inapproximability from different hardness assumptions prahladh harsha tifr 2011 school on...
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Inapproximability from different
hardness assumptions
Prahladh Harsha TIFR
2011 School on Approximability
Hardness of approximation
Worst case hardness
PCP theorem, Hardness of Label cover Unique Games Conjecture
Average case hardness
Feige’s Random-3-SAT assumption
Label CoverLabel Cover (LC)G – Bipartite graph1, 2 – labels(projection) constraint per edge ce:1 2
Edge e is satisfied if ce(σ1)=σ2
Goal: Find an assignment to vertices that satisfies the most edges
Gap(α,β)-LC: Distinguish between instances
• At least α fraction of constraints satisfied
• At most β fraction of constraints satisfied
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PCP TheoremLabel Cover (LC)G – Bipartite graph1, 2 – labels(projection) constraint per edge ce:1 2
Edge e is satisfied if ce(σ1)=σ2
PCP Theorem […., AS’92, ALMSS’92]
Gap(1,0.9999)-LC is NP-hard
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Stronger form of PCP Theorem
Label Cover (LC)G – Bipartite graph1, 2 – labels(projection) constraint per edge ce:1 2
Edge e is satisfied if ce(σ1)=σ2
PCP Theorem + Repetition Theorem [Raz’95]
For every constant δ there exists alphabets 1,2, Gap(1,δ)-LC is NP-hard
(starting point for all tight hardness of approximation reductions)
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Proving tight hardness results
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Outer Verifier
Dictatorship Test
[Fourier Analysis]+(composition)
Inner Verifier
Proving tight hardness results
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+direction reduction SETCOVER
Lattice probs.
Fourier analysis MAX3LIN, MAX3SAT,
CLIQUE
Fourier analysis MAXCUT, VERTEX-COVER
(with unique constraints)Unique Games Conjecture
Stronger form of PCP Theorem
Label Cover (LC)G – Bipartite graph1, 2 – labels(projection) constraint per edge ce:1 2
Edge e is satisfied if ce(σ1)=σ2
PCP Theorem + Repetition Theorem [Raz’95]
For every constant δ there exists alphabets 1,2, Gap(1,δ)-LC is NP-hard
(mother of all tight hardness of approximation reductions)
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Label Cover Constructions
Low Degree Test (LDT) [RS’92] Given function f:Fm F (F – field), check if f
is the evaluation of a low-degree polynomial without reading all of F
f:Fm F
Use fact that restriction of low-degree polynomial to a line is still low-degree
Label Cover for LDT
Points tablef:Fm F Lines table
flines
Constraint: flines(l)(x) = f(x)
Large Alphabet
Size
Label Cover -- LDT [AS’97, RS’97] Completeness: If f:Fm F is a low-degree polynomial, there
exists lines table flines such that
Pr[flines(l)(x) = f(x)] = 1 Soundness: If f:Fm F is “far” from being low-degree
polynomial, there for all lines table flines we have
Pr[flines(l)(x) = f(x)] ≤ δ
Label Cover for NP Encode problem in NP using polynomials
to lift the label cover for LDT to all of NP
Label Cover for NP [RS’97, AS’97]:
For every alphabet and error δ=1/log||, Gap(1,δ)-LC is NP-hard, if || > npolylog n
Caveat: Large Alphabet SizeRenders result “useless” for hardness results
Alphabet Reduction [MR’08, DH’09]
Alphabet Reduction: Label Cover instance with large alphabet size
Label Cover instance with small alphabet size
Idea: Recurse!![in the style of AS’92]
Use an “Inner” Label Cover to reduce alphabet of outer label cover
Alphabet Reduction
Alphabet Reduction
Alphabet Reduction Alphabet Size
Reduced
However
3-partite graph instead of bipartite
Idea: [2-query composition DH’09]
Combine leftmost and rightmost components by identifying nodes in left partition
(combine all left-neighbours of a right vertex)
Label Cover for NPPerforming alphabet reduction repeatedly:
Label Cover for NP [MR’08, DH’09]:
For every alphabet and error δ=1/log||, Gap(1,δ)-LC is NP-hard.
Advantages:• Sub-constant error achievable• Nearly linear sized reduction
Label Cover variants Some hardness reductions require more
structure of the label cover instance
[KH’04] Hardness of Balanced homogenous linear equations MAXBISECTION
Mixing property Bipartite graph is a good sampler
Smoothness
Open Questions Sliding Scale Conjecture [BGLR’93]
For every alphabet and error δ=1/poly||, Gap(1,δ)-LC is NP-hard
(current results only obtain δ=1/log|| )
Obtain polynomial sized mixing and smooth PCPs (current constructions require subexponential sized proofs)
Average Case Hardness
Assumptions
Inapproximability results based on Worst case hardness assumptions (so far)
Average case hardness assumptions Cryptographic assumptions Random 3SAT hardness (Feige)
Random 3SAT n variables x1, x2, …., xn
m = Cn clauses (xi v xj v xk) Chosen randomly and independently
C – small, satisfiable w.h.p C – large, unsatisfiable w.h.p
Random 3SAT – large C
For large C Typical – unsatisfiable In fact at most (7/8 + δ) clauses satisfiable Rare – satisfiable Rare even for (1-δ) clauses to be satisfied
Random 3SAT – large C Proofs of unsatisfiability (coNP proof)
When C > √n, can find short of proof of unsatisfiability w.h.p.
