Chapter 1 . . .
Communication Complexity
———– vs. ———–
Partition Numbers
(Based on 2 papers to appear at FOCS)
Chapter 1 . . .
Communication Complexity
———– vs. ———–
Partition Numbers
(Based on 2 papers to appear at FOCS)
Communication complexity
[Yao, ’79]
Alice
x
∈ {
0, 1
}
n
Bob
y
∈ {
0, 1
}
n
Deterministic protocols
1
1
1
1
1
1
1
1
1
1
1
1
1
1 1
1
1
1
1
1
1
1
1
1
1
1
1 1
1
1
1
1
1
1
1
1
1
1
1
1 1
1
1
0
0
0
0
0
0
0
0
0
0 0
0
0
0 0 0 0
0
0
0
Deterministic protocols
1
1
1
1
1
1
1
1
1
1
1
1
1
1 1
1
1
1
1
1
1
1
1
1
1
1
1 1
1
1
1
1
1
1
1
1
1
1
1
1 1
1
1
0
0
0
0
0
0
0
0
0
0 0
0
0
0 0 0 0
0
0
0
0
Deterministic protocols
1
1
1
1
1
1
1
1
1
1
1
1
1
1 1
1
1
1
1
1
1
1
1
1
1
1
1 1
1
1
1
1
1
1
1
1
1
1
1
1 1
1
1
0
0
0
0
0
0
0
0
0
0 0
0
0
0 0 0 0
0
0
0
0
0
1
0
1
0
1
A
B
B
A
Deterministic protocols
1
1
1
1
1
1
1
1
1
1
1
1
1
1 1
1
1
1
1
1
1
1
1
1
1
1
1 1
1
1
1
1
1
1
1
1
1
1
1
1 1
1
1
0
0
0
0
0
0
0
0
0
0 0
0
0
0 0 0 0
0
0
0
0
0
1
0
1
0
1
0
1
0
1
0
1
0
1
A
A
A
A
A
B
B
A
A
A
A
A
Deterministic protocols
1
1
1
1
1
1
1
1
1
1
1
1
1
1 1
1
1
1
1
1
1
1
1
1
1
1
1 1
1
1
1
1
1
1
1
1
1
1
1
1 1
1
1
0
0
0
0
0
0
0
0
0
0 0
0
0
0 0 0 0
0
0
0
0
0
1
0
1
0
1
0
1
0
1
0
1
0
1
A
A
A
A
A
B
B
B
B
B
B
B
B
B
B
A
A
A
A
A
Deterministic protocols
1
1
1
1
1
1
1
1
1
1
1
1
1
1 1
1
1
1
1
1
1
1
1
1
1
1
1 1
1
1
1
1
1
1
1
1
1
1
1
1 1
1
1
0
0
0
0
0
0
0
0
0
0 0
0
0
0 0 0 0
0
0
0
0
0
1
0
1
0
1
0
1
0
1
0
1
0
1
1 0 1 1 0 0 1 0 1 0 1 0 0 1 0 1
A
A
A
A
A
B
B
B
B
B
B
B
B
B
B
A
A
A
A
A
Deterministic protocols
1
1
1
1
1
1
1
1
1
1
1
1
1
1 1
1
1
1
1
1
1
1
1
1
1
1
1 1
1
1
1
1
1
1
1
1
1
1
1
1 1
1
1
0
0
0
0
0
0
0
0
0
0 0
0
0
0 0 0 0
0
0
0
0
0
1
0
1
0
1
0
1
0
1
0
1
0
1
1 0 1 1 0 0 1 0 1 0 1 0 0 1 0 1
A
A
A
A
A
B
B
B
B
B
B
B
B
B
B
A
A
A
A
A
B
B
B
B
B
B
B
B
B
B
P
cc
(
F
)
:
=
Deterministic communication complexity of
F
Partition number
χ
(
F
)
:
=
Least number of monochromatic
Deterministic protocols
1
1
1
1
1
1
1
1
1
1
1
1
1
1 1
1
1
1
1
1
1
1
1
1
1
1
1 1
1
1
1
1
1
1
1
1
1
1
1
1 1
1
1
0
0
0
0
0
0
0
0
0
0 0
0
0
0 0 0 0
0
0
0
0
Basic fact:
P
cc
(
F
)
≥
log
χ
(
F
)
[Kushilevitz–Nisan]:
P
cc
(
F
)
≤
O
(
log
χ
(
F
))
?
