Flexible optimization problems
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Flexible Optimization Problems. A. Akavia, S. Safra. Motivation. What do we do with all the NP-hard optimization problems ???. Relaxations: 2 Parameters. Optimization function  approximation Input  flexibility Example – graph coloring problem : Optimization function –

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Flexible optimization problems

Flexible Optimization Problems

A. Akavia, S. Safra


Motivation

Motivation

What do we do with all the NP-hard optimization problems ???


Relaxations 2 parameters

Relaxations: 2 Parameters

  • Optimization function approximation

  • Input flexibility

    Example – graph coloring problem:

  • Optimization function –

    • find an approximation of the min coloring.

  • Input flexibility –

    • find a k-coloring with few monochromatic edges.


Talk plan

Talk Plan

  • Approximation

  • Input flexibility

  • Flexible optimization problems

    • Examples

    • Definitions

    • Hardness results


Relaxation 1 approximation

Relaxation 1: Approximation

  • An approximation algorithm is an algorithm that returns an answer C which “g-approximates” the optimal solution C*.

  • C  g  C*(minimization)

  • 1/g  C*  C(maximization)


Relaxation 2 input flexibility example graph editing problems

example: 2-colorability

Relaxation 2: Input FlexibilityExample: Graph Editing problems

complexity and approximation results are w/r to the number of modifications

Input:

  • a graph G, and

  • a desired property

    Goal: find a small set of edge-modifications (addition/deletion/both) that transforms Ginto G’with the desired property


Our work flexible approximation problems

Our Work: Flexible-Approximation-Problems

  • Combining both relaxations –

    • approximating the optimization function,

    • while allowing input flexibility.

  • Example: Given a graph G, find a coloring with fewcolors, and perhaps few monochromatic edges.


Flexible optimization problems

Natural Flexible-Approximation-Problems:

  • Min Non-Deterministic Automaton

  • Min Synthesis Graph


Non deterministic finite automaton nfa

Non-Deterministic Finite Automaton (NFA)

  • Many applications:

    • program verification

    • speech recognition

    • natural language processing

Approximating the minimum NFA, which accepts a given language L is hard.


Flexible min nfa

Flexible Min NFA

Sometimes it suffices to find:

  • a smallNFA,(Approximation)

  • accepting a language “similar” to the input one.(Input Flexibility)


Example a utomata theoretic approach to program verification

Example: Automata-Theoretic Approach to Program Verification

  • ProgramP is correct with respect to a specificationsTif: L(P)  L(T)

  • In concurrent programming: processesP1,..., Pn are correct w/r to specificationTif: L(P1x…x Pn)  L(T)

P

. . .

P1

. . .

P2

. . .

P3

. . .

in the worst case, |P1x…xPn| is exponential in n


Example a utomata theoretic approach to program verification1

Example: Automata-Theoretic Approach to Program Verification

  • Coping with this state-explosion [K94]:

    • finding a small automaton P’ (Optimization)

    • such that L(P1x…x Pn)  L(P’)(Input Flexibility)

    • and then checking whether L(P')  L(T)

L(P1x…x Pn)  L(P’)

L(P1x…x Pn)  L(T)

L(P’)  L(T)


Combinatorial chemistry and flexible min synthesis graph

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Combinatorial Chemistry andFlexible Min Synthesis Graph

a multi dimensional space

attributecoordinate

molecule point

similarity (molecules)proximity (points)

  • Choose

  • Synthesize

  • Screen

*


Flexible optimization problems

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Node - grow step

Label - appended unit

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Produces each string s s.t.s = labels concatenation along a path

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Synthesis Graph

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Split Synthesis [F91,L91,CS99]


Flexible min synthesis graph

Flexible Min Synthesis Graph

Output: A smallsynthesis-graphproducing S’, which is similar to S

Input: A set of strings S

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  • Input flexibitily–

    • producing S’ and not S

  • Approximation –

    • finding a small synthesis graph

    • not the minimum


Definitions theorems and proofs

Definitions, Theorems and Proofs


Flexible optimization problems

  • Assume a distance function(x,x’)to be the smallest number of basic modifications (say, bit-changes) necessary in order to transform x to x’.

    the ball of radius d around a given input x is ball(x,d) = {x’ | (x,x’)d}


D g flexible approximation problem

d

(d,g)-flexible approximation problem

  • Assume:

    • a distance function , and

    • an optimizationfunction f.

  • In a (d,g)-flexible-approximation problem, given an input x, a solution y’ is returned s.t.

