty, Symmetric Markov Chains Are More Difficult to Predict

Roberto Araiza, Gang Xiang, Olga Kosheleva

University of Texas at El Paso El Paso, Texas 79968, USA raraiza@utep.edu, gxiang@utep.edu

Damjan ˇ Skulj

Faculty of Social Sciences University of Ljubljana 100 Ljubljana, Slovenia Damjan.Skulj@Fdv.Uni-Lj.Si

Outline Markov Chains are . . . Markov Chains under . . . From the . . . Symmetric Markov Chains In General, Symmetry . . . Proof of NP-Hardness Proof (cont-d) Acknowledgments Title Page ◭◭ ◮◮ ◭ ◮ Page 2 of 10 Go Back Full Screen Close Quit

1. Outline

• Markov chains: important tool for solving practical

problems.

• Traditional approach: assumes that we know the exact

transition probabilities pij.

• In reality: we often only know these transition proba-

bilities with interval (or fuzzy) uncertainty.

• Symmetry: in some situations, the Markov chain is

symmetric: pij = pji.

• In general: symmetry simplifies computations.
• New result: for Markov chains under interval and fuzzy

uncertainty, symmetry has the opposite effect – it makes the computational problems more difficult.

Outline Markov Chains are . . . Markov Chains under . . . From the . . . Symmetric Markov Chains In General, Symmetry . . . Proof of NP-Hardness Proof (cont-d) Acknowledgments Title Page ◭◭ ◮◮ ◭ ◮ Page 3 of 10 Go Back Full Screen Close Quit

2. Markov Chains are Important

• What is a Markov chain: when the probability pij of

going from state i to state j – depends only on these two states, and – does not depend on the previous history.

• Example: described gene-related processes in bioinfor-

matics.

• Reminder: for each state i, the probabilities pij of going

to different states j = 1, . . . , n should add up to one:

n

• j=1

pij = 1.

• Computational advantage: we can determine the prob-

abilities p(2)

ij of 2-step transitions as p(2) ij = n

• k=1

pik · pkj.

• Similarly, we can then define the probabilities of 3-step

transitions, etc.

Outline Markov Chains are . . . Markov Chains under . . . From the . . . Symmetric Markov Chains In General, Symmetry . . . Proof of NP-Hardness Proof (cont-d) Acknowledgments Title Page ◭◭ ◮◮ ◭ ◮ Page 4 of 10 Go Back Full Screen Close Quit

3. Markov Chains under Interval and Fuzzy Uncer- tainty

• In practice, we often do not know the exact values of

the transition probabilities pij.

• Instead, we sometimes know the intervals [pij, pij] of

possible values of pij.

• Even more generally, we know fuzzy numbers µij which

describe these probabilities.

• A natural question: what can we conclude about the

2-step transition probabilities?

• Interval case: we would like to know the intervals

p(2)

ij =

n

• k=1

pik · pkj : pab ∈ pab,

n

• b=1

pab = 1

• .
• Fuzzy case: we would like to know the fuzzy sets cor-

responding to p(2)

ij .

Outline Markov Chains are . . . Markov Chains under . . . From the . . . Symmetric Markov Chains In General, Symmetry . . . Proof of NP-Hardness Proof (cont-d) Acknowledgments Title Page ◭◭ ◮◮ ◭ ◮ Page 5 of 10 Go Back Full Screen Close Quit

4. From the Computational Viewpoint, It Is Sufficient to Consider Interval Uncertainty

• Known: a fuzzy number µij(p) can be described by its

α-cuts pij(α)

def

= {p | µij(p) ≥ α}.

• Conclusion: a fuzzy number µij(p) can be viewed as a

family of nested intervals pij(α).

• Our objective: compute the fuzzy number µ(2)

ij

corre- sponding to the desired value p(2)

ij .

• Known: each α-cut of µ(2)

ij can be computed based on

the α-cuts pij(α) of the inputs.

• Example: to describe 10 different levels of uncertainty,

we solve 10 interval computation problems.

• Conclusion: from the computational viewpoint, it is

sufficient to produce an efficient algorithm for the in- terval case.