P
cc
(
F
)
:
=
Deterministic communication complexity of
F
Partition number
χ
(
F
)
:
=
Least number of monochromatic
Deterministic protocols
1
1
1
1
1
1
1
1
1
1
1
1
1
1 1
1
1
1
1
1
1
1
1
1
1
1
1 1
1
1
1
1
1
1
1
1
1
1
1
1 1
1
1
0
0
0
0
0
0
0
0
0
0 0
0
0
0 0 0 0
0
0
0
0
Basic fact:
P
cc
(
F
)
≥
log
χ
(
F
)
[Kushilevitz–Nisan]:
P
cc
(
F
)
≤
O
(
log
χ
(
F
))
?
P
cc
(
F
)
:
=
Deterministic communication complexity of
F
Partition number
χ
(
F
)
:
=
Least number of monochromatic
rectangles required to partition the communication matrix
Our results
Our results
I
Theorem 1:
∃
F
:
P
cc
(
F
)
≥
Ω
˜
(
log
1.5
χ
(
F
))
I
Theorem 2:
∃
F
:
P
cc
(
F
)
≥
Ω
˜
(
log
2
χ1
(
F
))
I
Corollary:
∃
F
:
P
cc
(
F
)
≥
Ω
˜
(
log
2
rank
(
F
))
I
Theorem 3:
∃
F
:
coNP
cc
(
F
)
≥
Ω
(
log
1.128
χ1
(
F
))
=
Our results
I
Theorem 1:
∃
F
:
P
cc
(
F
)
≥
Ω
˜
(
log
1.5
χ
(
F
))
I
Previously:
[Aho–Ullman–Yannakakis,
STOC’83]:
∀F
:
P
cc
(
F)
≤
O(log
2
χ(F))
[Kushilevitz–Linial–Ostrovsky,
STOC’96]:
∃F
:
P
cc
(
F)
≥
2
·
log
χ
(
F)
I
Theorem 2:
∃
F
:
P
cc
(
F
)
≥
Ω
˜
(
log
2
χ1
(
F
))
I
Corollary:
∃
F
:
P
cc
(
F
)
≥
Ω
˜
(
log
2
rank
(
F
))
I
Theorem 3:
∃
F
:
coNP
cc
(
F
)
≥
Ω
(
log
1.128
χ1
(
F
))
=
One-sided
partition numbers:
χ
(
F
) =
χ
1
(
F
) +
χ
0
(
F
)
,
χ
i
(
F
)
:
=
least number of rectangles
needed to partition
F
−
1
(
i
)
Our results
I
Theorem 1:
∃
F
:
P
cc
(
F
)
≥
Ω
˜
(
log
1.5
χ
(
F
))
I
Theorem 2:
∃
F
:
P
cc
(
F
)
≥
Ω
˜
(
log
2
χ1
(
F
))
I
Corollary:
∃
F
:
P
cc
(
F
)
≥
Ω
˜
(
log
2
rank
(
F
))
I
Theorem 3:
∃
F
:
coNP
cc
(
F
)
≥
Ω
(
log
1.128
χ1
(
F
))
=
One-sided
partition numbers:
χ
(
F
) =
χ
1
(
F
) +
χ
0
(
F
)
,
χ
i
(
F
)
:
=
least number of rectangles
needed to partition
F
−
1
(
i
)
Our results
I
Theorem 1:
∃
F
:
P
cc
(
F
)
≥
Ω
˜
(
log
1.5
χ
(
F
))
I
Theorem 2:
∃
F
:
P
cc
(
F
)
≥
Ω
˜
(
log
2
χ1
(
F
))
I
Previously:
Clique vs. Independent Set
[Yannakakis,
STOC’88]:
∀F
:
P
cc
(
F)
≤
O(log
2
χ1
(F))
I
Corollary:
∃
F
:
P
cc
(
F
)
≥
Ω
˜
(
log
2
rank
(
F
))
I
Theorem 3:
∃
F
:
coNP
cc
(
F
)
≥
Ω
(
log
1.128
χ1
(
F
))
=
Our results
I
Theorem 1:
∃
F
:
P
cc
(
F
)
≥
Ω
˜
(
log
1.5
χ
(
F
))
I
Theorem 2:
∃
F
:
P
cc
(
F
)
≥
Ω
˜
(
log
2
χ1
(
F
))
I
Corollary:
∃
F
:
P
cc
(
F
)
≥
Ω
˜
(
log
2
rank
(
F
))
Observation:
χ1
(
F
)
≥
rank
(
F
)
I
Theorem 3:
∃
F
:
coNP
cc
(
F
)
≥
Ω
(
log
1.128
χ1
(
F
))
=
Our results
I
Theorem 1:
∃
F
:
P
cc
(
F
)
≥
Ω
˜
(
log
1.5
χ
(
F
))
I
Theorem 2:
∃
F
:
P
cc
(
F
)
≥
Ω
˜
(
log
2
χ1
(
F
))
I
Corollary:
∃
F
:
P
cc
(
F
)
≥
Ω
˜
(
log
2
rank
(
F
))
I
Previously:
Log-rank conjecture
[Lov´asz–Saks,
FOCS’88]:
∀F
:
P
cc
(
F)
≤
log
O
(
1
)
rank
(F)
[Kushilevitz–Nisan–Wigderson,
FOCS’94]:
∃F
:
P
cc
(
F)
≥
Ω
(log
1.63
rank(
F))
I
Theorem 3:
∃
F
:
coNP
cc
(
F
)
≥
Ω
