AI 中文总结
研究由(L,V,W)参数化的一维精确覆盖问题，给出O(n)算法，通过差分递归和余数匹配求下界，证明重基min_c a_c≥B(L,V,W)时界精确且阈值尖锐，贡献了算法、证明及阈值。
AI 中文摘要
我们研究了一个由三个整数(L,V,W)参数化的一维精确覆盖问题：给定一个整数序列a₀,…,aₙ₋₁，将其写成长度为L的“水平”块[1,…,1]、值为V的“垂直”块和值为W的块的非负整数组合，同时最小化值为W的块的数量。我们给出了一个O(n)算法，通过逐类差分递归来消除水平块耦合，然后在最后L列上对模V的余数进行匹配。通过模L类不变量，我们证明了其输出始终是最优值的有效下界。然后我们证明了主要结果：一旦序列足够密集——一个重基min_c a_c≥B(L,V,W)，其中B(L,V,W)=⌈((L - 1)lcm(V,W))/(LW)-1⌉W+(L - 1)(V - 1)，该界就是精确的。精确性证明是对精确动态规划的一种分支切割论证：两个结构等价性（模V的水平到垂直交换和模lcm(V,W)/W的垂直约简）将动态规划简化为余数计算，并且B的两个求和项恰好是防止两个等价性产生负余数的储备。我们进一步表明该阈值是尖锐的：对于(L,V,W)=(3,6,4)，B = 14，并且序列(17,16,13,16,17)，其中min_c a_c = 13，使得算法严格少计，所以B - 1是不够的。一个独立的精确动态规划在许多参数三元组上与算法在每个测试的min_c a_c≥B的序列上一致，并且发布了测试工具test_general.c用于重现。贡献在于算法、分支切割精确性证明和尖锐阈值B(L,V,W)。
英文摘要
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 $B(L,V,W)$.