Outline Markov Chains are . . . Markov Chains under . . . From the . . . Symmetric Markov Chains In General, Symmetry . . . Proof of NP-Hardness Proof (cont-d) Acknowledgments Title Page ◭◭ ◮◮ ◭ ◮ Page 6 of 10 Go Back Full Screen Close Quit

5. Symmetric Markov Chains

• Known: efficient algorithms for computing 2-step tran-

sition probabilities under interval uncertainty.

• Symmetric Markov chains: in some practical situa-

tions, we have symmetric (T-invariant) Markov chains.

• Definition: the probability pij of going from state i to

state j is always equal to the probability of going from state j to state i: pij = pji.

• Example: mutations and other transitions in bioinfor-

matics.

• Symmetric Markov chains under interval uncertainty:

in the formula describing this range, we impose this additional symmetry requirement: p(2)

ij,sym =

n

• k=1

pik · pkj : pab ∈ pab, pab = pba,

n

• b=1

pab = 1

• .

Outline Markov Chains are . . . Markov Chains under . . . From the . . . Symmetric Markov Chains In General, Symmetry . . . Proof of NP-Hardness Proof (cont-d) Acknowledgments Title Page ◭◭ ◮◮ ◭ ◮ Page 7 of 10 Go Back Full Screen Close Quit

6. In General, Symmetry Helps, but Not for Markov Chains under Interval Uncertainty

• Known: symmetry assumption usually enables us to

speed up computations.

• First reason: because of symmetry, we need to store

fewer data values pij.

• Second reason: for symmetric matrices (like pij), there

are often faster algorithms.

• It is reasonable to expect: that under interval uncer-

tainty, symmetry will also be helpful.

• Our result: contrary to these expectations, under inter-

val uncertainty, symmetry makes Markov chain com- putations more complex.

• Precise result: computing the (endpoints of the) exact

range p(2)

ij,sym of 2-step probabilities is NP-hard.

Outline Markov Chains are . . . Markov Chains under . . . From the . . . Symmetric Markov Chains In General, Symmetry . . . Proof of NP-Hardness Proof (cont-d) Acknowledgments Title Page ◭◭ ◮◮ ◭ ◮ Page 8 of 10 Go Back Full Screen Close Quit

7. Proof of NP-Hardness

• Main idea: we reduce, to our problem, a known NP-

hard subset problem: – given n positive integers s1, . . . , sn, – find the values εi ∈ {−1, 1} for which

n

• i=1

εi · si = 0.

• Reduction: to each instance of the subset problem, we

assign a Markov chain with [p1i, p1i] = 1 n − α · si, 1 n + α · si

• for some small α > 0.
• How to select α: we want to to guarantee that proba-

bilities are non-negative: p1i = 1 n − α · si ≥ 0.

• Example: α =

1 n · max

i

si .

Outline Markov Chains are . . . Markov Chains under . . . From the . . . Symmetric Markov Chains In General, Symmetry . . . Proof of NP-Hardness Proof (cont-d) Acknowledgments Title Page ◭◭ ◮◮ ◭ ◮ Page 9 of 10 Go Back Full Screen Close Quit

8. Proof (cont-d)

• Due to symmetry: p(2)

11,sym = n

• i=1

p1i · pi1 =

n

• i=1

(p1i)2.

• Auxiliary notation: ∆i

def

= 1 α ·

• p1i − 1

n

• , then

p1i = pi1 = 1 n + α · ∆i and ∆i ∈ [−si, si].

• Since

n

• i=1

p1i = 1, we have

n

• i=1

∆i = 0, and p(2)

11,sym = n

• i=1

1 n + α · ∆i 2 = 1 n + α2 ·

n

• i=1

∆2

i.

• Easy to prove: p(2)

11,sym = 1

n+α2·

n

• i=1

s2

i is possible ⇔ the

• riginal instance of a subset problem has a solution.
• Reduction is proven, so our problem is indeed NP-hard.

Outline Markov Chains are . . . Markov Chains under . . . From the . . . Symmetric Markov Chains In General, Symmetry . . . Proof of NP-Hardness Proof (cont-d) Acknowledgments Title Page ◭◭ ◮◮ ◭ ◮ Page 10 of 10 Go Back Full Screen Close Quit

9. Acknowledgments This work was supported in part:

• by the Texas Department of Transportation grant No.

0-5453 and

• by the Texas Advanced Research Program Grant No.

003661-0008-2006.