(
log
1.128
χ
1
(
F
))
=
Our results
I
Theorem 1:
∃
F
:
P
cc
(
F
)
≥
Ω
˜
(
log
1.5
χ
(
F
))
I
Theorem 2:
∃
F
:
P
cc
(
F
)
≥
Ω
˜
(
log
2
χ1
(
F
))
I
Corollary:
∃
F
:
P
cc
(
F
)
≥
Ω
˜
(
log
2
rank
(
F
))
I
Theorem 3:
∃
F
:
coNP
cc
(
F
)
≥
Ω
(
log
1.128
χ1
(
F
))
=
Nondeterministic protocols
Algorithmic definition:
1
Players
guess
a proof string
p
∈ {0, 1}
C
2
Alice
accepts depending on
(
x
,
p
)
3
Bob
accepts depending on
(
y
,
p
)
4
(
x
,
y
)
is
accepted
iff both players accept
NP
cc
(
F
)
:
=
Least
C
for which there is an above
type protocol accepting
F
−
1
(
1
)
Combinatorial definition:
NP
cc
(
F
)
:
=
log Cov
1
(
F
)
Cov
1
(
F
)
:
=
Least number of monochromatic
rectangles needed to cover
F
−
1
(
1
)
Nondeterministic protocols
1
1
1
1
1
1
1
1
1
1
1
1
1
1 1
1
1
1
1
1
1
1
1
1
1
1
1 1
1
1
1
1
1
1
1
1
1
1
1
1 1
1
1
0
0
0
0
0
0
0
0
0
0 0
0
0
0 0 0 0
0
0
0
0
NP
cc
=
log Cov
1
(
F
)
UP
cc
=
log
χ
1
(
F
)
Nondeterministic protocols
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1 1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 0
0
0
0
0
0
0
NP
cc
=
log Cov
1
(
F
)
UP
cc
=
log
χ
1
(
F
)
2UP
cc
=
log
χ
(
F
)
Nondeterministic protocols
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1 1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 0
0
0
0
0
0
0
NP
cc
=
log Cov
1
(
F
)
UP
cc
=
log
χ
1
(
F
)
2UP
cc
=
log
χ
(
F
)
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1 1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 0
0
0
0
0
0
0
[Yannakakis,
STOC’88
]:
P
cc
≤
(
UP
cc
)
2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1 1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 0
0
0
0
0
0
0
B
B
B
[Yannakakis,
STOC’88
]:
P
cc
≤
(
UP
cc
)
2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1 1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 0
0
0
0
0
0
0
A
A
A
B
B
B
A
A
A
B
B
B
A
A
A
B
B
B
[Yannakakis,
STOC’88
]:
P
cc
≤
(
UP
cc
)
2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1 1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 0
0
0
0
0
0
0
A
A
A
B
B
B
A
A
A
B
B
B
A
A
A
B
B
B
x
y
[Yannakakis,
STOC’88
]:
P
cc
≤
(
UP
cc
)
2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1 1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
A
A
A
B
B
B
A
A
A
B
B
B
A
A
A
B
B
B
x
y
x
y
;
[Yannakakis,
STOC’88
]:
P
cc
≤
(
UP
cc
)
2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1 1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
A
A
A
B
B
B
A
A
A
B
B
B
A
A
A
B
B
B
x
y
x
y
[Yannakakis,
STOC’88
]:
P
cc
≤
(
UP
cc
)
2
UP
cc
=
k
UP
cc
=
k
−
1
Our results
I
Theorem 1:
∃
F
:
P
cc
(
F
)
≥
Ω
˜
(
2UP
cc
(
F
)
1.5
)
I
Theorem 2:
∃
F
:
P
cc
(
F
)
≥
Ω
˜
(
UP
cc
(
F
)
2
)
Our results
I
Theorem 1:
∃
F
:
P
cc
(
F
)
≥
Ω
