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![Page 1: Inapproximability from different hardness assumptions Prahladh Harsha TIFR 2011 School on Approximability.](https://reader030.fdocuments.net/reader030/viewer/2022032703/56649d1a5503460f949f04b9/html5/thumbnails/1.jpg)
Inapproximability from different
hardness assumptions
Prahladh Harsha TIFR
2011 School on Approximability
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Hardness of approximation
Worst case hardness
PCP theorem, Hardness of Label cover Unique Games Conjecture
Average case hardness
Feige’s Random-3-SAT assumption
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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
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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
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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
(mother of all tight hardness of approximation reductions)
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Label Cover Constructions
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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
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Label Cover for LDT
Points tablef:Fm F Lines table
flines
Constraint: flines(l)(x) = f(x)
Large Alphabet
Size
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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
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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
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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
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Alphabet Reduction
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Alphabet Reduction
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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)
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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
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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
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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)
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Average Case Hardness
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Assumptions
Inapproximability results based on Worst case hardness assumptions (so far)
Average case hardness assumptions Cryptographic assumptions Random 3SAT hardness (Feige)
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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
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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
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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
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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 )
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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.
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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
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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)
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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
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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)
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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
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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
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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
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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”
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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”
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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, ….
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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
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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)
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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))
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Open Problems (Dis)prove Feige’s hypothesis
Connections between average complexity and approximation complexity
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