    • y’ is feasible for some x’ball(x,d), and

    • f(x’,y’) is g-approximateto the optimum (for x).

f(x’,y’) ≤ gf(x,y*)

x’

x


Biclique edge cover definition

Biclique Edge Cover - Definition

Input: Bipartite graph G

Goal: Cover all edges by bicliques(i.e. complete bipartite subgraphs)


D g biclique edge cover

(d,g)-Biclique Edge Cover

  • Assume:

    • (G,G’) = symmetric edge difference

    • f(G,y) = no. of bicliques in the cover y.

G

G’

f(G’,y)=2

f(G,y)=3

(G,G’)=7


D g biclique edge cover1

(d,g)-Biclique Edge Cover

  • In a (d,g)-Biclique Edge Cover ((d,g)-BEC) given an input G, a solution y’ is returned s.t.

    • y’ is the no. of bicliques in a cover ofG’ball(x,d), and

    • f(G’,y’) is g-approximateto the min-BECfor G.

G

G’


Hardness of d g biclique edge cover

Hardness of (d,g)-Biclique Edge Cover

Thm: >0(d,g)-Biclique-Edge-Cover problem is hard for any g=O(|V|1/5-) and d=O(g), unless NP=ZPP.

Proof: later.


D g non deterministic finite automaton nfa

(d,g)-Non-Deterministic Finite Automaton (NFA)

  • Assume:

    • (L,L’) = symmetric difference between L and L’

    • f(L,A) = no. of states in an NFA A accepting L.

  • In a (d,g)-NFA, given a language L, a solution y’ is returned s.t.

    • y’ is the no. of states in an NFA accepting L’ball(L,d), and

    • f(L’,y’) is g-approximateto the optimum for L.


Hardness of d g nfa

Hardness of (d,g)-NFA

Thm: >0 (d,g)-NFA problem is hard for any g=O(|L|1/10-) and d=O(g), unless NP=ZPP


Reduction outlines

qF

q0

{uv| (u,v)E}

Bipartite graph G

Strings set S

NFA

Reduction Outlines

  • Reduction from Biclique-Edge-Cover:


Proof

v5

v4

v3

v2

v1

Define L

Define L = {vu| (v,u)E}

u5

u4

u3

u2

u1

qF

q0

A - an automaton acceptin L

Proof

Reduction from flexible approximation Biclique Edge Cover:

k-biclique edge cover of G’in d-distance from G 

NFA with k+2 states,accepting L’ in d-distance from L

Reduction from Biclique Edge Cover:

k-biclique edge cover of G 

NFA with k+2 states,accepting L

(d,g)-BECis hard for

g=O(|V|1/5-) and d=O(g)

 (d,g)-NFAis hard for

g=O(|L|1/10-) and d=O(g)

G=(V,U,E) a bipartite graph.


Reminder

Reminder

  • A synthesis-graph Hproduces a string s if:s is a label concatenation along a path from first to last layers in H.


D g synthesis graph

(d,g)-Synthesis Graph

Given a set of strings S, output a small synthesis-graph producing S’, which is similar to S

  • Assume:

    • (S,S’) = symmetric difference between S and S’

    • f(S,H) = no. of internal nodes in a synthesis graph H that produces the strings S.

  • In a (d,g)-Synthesis Graph, given an input S, a solution y’ is returned s.t.

    • y’ is the no. of internal nodes in a synthesis graph H producing S’ball(S,d), and

    • f(S’,y’) g-approximatethe optimum for S.


Hardness of d g synthesis graph

Hardness of (d,g)-Synthesis Graph

Thm: >0 (d,g)-Synthesis Graph problem is hard for any g=O(|S|1/10-) and d=O(g), unless NP=ZPP

Proof:


Reduction outlines1

A

A

{uAv| (u,v)E}

A

Bipartite graph G

Synthesis graph H

Strings set S

Reduction Outlines

  • Reduction from Biclique-Edge-Cover:


Proof1

Define S = {vAu| (v,u)E}

Define S

Proof

Reduction from flexible approximation Biclique Edge Cover:

k-biclique edge coverof G’in d-distance from G synthesis graph H with k internal nodes producing S’ in d-distance from S

Reduction from Biclique Edge Cover [CS99]:

k-biclique edge coverof G synthesis graph H with k internal nodes producing S

A

A

(d,g)-BECis hard for

g=O(|V|1/5-) and d=O(g)

 (d,g)-SGis hard for

g=O(|S|1/10-) and d=O(g)

A

G=(V,U,E) a bipartite graph.

H - a graph constructing S


Hardness proof of d g biclique edge cover d g bec

Hardness Proof of(d,g)-Biclique-Edge-Cover((d,g)-BEC)


D g biclique edge cover d g bec

(d,g)-Biclique Edge Cover ((d,g)-BEC)

  • In a (d,g)-Biclique Edge Cover ((d,g)-BEC) given an input G, a solution y’ is returned s.t.