˜
(
2UP
cc
(
F
)
1.5
)
I
Theorem 2:
∃
F
:
P
cc
(
F
)
≥
Ω
˜
(
UP
cc
(
F
)
2
)
I
Theorem 3:
∃
F
:
coNP
cc
(
F
)
≥
Ω
(
UP
cc
(
F
)
1.128
)
P
coNP
Our results
I
Theorem 1:
∃
F
:
P
cc
(
F
)
≥
Ω
˜
(
2UP
cc
(
F
)
1.5
)
I
Theorem 2:
∃
F
:
P
cc
(
F
)
≥
Ω
˜
(
UP
cc
(
F
)
2
)
I
Theorem 3:
∃
F
:
coNP
cc
(
F
)
≥
Ω
(
UP
cc
(
F
)
1.128
)
P
coNP
Clique vs. Independent Set
[Yannakakis,
STOC’88
]:
Clique vs. Independent Set
is
complete
for
UP
cc
CIS
G
on a graph
G
= (
V
,
E
)
:
Alice
holds a clique
C
⊆
V
Bob
holds an independent set
I
⊆
V
Output
|
C
∩
I
| ∈ {
0, 1
}
UP
cc
reduces to CIS
[Yannakakis,
STOC’88
]
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1 1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 0
0
0
0
0
0
0
;
Fix a partition of
F
−
1
(
1
)
.
Construct
G
= (
V
,
E
)
where
V
=
{rectangles}
and
{
u
,
v
} ∈
E
iff
u
and
v
share a row
F
reduces to CIS
G
:
Alice
(
Bob
) maps her
row
(
column
)
UP
cc
reduces to CIS
[Yannakakis,
STOC’88
]
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1 1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 0
0
0
0
0
0
0
;
Fix a partition of
F
−
1
(
1
)
.
Construct
G
= (
V
,
E
)
where
V
=
{rectangles}
and
{
u
,
v
} ∈
E
iff
u
and
v
share a row
More on CIS
Yannakakis’s motivation:
Size of LPs for the vertex packing polytope of
G
Breakthrough: [Fiorini et al., STOC’12]
For an
n
-node graph:
UP
cc
(
CIS
G
) =
log
n
∀
G
:
P
cc
(
CIS
G
)
≤
O
(
log
2
n
)
∀
G
:
coNP
cc
(
CIS
G
)
≤
O
(
log
2
n
)
∀
G
:
Yannakakis’s question:
coNP
cc
(
CIS
G
)
≤
O
(
log
n
)
?
∀
G
:
Alon–Saks–Seymour conjecture:
chr
(
G
)
≤
bp
(
G
) +
1
?
More on CIS
Yannakakis’s motivation:
Size of LPs for the vertex packing polytope of
G
Breakthrough: [Fiorini et al., STOC’12]
For an
n
-node graph:
UP
cc
(
CIS
G
) =
log
n
∀
G
:
P
cc
(
CIS
G
)
≤
O
(
log
2
n
)
∀
G
:
coNP
cc
(
CIS
G
)
≤
O
(
log
2
n
)
∀
G
:
Yannakakis’s question:
coNP
cc
(
CIS
G
)
≤
O
(
log
n
)
?
∀
G
:
Alon–Saks–Seymour conjecture:
chr
(
G
)
≤
bp
(
G
) +
1
?
∀
G
:
chr
(
G
)
:
=
Chromatic number of
G
More on CIS
Yannakakis’s motivation:
Size of LPs for the vertex packing polytope of
G
Breakthrough: [Fiorini et al., STOC’12]
For an
n
-node graph:
UP
cc
(
CIS
G
) =
log
n
∀
G
:
P
cc
(
CIS
G
)
≤
O
(
log
2
n
)
∀
G
:
coNP
cc
(
CIS
G
)
≤
O
(
log
2
n
)
∀
G
:
Yannakakis’s question:
coNP
cc
(
CIS
G
)
≤
O
(
log
n
)
?
∀
G
:
Alon–Saks–Seymour conjecture:
chr
(
G
)
≤
bp
(
G
) +
1
?
∀
G
:
More on CIS
Yannakakis’s motivation:
Size of LPs for the vertex packing polytope of
G
Breakthrough: [Fiorini et al., STOC’12]
For an
n
-node graph:
UP
cc
(
CIS
G
) =
log
n
∀
G
:
P
cc
(
CIS
G
)
≤
O
(
log
2
n
)
∀
G
:
coNP
cc
(
CIS
G
)
≤
O
(
log
2
n
)
∀
G
:
Yannakakis’s question:
coNP
cc
(
CIS
G
)
≤
O
(
log
n
)
?
