Circuit Bounds for Bipartite Matching in Planar Grid Graphs
Aayush Ojha, 13009
Sarthak Garg, 13624
Project Advisor:- Dr. Raghunath Tewari
Preliminaries
- Grid Graphs:- Graphs that can be embedded in a planar integral lattice and all edges are between neighboring points in lattice.
- :- Family of Circuits with AND, OR, and NOT gates and having unbounded fan-in for AND and OR gates. Constant depth.
- :- All gates from and gates for all . "AC with counters". Constant depth.
- Semigroups:- A set along with an associative binary operation over it.
- Monoids:- Semigroups with identity.
- Solvable Monoid:- All groups contained in Monoid are solvable.
- Aperiodic Monoid:- All groups contained in Monoid are trivial.
AC^0
AC^0
Mod\, m
ACC^0
m
Preliminaries
Monoid Word Problem:- Given
find ,
where is a monoid and is monoid product.
Non Uniform DFA over Monoids:- For a length n, let advice be a mapping from alphabets to monoid elements then for a string
we output .
iff is recognized by NUDFA over aperiodic monoid.
iff is recognized by NUDFA over solvable monoid.
x_1, x_2,\ldots,x_n \in \mathcal{M}
x_1\circ x_2 \circ \ldots x_n
\mathcal{M}
\circ
a_1a_2a_3\ldots a_n
f(a_1)\circ f(a_2) \circ \ldots \circ f(a_n)
f
L \in AC^0
L
L \in ACC^0
L
Preliminaries
Concatenation of grid graphs
G_1
G_2
G_1 \oplus G_2
Motivation
Recently, It was shown that Bipartite Matching for Grid Graph is
in by Hansen et al.
They reduced it to Monoid word problem using NUDFA over Monoid. Then they showed that underlying Monoid is solvable. Although this might be a loose result.
So we aim to show that underlying Monoid is aperiodic. This will put Bipartite Matching for Grid Graph in .
Known lower bound for this problem is thus if proved this will
be a tight bound.
ACC^0
AC^0
AC^0
Conditions for Aperiodic Monoid
Underlying Monoid:-
\mathcal{M} = \{ G^{\mathcal{M}}: \text{G is an odd length grid layered planar graph} \} \cup \{1\} \cup \{0\}
G^{\mathcal{M}} = (X,Y,R)
(A,B) \in R \text{ there is a matching for all vertices except } \overline{A} \text{ and } B
X\text{ is set of vertices in leftmost layer of } G
Y\text{ is set of vertices in rightmost layer of } G
Binary Operation:-
(X_1,Y_1,R)(X_2,Y_2,S) = (X_1,Y_2,R \circ S) \text{ for } Y_1 = X_2.
\text{Otherwise, } (X_1,Y_1,R)(X_2,Y_2,S) = 0
1x=x=x1
0x=0=x0
Conditions for Aperiodic Monoid
Monoid Product is corresponding to graph concatenation
\text{Let } S \text{ be any grid graph of } w \text{ rows and } l \text{(odd) columns}
G = S \oplus S \oplus \ldots
G \text{ is an infinite graph. We call each } S \text{ a block.}
\text{Suppose }G \text{ has a perfect matching and }M_i\text{ be matching edges in } i_{th} \text{ block.}
\tilde{G_i} = M_i \cup M_{i+1} \text{. We say } M_i \text{ matching and } M_{i+1} \text{ projected edges in } \tilde{G_i}
\tilde{G} = \tilde{G_1} \oplus \tilde{G_2} \oplus \ldots
Conditions for Aperiodic Monoid
\tilde{G} = \tilde{G_1} \oplus \tilde{G_2} \oplus \ldots
\text{Non-existence of infinite path in } \tilde{G}
\forall m \exists( n = km) \text{ st. no path exists from first layer }
\text{ to rightmost layer of } n_{th} \text{ block}
Important Properties
\tilde{G} \text{ can have infinite paths, path segments and closed loops}
\text{For every matching edge in } i_{th}\text{ block there is a corresponding }
\text{projected edge in } (i-1)_{th}\text{ block.}
\text{Direction of a matching edge and projected edge is opposite.}
\text{For every path, P, we have a projected path } \hat{P} \text{ formed by}
\text{1) Corresponding matching edges to projected edges in P}
\text{2) Forward projecting every matching edge in P to next block}
Closed Region
\text{A set of points that can be matched with themselves.}
\text{Every Closed Region must have even number of points.}
\text{A path which always moves forward is } H\text{-monotone}
H-monotone and V-monotone Paths
\text{A path which continuously goes down to bottom row and then }
\text{continuously up to top row and so on is } V\text{-monotone.}
Our Results
\text{No infinite Paths for } n=2 \text{ and } 3
\text{Non-existence of infinite } H\text{-monotone and }V\text{-monotone paths}
\text{Non-existence of infinite } V\text{-monotone paths}
\text{Non-existence of infinite paths which goes from top row to bottom row}
\text{Non-existence of infinite } H\text{-monotone and }V\text{-monotone paths}
\text{Consider a segment } P\text{ of such a path going from the top to the bottom row.}
\text{This segment leads to the formation of a closed region }C \text{.}
\text{The left boundary of } C \text{ is formed by the matching edges of } P \text{.}
\text{The right boundary of } C \text{ is formed by the matching edges of } \hat{P} \text{.}
\text{We will show that the number of vertices in } C\text{ is odd.}
\text{The segment } P \text{ enters and exits the } i^{th} \text{ row exactly once.}
C(i) = \text{number of vertices in } C \text{ present in the } i^{th} \text{ row.}
C \text{ is set of points enclosed by two boundaries.}
\text{Non-existence of infinite } H\text{-monotone and }V\text{-monotone paths}
\text{ Case 1: Enters by a matching edge and exits by a matching edge}
C(i) \equiv 0 \text{ mod } 2
\text{ Case 2: Enters by a matching edge and exits by a projected edge}
C(i) \equiv 1 \text{ mod } 2
\text{ Case 3: Enters by a projected edge and exits by a matching edge}
C(i) \equiv 1 \text{ mod } 2
\text{ Case 4: Enters by a projected edge and exits by a projected edge}
C(i) \equiv 0 \text{ mod } 2
\text{Non-existence of infinite } H\text{-monotone and }V\text{-monotone paths}
\text{Except last row}
\text{ if } P \text{ enters with projected edge}
\text{Last row suffers a parity change by the unmatched last point in } P
\text{ if } P \text{ enters with matching edge}
\text{Region is not closed, so we add a Projected horizontal edge to } P
\text{Last row suffers a parity change by the unmatched last point in } P
\text{Thus, } C \text{ contains odd number of points.}
\text{Non-existence of infinite } V\text{-monotone paths}
\text{Same proof works }
\text{In each row, we either always go left or always go right.}
\text{Non-existence of infinite paths which goes from top row to bottom row}
Top Row
Bottom Row
i^{th}\text{ row}
\text{Define } C(i) \text{ as shown}
Future Works
\text{1) Generalising to any grid graph (Our results are for complete grid graphs)}
\text{3) Identifying more Closed Regions and their analysis}
\text{4) Structure of Closed Regions, having even vertices but no perfect matching}
\text{5) Show problem to be }AC^0\text{-Hard.}
\text{2) Showing that all groups } \mathcal{G} \subset \mathcal{M} \text{ have order } m^{k} \text{.}
\text{ This will put problem in } AC^0[m]
References
NC^1