A Linear-Time Residue Bound for a One-Dimensional (L,V,W) Block-Cover Problem, and a Sharp Heavy-Base Threshold for its Exactness

Abstract

We study a one-dimensional exact-cover problem parameterized by three integers $(L,V,W)$: given an integer profile $a_0,\dots,a_{n-1}$, write it as a nonnegative integer combination of a length-$L$ ``horizontal'' block $[1,\dots,1]$, a value-$V$ ``vertical'' block, and a value-$W$ block, while minimizing the number of value-$W$ blocks. We give an $O(n)$ algorithm that eliminates the horizontal-block coupling by a class-wise difference recurrence and then matches residues modulo $V$ on the last $L$ columns. We prove that its output is always a valid \emph{lower bound} on the optimum, via a mod-$L$ class invariant. We then prove the main result: once the profile is dense enough --- a \emph{heavy base} $\min_c a_c \ge B(L,V,W)$ with \[ B(L,V,W)=\Big\lceil \tfrac{(L-1)\lcm(V,W)}{LW}-1\Big\rceil\,W+(L-1)(V-1), \] the bound is \emph{exact}. The exactness proof is a branch-cut argument on the exact dynamic program: two structural equivalences (a horizontal-to-vertical exchange modulo $V$, and a vertical reduction modulo $\lcm(V,W)/W$) collapse the DP to the residue computation, and the two summands of $B$ are exactly the reserves that keep both equivalences from producing a negative residual. We further show the threshold is sharp: for $(L,V,W)=(3,6,4)$, $B=14$, and the profile $(17,16,13,16,17)$ with $\min_c a_c=13$ makes the algorithm strictly undercount, so $B-1$ does not suffice. An independent exact dynamic program agrees with the algorithm on every tested profile with $\min_c a_c\ge B$ across many parameter triples, and the test harness \texttt{test\_general.c} is released for reproduction. The contribution is the algorithm, the branch-cut exactness proof, and the sharp threshold $B(L,V,W)$.