∀
G
:
Polynomial Alon–Saks–Seymour conjecture:
chr
(
G
)
≤
poly
(
bp
(
G
)
)
?
More on CIS
Yannakakis’s motivation:
Size of LPs for the vertex packing polytope of
G
Breakthrough: [Fiorini et al., STOC’12]
For an
n
-node graph:
UP
cc
(
CIS
G
) =
log
n
∀
G
:
P
cc
(
CIS
G
)
≤
O
(
log
2
n
)
∀
G
:
coNP
cc
(
CIS
G
)
≤
O
(
log
2
n
)
∀
G
:
Yannakakis’s question:
coNP
cc
(
CIS
G
)
≤
O
(
log
n
)
?
∀
G
:
Polynomial Alon–Saks–Seymour conjecture:
chr
(
G
)
≤
poly
(
bp
(
G
)
)
?
∀
G
:
[Alon–Haviv]
=
⇒
=
More on CIS
Yannakakis’s motivation:
Size of LPs for the vertex packing polytope of
G
Breakthrough: [Fiorini et al., STOC’12]
For an
n
-node graph:
UP
cc
(
CIS
G
) =
log
n
∀
G
:
P
cc
(
CIS
G
)
≤
O
(
log
2
n
)
∀
G
:
coNP
cc
(
CIS
G
)
≤
O
(
log
2
n
)
∀
G
:
Yannakakis’s question:
coNP
cc
(
CIS
G
)
≤
O
(
log
n
)
?
∀
G
:
Polynomial Alon–Saks–Seymour conjecture:
chr
(
G
)
≤
poly
(
bp
(
G
)
)
?
∀
G
:
[Alon–Haviv]
=
⇒
=
More on CIS
Yannakakis’s motivation:
Size of LPs for the vertex packing polytope of
G
Breakthrough: [Fiorini et al., STOC’12]
For an
n
-node graph:
UP
cc
(
CIS
G
) =
log
n
∀
G
:
P
cc
(
CIS
G
)
≤
O
(
log
2
n
)
∀
G
:
coNP
cc
(
CIS
G
)
≤
O
(
log
2
n
)
∀
G
:
Yannakakis’s question:
coNP
cc
(
CIS
G
)
≤
O
(
log
n
)
?
∀
G
:
Polynomial Alon–Saks–Seymour conjecture:
chr
(
G
)
≤
poly
(
bp
(
G
)
)
?
∀
G
:
[Alon–Haviv]
=
⇒
=
⇒
[Bousquet et al.]
Communication:
I
Theorem 1:
∃
F
:
P
cc
(
F
)
≥
Ω
˜
(
2UP
cc
(
F
)
1.5
)
I
Theorem 2:
∃
F
:
P
cc
(
F
)
≥
Ω
˜
(
UP
cc
(
F
)
2
)
Communication:
I
Theorem 1:
∃
F
:
P
cc
(
F
)
≥
Ω
˜
(
2UP
cc
(
F
)
1.5
)
I
Theorem 2:
∃
F
:
P
cc
(
F
)
≥
Ω
˜
(
UP
cc
(
F
)
2
)
I
Theorem 3:
∃
F
:
coNP
cc
(
F
)
≥
Ω
(
UP
cc
(
F
)
1.128
)
2-step strategy:
Decision tree:
I
Theorem 1:
∃
f
:
P
dt
(
f
)
≥
Ω
˜
(
2UP
dt
(
f
)
1.5
)
I
Theorem 2:
∃
f
:
P
dt
(
f
)
≥
Ω
˜
(
UP
dt
(
f
)
2
)
I
Theorem 3:
∃
f
:
coNP
dt
(
f
)
≥
Ω
(
UP
dt
(
f
)
1.128
)
2-step strategy:
Decision tree models
f
:
{0, 1}
n
→ {0, 1}
P
dt
(
f
)
=
Deterministic
query complexity
NP
dt
(
f
)
=
Nondeterministic
query complexity
=
1-certificate complexity
=
DNF width
Quadratic
P
-vs-
UP
gap
0
1 1 1 1 1
1
0 0
1 1 1
0
1
0
1
0
1
1
0
1 1 1
0
1 1 1 1 1
0
1 1
0
1 1 1
Warm-up example
f
:
M
is
k
×
k
matrix with entries in
{
0, 1
}
f
(
M
) =
1
⇐⇒
M
contains a
unique all-1 column
NP
dt
=
2
k
−
1
Quadratic
P
-vs-
UP
gap
1
1
0
1
0
0
1
1
0
0
1
Warm-up example
f
:
M
is
k
×
k
matrix with entries in
{
0, 1
}
f
(
M
) =
1
⇐⇒
M
contains a
unique all-1 column
NP
dt
=
2
k
−
1
Quadratic
P
-vs-
UP
gap
?
