Quantum Lower Bound for Graph Collision Implies Lower Bound for Triangle Detection

We show that an improvement to the best known quantum lower bound for GRAPH-COLLISION problem implies an improvement to the best known lower bound for TRIANGLE problem in the quantum query complexity model. In GRAPH-COLLISION we are given free access to a graph $(V,E)$ and access to a function $f:V\rightarrow \{0,1\}$ as a black box. We are asked to determine if there exist $(u,v) \in E$, such that $f(u)=f(v)=1$. In TRIANGLE we have a black box access to an adjacency matrix of a graph and we have to determine if the graph contains a triangle. For both of these problems the known lower bounds are trivial ($\Omega(\sqrt{n})$ and $\Omega(n)$, respectively) and there is no known matching upper bound.


Introduction
By Q(f ) we denote the bounded-error quantum query complexity of a function f . We consider the quantum query complexity for some graph problems.
Definition 1. In TRIANGLE problem it is asked whether an n-vertex graph G = (V, E) contains a triangle, i.e. a complete subgraph on three vertices. The adjacency matrix of the graph is given in a black box which can be queried by asking if (x, y) ∈ E.
Recently there have been several improvements in the algorithms for the TRIAN-GLE problem in the quantum black box model. The problem was first considered by Buhrman et al. (2005) who gave an O(n + √ nm) algorithm where n is the number of vertices and m -the number of edges. Later in 2007 Magniez et al. gave anÕ(n 13/10 ) algorithm based on quantum walks. Introducing a novel concept -learning graphs, and using a new technique Belovs (2012b) was able to reduce the complexity to O(n 35/27 ). Lee, Magniez, and Santha (2013) using a more refined learning graph approach reduced the complexity toÕ(n 9/7 ). Currently the best known algorithm is by Le Gall (2014) who exhibited a quantum algorithm which solves the TRIANGLE problem with query complexityÕ(n 5/4 ). Classically the query complexity of TRIANGLE is Θ(n 2 ); however, it is an open question whether TRIANGLE can be computed in time better than O(n ω ) where ω is the matrix multiplication constant.
Definition 2. In GRAPH-COLLISION G problem a known n-vertex undirected graph G = (V, E) is given and a coloring function f : V → {0, 1} whose values can be obtained by querying the black box for the value of f (x) of a given x ∈ V . We say that a vertex x ∈ V is marked iff f (x) = 1. The value of the GRAPH-COLLISION G instance is 1 iff there exists an edge whose both vertices are marked, By Q(GRAPH-COLLISION) we mean the complexity of solving GRAPH-COLLI-SION G for the hardest n-vertex graph G.
There has been an increased interest in the quantum query complexity of the GRAPH-COLLISION problem, mainly because algorithms for solving GRAPH-COLLI-SION are used as a subroutine in algorithms for the TRIANGLE problem by Magniez, Santha, and Szegedy (2007) and Boolean matrix multiplication by Jeffery, Kothari, and Magniez (2012).
The best known quantum algorithm for GRAPH-COLLISION for an arbitrary nvertex graph has complexity O(n 2/3 ) by Magniez, Santha, and Szegedy (2007). However, for some graph classes there are algorithms with complexity O(  (2012)). It is an open question whether for every n-vertex graph G GRAPH-COLLISION G can be solved with O( √ n) queries. Contrary to the improvements in the algorithms for these two problems, the best known lower bounds for Q(GRAPH-COLLISION) and Q(TRIANGLE) are still the trivial Ω( √ n) and Ω(n) respectively, which follow from the reduction to search problem. Nonetheless these lower bounds seem hard to improve with the current techniques.
As mentioned before, algorithms for GRAPH-COLLISION have been used as a subroutine for constructing algorithms for the TRIANGLE problem, therefore an improved algorithm for GRAPH-COLLISION would result in an improved algorithm for TRIAN-GLE. In this paper we show a reduction in the opposite direction-that an improvement in the lower bound on Q(GRAPH-COLLISION) would imply an improvement in the lower bound on Q(TRIANGLE).

Result
Theorem 1. If there is a graph G = (V, E) with n vertices such that GRAPH-COLLISION G has quantum query complexity t then TRIANGLE problem has quantum query complexity at least Ω(t √ n).
Proof. We show how to transform the graph G into a graph G with 3n vertices so that it is hard to decide if G contains a triangle. More precisely, we construct the graph G in such a way that solving the TRIANGLE problem on G is equivalent to solving OR function from the results of n independent instances of GRAPH-COLLISION G . First, we want to get rid of any triangles in G, therefore we transform G into an equivalent bipartite graph G 2 = (V 2 , E 2 ) with 2n vertices by setting V 2 = {v 1 , v 2 | v ∈ V } and E 2 = {(x 1 , y 2 ) | (x, y) ∈ E}. The graph G 2 is equivalent to G in the following sense-if we mark the vertices v 1 and v 2 in G 2 for every marked vertex v in G, then G 2 has a collision iff G has a collision. However, the graph G 2 does not contain any triangle (since it is bipartite).
Next, we add n isolated vertices z 1 , . . . , z n to G 2 thereby obtaining a graph G . Let f 1 , . . . , f n : V → {0, 1} be the colorings from n independent GRAPH-COLLISION G instances. We add the edges (z i , v 1 ) and (z i , v 2 ) to G iff v ∈ V is marked by the respective coloring, i.e., iff f i (v) = 1.
See Fig. 1 for an example.  The only possible triangles in the graph G can be of the form {z i , v 1 , w 2 } for some i ∈ {1, . . . , n} and v, w ∈ V . Moreover, there is a triangle {z i , v 1 , w 2 } iff f i is such coloring that G has a collision (v, w), i.e., iff f i (v) = f i (w) = 1. Therefore detecting a triangle in G is essentially calculating OR function from the results of n instances of GRAPH-COLLISION G .
Setting f = OR and g = GRAPH-COLLISION G gives the desired bound.
As the next corollary shows, a better lower bound on GRAPH-COLLISION implies a better lower bound on the TRIANGLE problem.