For large constant < C < √n, though unsatisfiable, current techniques do not prove unsatisfiability
Feige’s Random 3SAT hypothesis For all 0<δ<1/8, there exists a large
constant C, and there does NOT exist a polynomial time algorithm that
INPUT: Random 3CNF formula with n variables and Cn clauses
OUTPUT: “typical” or “rare” on most inputs (> 50%) output typical but
never outputs “typical” on a rare instance (i.e, (1-δ) satisfiable )
MAX3SAT approximability
Feige’s Random 3SAT hypothesis
MAX3SAT is inapproximable (within
polynomial time) to a factor better than (7/8 +δ), for all δ>0.
MAX3AND
INPUT Boolean formula on n variables x1, x2, …., xn
m ANDs of 3 literals (xi ∧ xj ∧ xk)
OUTPUT Assignment
OBJECTIVE maximizes number of ANDs being satisfied
MAX3AND - approximability Feige’s Random 3SAT hypothesis
Not possible to approximate better than a factor (1/2 +δ) In particular, can’t distinguish between
MAX3AND instances > (1/4 - δ) satisfiable < (1/8 + δ ) satisfiable
(even for random MAX3AND instances)
Why MAX3AND? Gives inapproximability results
MAX-COMPLETE BIPARITITE GRAPH MINBISECTION DENSE k-SUBGRAPH 2-CATALOG SEGMENTATION
(not approximable beyond a particular constant)Previously, no known inapproximability results
Max Complete Bipartite Graph INPUT
n x n bipartite graph OUTPUT
k x k complete bipartite subgraph OBJECTIVE
Maximize k
Feige’s Hypothesis implies can’t approximate better than (1/2 + δ) (reduction from hardness of randomMAX3AND)
Reduction from MAX3AND Given a random MAX3AND with m ANDs construct bipartite graph
• Vertices on each side• m ANDS
• Edges• If two ANDS can
be satisfied simulatenously
Reduction from MAX3AND (contd) MAX3AND instance – (1/4 -δ ) satsifiable
Corresponding vertices from a kxk complete bipartite graph with k = (1/4 -δ )m
random MAX3AND instance – (1/8+δ ) satisfiable Easy to check that for a random MAX3AND
instance, whp every (1/8 +δ)m ANDs involve at least (n+1) literals
Any kxk bipartite graph with k > (1/8+δ)m involves a variable and its negation and hence not complete
Hardness of random MAX3AND Algorithm for random 3SAT (refuting
Feige’s hypothesis)
Idea: View input random 3CNF formula as a 3AND formula and use algorithm for random MAX3AND 3CNF - (7/8 +δ)-satisfiable
3AND formula – (1/8 +δ)-satisfiable 3CNF – (1-δ) satisfiable
3AND formula – (1/4 -δ)-satisfiable
Not exactly true, instead will use simple checks and SDPs to detect non-typical behavior
Hardness of MAX3AND (contd) Input: random 3SAT instance Algorithm
1. If any literal does not occur (3C/2±δ) times, output “rare”
2. Construct graphs G12, G23, G31whose vertices are all 2n literals and edges as follows:G12 :(xi,xj) is there is a clause of the form (xi,v xj v xk)
3. Run MAXCUT SDP on all 3 graphs, if SDP outputs larger than (1/2+δ)m, output “rare”
4. Run MAX3AND algorithm on instance If output > (1/4-δ), output “rare” If output < (1/8+δ), output “typical”
Hardness of MAX3AND (contd) Typical Instances ( < (7/8+δ)-satisfiable)
Easy to check algorithm outputs “typical” on most typical instances
Rare instance (> (1-δ)-satisfiable) Literal-occurrence and SDP checks ensure that
when viewed as a NOT-ALL-EQUAL-SAT instance, no assignment satisfies > (3/4+δ) clauses
Hence, at least (1/4-δ)-clauses are satisfied as ANDs for which the algorithm outputs “rare”
Random 3SAT assumption Feige’s hypothesis
MAX3AND inapproximable to better than ½ (even on random instances)
Inapproximability results COMPLETE-BIPARTITE-GRAPH, MINBISECTION,
DENSEST k-SUBGRAPH, ….
Other assumptions? Maximal number of equations satisfiable in
a random linear system [Ale’03]
Implies Feige’s hypothesis
Inapproximability of nearest-codeword-problem to within n1-δ
Hard to distinguish low-rigidity matrices and random matrices
Quasirandom PCPs [Kho’04] Suffices to having following quasi-
randomess of 3SAT For any set of half of the variables, (1/8±δ)-
fraction of clauses have all 3 variables from this set
Khot constructed PCPs with this quasirandom property leading to inapproximability results for earlier problems (based on worst case hardness)
Quasirandom PCPs PCPs which exhibit very different query
behaviour on YES and NO instance PCP verifier makes d queries
NO instances: For any set of half the proof locations, the probability that all the d queries are in the set ≈ 2-d
YES instances: There is a set of half the proof location, which the verifier queries more frequently ( > 2-(d-1))
Open Problems (Dis)prove Feige’s hypothesis
Connections between average complexity and approximation complexity
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