Warm-up example
f
:
M
is
k
×
k
matrix with entries in
{
0, 1
}
f
(
M
) =
1
⇐⇒
M
contains a
unique all-1 column
NP
dt
=
2
k
−
1
Quadratic
P
-vs-
UP
gap
1
Warm-up example
f
:
M
is
k
×
k
matrix with entries in
{
0, 1
}
f
(
M
) =
1
⇐⇒
M
contains a
unique all-1 column
NP
dt
=
2
k
−
1
Quadratic
P
-vs-
UP
gap
1 1 1 1 1 1
1
1 1 1 1
1
1
1
1
1 1
1 1 1
?
Warm-up example
f
:
M
is
k
×
k
matrix with entries in
{
0, 1
}
f
(
M
) =
1
⇐⇒
M
contains a
unique all-1 column
NP
dt
=
2
k
−
1
Quadratic
P
-vs-
UP
gap
1 1 1 1 1 1
1
1 1 1 1
1
1
0
1
1
1 1
1 1 1
Warm-up example
f
:
M
is
k
×
k
matrix with entries in
{
0, 1
}
f
(
M
) =
1
⇐⇒
M
contains a
unique all-1 column
NP
dt
=
2
k
−
1
Quadratic
P
-vs-
UP
gap
1 1 1 1 1 1
1 1 1 1 1 1
0
1 1 1
0
1
1
0
1 1 1
0
1 1 1
1 1
1 1
0
1 1 1
?
Warm-up example
f
:
M
is
k
×
k
matrix with entries in
{
0, 1
}
f
(
M
) =
1
⇐⇒
M
contains a
unique all-1 column
NP
dt
=
2
k
−
1
Quadratic
P
-vs-
UP
gap
1
1
0
1
0
0
1
0
1
0
1
Actual gap example
f
:
M
is
k
×
k
matrix with entries in
{
0, 1
}×
([
k
]
×
[
k
]
∪ {⊥})
f
(
M
) =
1
⇐⇒
M
contains a
unique all-1 column
that has a
linked list
through
0
’s in other columns
UP
dt
≈
2
k
−
1
Quadratic
P
-vs-
UP
gap
1 1 1
1
1
1
1
1
1 1
1 1
1 1 1
?
Actual gap example
f
:
M
is
k
×
k
matrix with entries in
{
0, 1
}×
([
k
]
×
[
k
]
∪ {⊥})
f
(
M
) =
1
⇐⇒
M
contains a
unique all-1 column
that has a
linked list
through
0
’s in other columns
UP
dt
≈
2
k
−
1
Quadratic
P
-vs-
UP
gap
1 1 1
1
1
1
0
1
1
1 1
1 1
1 1 1
Actual gap example
f
:
M
is
k
×
k
matrix with entries in
{
0, 1
}×
([
k
]
×
[
k
]
∪ {⊥})
f
(
M
) =
1
⇐⇒
M
contains a
unique all-1 column
that has a
linked list
through
0
’s in other columns
UP
dt
≈
2
k
−
1
Quadratic
P
-vs-
UP
gap
1 1 1 1
1
1 1 1
1
0
1
1
1
1 1
1 1
1 1
1 1 1
?
Actual gap example
f
:
M
is
k
×
k
matrix with entries in
{
0, 1
}×
([
k
]
×
[
k
]
∪ {⊥})
f
(
M
) =
1
⇐⇒
M
contains a
unique all-1 column
that has a
linked list
through
0
’s in other columns
UP
dt
≈
2
k
−
1
Quadratic
P
-vs-
UP
gap
1 1 1 1
1
1 1 1
1
0
1
1
0
1
1 1
1 1
1 1
1 1 1
Actual gap example
f
:
M
is
k
×
k
matrix with entries in
{
0, 1
}×
([
k
]
×
[
k
]
∪ {⊥})
f
(
M
) =
1
⇐⇒
M
contains a
unique all-1 column
that has a
linked list
through
0
’s in other columns
UP
dt
≈
2
k
−
1
Quadratic
P
-vs-
UP
gap
1 1 1 1 1 1
1 1 1
1
0
1 1
1
0
1
1
1 1 1 1 1 1
1 1
?
