[Paper Review] Searching for partial Hadamard matrices

Víctor Álvarez, José-Andrés Armario+5 more|arXiv (Cornell University)|Jan 19, 2012

graph theory and CDMA systems12 references3 citations

TL;DR

This paper proposes three algorithms for constructing large partial Hadamard matrices by searching for large cliques in a subgraph $G_t$ of Ito’s Hadamard Graph $\Delta(4t)$. The key contribution is that the best heuristic algorithm can produce partial Hadamard matrices exceeding half the size of a full $4t \times 4t$ Hadamard matrix, surpassing prior asymptotic bounds, especially when initialized with existing partial matrices from De Launey and Gordon's methods.

ABSTRACT

Three algorithms looking for pretty large partial Hadamard matrices are described. Here "large" means that hopefully about a third of a Hadamard matrix (which is the best asymptotic result known so far, [dLa00]) is achieved. The first one performs some kind of local exhaustive search, and consequently is expensive from the time consuming point of view. The second one comes from the adaptation of the best genetic algorithm known so far searching for cliques in a graph, due to Singh and Gupta [SG06]. The last one consists in another heuristic search, which prioritizes the required processing time better than the final size of the partial Hadamard matrix to be obtained. In all cases, the key idea is characterizing the adjacency properties of vertices in a particular subgraph G_t of Ito's Hadamard Graph Delta (4t) [Ito85], since cliques of order m in G_t can be seen as (m+3)*4t partial Hadamard matrices.

Motivation & Objective

• To develop efficient algorithms for constructing large partial Hadamard matrices, which are useful in applications requiring orthogonal rows but not full Hadamard matrices.
• To overcome the computational infeasibility of exhaustive search by introducing heuristic and genetic algorithm-based approaches for clique detection in $G_t$.
• To demonstrate that partial Hadamard matrices larger than half the size of a full $4t \times 4t$ Hadamard matrix can be constructed, surpassing known asymptotic bounds.

Proposed method

• The algorithms are based on modeling the search for partial Hadamard matrices as a clique-finding problem in a subgraph $G_t$ of Ito’s Hadamard Graph $\Delta(4t)$, where vertices represent $\{+1,-1\}$-vectors with specific orthogonality and distribution constraints.
• The first method performs a local exhaustive search for cliques in $G_t$, ensuring correctness but at high computational cost.
• The second method adapts Singh and Gupta’s genetic algorithm for the maximum clique problem, though it remains computationally expensive due to reliance on the exhaustive subroutine.
• The third method, Algorithm 4.3, is a heuristic prioritizing runtime efficiency over clique size, enabling faster exploration of larger cliques.
• Clique extensions are performed by iteratively adding vertices that maintain pairwise orthogonality, using the structure of $G_t$ to limit search space.
• The algorithms are initialized with partial Hadamard matrices from De Launey (2000) and De Launey-Gordon (2001), which are then extended to larger depths.

Experimental results

Research questions

• RQ1 Can heuristic and genetic algorithms outperform exhaustive search in constructing large partial Hadamard matrices?
• RQ2 Is it possible to construct partial Hadamard matrices with depth exceeding half of a full $4t \times 4t$ Hadamard matrix using clique-based methods in $G_t$?
• RQ3 How effective is the proposed heuristic algorithm (Algorithm 4.3) in extending existing partial Hadamard matrices from prior constructions?
• RQ4 Can the structure of $G_t$ be exploited to design efficient clique-finding algorithms that avoid exhaustive search?
• RQ5 What is the impact of initial clique quality on the final depth of the constructed partial Hadamard matrix?

Key findings

• Algorithm 4.3 successfully extended a 5-vertex clique in $G_4$ to a 12-vertex clique, yielding a $15 \times 16$ partial Hadamard matrix, exceeding half the size of a full $16 \times 16$ Hadamard matrix.
• For $t=6$, the algorithm extended a 9-vertex clique to a 15-vertex clique in 1 minute 7 seconds, producing a $18 \times 24$ partial Hadamard matrix.
• In $t=8$, the algorithm extended a 13-vertex clique to a 17-vertex clique, resulting in a $20 \times 32$ partial Hadamard matrix, again surpassing the half-size threshold.
• For $t=7$, no extension was possible, suggesting the input clique was maximal, indicating potential limits in extendability for certain initial configurations.
• The results show that when initialized with high-quality partial matrices from De Launey (2000) and De Launey-Gordon (2001), the heuristic algorithm can produce partial Hadamard matrices with depth greater than $2t$, i.e., more than half of a $4t \times 4t$ matrix.
• The cocyclic framework was shown to be inherently limited, as no partial Hadamard matrix from a cocyclic matrix can exceed half the size of the original matrix due to row pairing constraints.