The minimum distance problem (MDP) for low-density parity-check (LDPC) codes is a central problem in coding theory and is closely related to the analysis of low-weight codewords and error-floor behavior. Although the unrestricted MDP is computationally intractable, its complexity under degree constraints that commonly occur in LDPC code design has remained less clear. In this paper, we study the MDP for left regular and biregular Tanner graphs. For every fixed $J\geq3$, we prove that the standard at-most-weight problem is $\mathrm{NP}$-complete for $J$-left regular Tanner graphs and that its exact-weight variant is $\mathrm{W}[1]$-complete when parameterized by the prescribed weight. For biregular Tanner graphs, we prove $\mathrm{NP}$-completeness for $(3,K)$-regular instances for every fixed $K\geq 3$ by replacing degree-two auxiliary completion blocks with a single-port high-girth gadget. A nonzero relative support inside this gadget induces an essentially cubic graph, so the Moore bound gives an exponential lower bound in the girth and allows a polynomial-size Karp reduction. Combining this right-degree amplification with a replica-and-global-check left-degree amplification yields $\mathrm{NP}$-completeness for $(J,K)$-regular Tanner graphs for every fixed $J,K\geq 3$. The reductions are based on a degree-preserving transformation framework consisting of hyperedge decomposition, check node splitting, and controlled variable replication. These transformations relate different degree distributions while preserving explicit maps among nonzero codewords, even covers, and nonempty $(a,0)$-trapping sets. The results delineate the computational limits of computing minimum distance exactly under natural regularity constraints.

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