1 1 1
Actual gap example
f
:
M
is
k
×
k
matrix with entries in
{
0, 1
}×
([
k
]
×
[
k
]
∪ {⊥})
f
(
M
) =
1
⇐⇒
M
contains a
unique all-1 column
that has a
linked list
through
0
’s in other columns
UP
dt
≈
2
k
−
1
Quadratic
P
-vs-
UP
gap
1 1 1 1 1 1
1 1 1
1
0
1 1
1
0
1
1
1 1 1 1 1 1
1 1
0
1 1 1
Actual gap example
f
:
M
is
k
×
k
matrix with entries in
{
0, 1
}×
([
k
]
×
[
k
]
∪ {⊥})
f
(
M
) =
1
⇐⇒
M
contains a
unique all-1 column
that has a
linked list
through
0
’s in other columns
UP
dt
≈
2
k
−
1
Quadratic
P
-vs-
UP
gap
1 1 1 1 1 1
1 1 1
1 1
0
1 1 1
0
1
1
0
1 1 1
0
1 1 1 1 1 1
1 1
0
1 1 1
?
Actual gap example
f
:
M
is
k
×
k
matrix with entries in
{
0, 1
}×
([
k
]
×
[
k
]
∪ {⊥})
f
(
M
) =
1
⇐⇒
M
contains a
unique all-1 column
that has a
linked list
through
0
’s in other columns
UP
dt
≈
2
k
−
1
Quadratic
P
-vs-
UP
gap
1 1 1 1 1 1
1 1 1
1 1
0
1 1 1
0
1
1
0
1 1 1
0
1 1 1 1 1 1
1 1
0
1 1 1
?
Actual gap example
f
:
M
is
k
×
k
matrix with entries in
{
0, 1
}×
([
k
]
×
[
k
]
∪ {⊥})
f
(
M
) =
1
⇐⇒
M
contains a
unique all-1 column
that has a
linked list
through
0
’s in other columns
UP
dt
≈
2
k
−
1
P
dt
≈
k
2
Other separations inspired by our function
[Ambainis–Balodis–Belovs–Lee–Santha–Smotrovs]
(also [Mukhopadhyay–Sanyal]):
P
dt
(
f
)
≥
ZPP
dt
(
f
)
2
Counterexample to
Saks–Wigderson’86
!
P
dt
(
f
)
≥
BQP
dt
(
f
)
4
ZPP
dt
(
f
)
≥
RP
dt
(
f
)
2
[Ben-David]:
Other query separations
I
Theorem 2:
UP
dt
(
f
) =
k
P
dt
(
f
) =
k
2
(Previous slide)
⇐
I
Theorem 1:
2UP
dt
(
AND
◦
f
k
) =
k
2
P
dt
(
AND
◦
f
k
) =
k
3
Power
1.5
gap—cf.
log
3
4
≈
1.26
from
[Savick ´y’03 / Belovs’06]
I
Theorem 3:
∃
f
:
coNP
dt
(
f
)
≥
Ω
(
UP
dt
(
f
)
1.128
)
Composed functions
f
◦
g
n
f
f
z
1
z
2
z
3
z
4
z
5
g
g
g
g
g
x
1
y
1
x
2
y
2
x
3
y
3
x
4
y
4
x
5
y
5
Compose with g
n
Examples:
•
Set-disjointness:
OR
◦
AND
n
•
Inner-product:
XOR
◦
AND
n
•
Equality:
AND
◦ ¬
XOR
n
Simulation Theorem Template:
Simulate
cost-
C
protocol for
f
◦
g
n
in model
M
cc
using
height-
C
decision tree for
f
in model
M
dt
Composed functions
f
◦
g
n
f
f
z
1
z
2
z
3
z
4
z
5
g
g
g
g
g
x
1
y
1
x
2
y
2
x
3
y
3
x
4
y
4
x
5
y
5
Compose with
g
n
In general:
g
:
{
0, 1
}
b
× {
0, 1
}
b
→ {
0, 1
}
is a small gadget
Alice
holds
x
∈
(
{
0, 1
}
b
)
n
Bob
holds
y
∈
(
{
0, 1
}
b
)
n
Inputs
x
and
y
encode
z
:
=
g
n
(
x
,
y
)
Simulation Theorem Template:
Simulate
cost-
C
protocol for
f
◦
g
n
in model
M
cc
using
height-
C
decision tree for
f
in model
M
dt
Composed functions
f
◦
g
n
f
f
z
1
z
2
z
3
z
4
z
5
g
g
g
g
g
x
1
y
1
x
2
y
2
x
3
y
3
x
4
y
4
x
5
y
5
Compose with
g
n
In general:
g
:
{
0, 1
}
b
× {
0, 1
}
b
→ {
0, 1
}
is a small gadget
Alice
holds
x
∈
(
{
0, 1
}
b
)
n
Bob
holds
y
∈
(
{
0, 1
}
b
)
n
Inputs
x
and
y
encode
z
:
=
g
n
(
x
,
y
)
Simulation Theorem Template:
Simulate
cost-
C
protocol for
f
◦
g
n
in model
M
cc
using
height-
C
decision tree for
f
in model
M
dt
Composed functions
f
◦
g
n
OR
OR OR OR OR OR
OR
z
1
z
2
z
3
z
4
z
5
x
1
y1
x
2
y2
x
3
y3
x
4
y4
x
5
y5
Bad example:
Gadget must be
chosen carefully!