    • y’ is the no. of bicliques in the cover of G’ball(G,d), and

    • f(G’,y’) is g-approximateto the min-BECfor G.

G

f(G,y)=3


Hardness of d g bec

Hardness of (d,g)-BEC

Thm: (d,g)-BEC is hard for any g=O(|V|1/5-) and d=O(g), unless NP=ZPP.

Proof:


Flexible vs non flexible solutions

flexiblevs. non-flexiblesolutions

Lemma: from any solution y’ to G’ball(G,d),we may construct a solution y to G,s.t. f(G,y) ≤ f(G’,y’) + d

G

G’


Calculating approximation factor

Calculating Approximation Factor

Claim: Let G be a graph, if v’=f(G’,y’)’g-approximate(d,g)-BEC,where d=O(g),then y, f(G,y)v’+d, s.t. v=v’+dO(g)-approximateBEC

Proof: By the lemma y, f(G,y)a’+d, and v  O(g)optG

v = v’ + d

 goptG + d

 2goptG(since d = O(g))

Assume BEC is hard for g=O(|V|1/5-)

then:

(d,g)-BEC is hard for g=O(|V|1/5-) and d=O(g)


Hardness of approximation of bec

Hardness of Approximation of BEC

Proof Outlines:

  • Construction [Simon90]

  • Graph coloring is hard to approximate by g=O(|V|1-) [FK98]

  • Simple calculation, improves Simon’s bound from a constant factor to g=O(|V|1/5-).


Clique cover problem

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Clique Cover Problem

Input: A graph G

Goal: Cover all vertices by cliques

Clique Cover is equivalent to Graph Coloring


Construction simon 90

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Construction [Simon ‘90]

GI

G


Propositions

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GI

Propositions

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  • a clique-cover in G translates into abiclique-cover of the horizontaledges in GI, and vice versa

  • a clique-cover in G translates into t2biclique-covers of the horizontaledges in H.

No. of GI’s copies in H.


Propositions1

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Propositions

G=(V,E)

  • 2|E|t bicliques are necessary and sufficient in order to cover the diagonal edges of H.


Propositions2

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Propositions

  • horizontaledges in different copies of GIcannotbe members of the samebiclique.

Ascending, hence cannot connect copies within the same row.


Lemma

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Lemma

G=(V,E)

  • sclique-cover in G(t2s + 2|E|t)biclique-cover in H

  • rbiclique-cover in H(r – 2|E|t) / t2clique-cover in G


Calculating approximation factor1

Calculating Approximation Factor

  • Given a solution with rbicliques for H, we may construct a solution with scliques for G, and

  • it is easy to verify that

    • if ris O(|VH|x)-approximate to ropt

    • then sisO(|V|5x)-approximate to sopt

Clique-Cover is hard to approximate by O(|V|1-), >0, unless NP=ZPP (from [FK98])

 BEC is hard to approx within g=O(|VH|1/5-), >0, , unless NP=ZPP.


Related work

Related Work

  • Property testing [GGR]

    • Sample size

    • Typically seeking an exact solution and not an approximated one

  • Bi-criteria optimization[MRSR]

    • optimization (vs. relaxation).

      • Might be easy when only one criterion is considered.

      • Bi-criteria: hard with one criterionhard in the bi-criterionversion.

    • Typically hostile objectives.


Discussion

relevant case for (d,g)-SG

hard

easy

d=0

d=O(|V|1/5-)

d=O(|V|)

d=O(|E|)

d=|V|2-|E|

Graph coloring is easy

Discussion

  • Similar proof is valid for any problem, which is

    • hard to approximate, and

    • “v  vd + d”

  • The gap is still too large:


Future work

Future Work

  • Improving our results

  • Extending our results to other problems

  • Achieving positive results

    • Parameterized polynomial solution

    • Approximation algorithm


Property testing ggr

Property Testing[GGR]

  • Distinguish, by a small no. of probes, between instances x that

    • satisfy a given property accept

    • no set of |x| modifications causes x to satisfy the property reject

back


Bi criteria optimization mrsr

Bi-criteria Optimization [MRSR]

  • Bi-criteria network design problem is a tuple (A,B,S)

    • A,B are two minimization objectives

    • S specify a membership requirement in a class of sub-graphs

    • The problem specifies a budget value on objective A

    • And seeks minimum over B (within the budget)

  • A (k,l)-approximation algorithm for an (A,B,S)-bi-criteria optimization problem is a poly.-time algorithm, that produces a solution that belongs to the sub-graph class S, in which

    • the objective A is at most k times the budget, and

    • the objective B is at most l times the minimum

back


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