In general:
g
:
{
0, 1
}
b
× {
0, 1
}
b
→ {
0, 1
}
is a small gadget
Alice
holds
x
∈
(
{
0, 1
}
b
)
n
Bob
holds
y
∈
(
{
0, 1
}
b
)
n
Inputs
x
and
y
encode
z
:
=
g
n
(
x
,
y
)
Simulation Theorem Template:
Simulate
cost-
C
protocol for
f
◦
g
n
in model
M
cc
using
height-
C
decision tree for
f
in model
M
dt
Known simulation theorems
Model
Gadget
Reference
P
g
(
x
,
y
)
:
=
y
x
where
|
y
|
=
n
Θ
(
1
)
[Raz–McKenzie,
FOCS’97
]
NP
g
(
x
,
y
)
:
=
h
x
,
y
i
mod 2
where
|
x
|
,
|
y
|
=
Θ
(
log
n
)
[GLMZW,
STOC’15
]
PP
Constant-size
g
[Sherstov,
STOC’08
],
[Shi–Zhu,
QIC’09
]
Simulation for P (Our formulation):
Known simulation theorems
Model
Gadget
Reference
P
g
(
x
,
y
)
:
=
y
x
where
|
y
|
=
n
Θ
(
1
)
[Raz–McKenzie,
FOCS’97
]
NP
g
(
x
,
y
)
:
=
h
x
,
y
i
mod 2
where
|
x
|
,
|
y
|
=
Θ
(
log
n
)
[GLMZW,
STOC’15
]
PP
Constant-size
g
[Sherstov,
STOC’08
],
[Shi–Zhu,
QIC’09
]
P
BPP
NP
MA
SBP
WAPP
PostBPP
PP
corruption
smooth rectangle
approx rank
+Communication:
I
Theorem 1:
∃
F
:
P
cc
(
F
)
≥
Ω
˜
(
2UP
cc
(
F
)
1.5
)
I
Theorem 2:
∃
F
:
P
cc
(
F
)
≥
Ω
˜
(
UP
cc
(
F
)
2
)
I
Theorem 3:
∃
F
:
coNP
cc
(
F
)
≥
Ω
(
UP
cc
(
F
)
1.128
)
Decision tree:
I
Theorem 1:
∃
f
:
P
dt
(
f
)
≥
Ω
˜
(
2UP
dt
(
f
)
1.5
)
I
Theorem 2:
∃
f
:
P
dt
(
f
)
≥
Ω
˜
(
UP
dt
(
f
)
2
)
I
Theorem 3:
∃
f
:
coNP
dt
(
f
)
≥
Ω
(
UP
dt
(
f
)
1.128
)
Future directions
In progress:
•
Find an
F
with
BPP
cc
(
F
)
2UP
cc
(
F
)
Solved for query complexity:
[Kothari–Racicot-Desloges–Santha, RANDOM’15]
Open problems:
•
Simulation theorem for
BPP
•
Improve gadget size down to
b
=
O
(
1
)
(Gives new proof of
Ω
(
n
)
bound for disjointness)
Big challenges:
•
Log-rank conjecture
•
Lower bounds against
PH
cc
(or
AM
cc
)
Future directions
In progress:
•
Find an
F
with
BPP
cc
(
F
)
2UP
cc
(
F
)
Solved for query complexity:
[Kothari–Racicot-Desloges–Santha, RANDOM’15]
Open problems:
•
Simulation theorem for
BPP
•
Improve gadget size down to
b
=
O
(
1
)
(Gives new proof of
Ω
(
n
)
bound for disjointness)
Big challenges:
•
Log-rank conjecture
•
Lower bounds against
PH
cc
(or
AM
cc
)
