On the Variants of Subset Sum: Projected and Unbounded

In this work, we focus on variants of Subset Sum. We first define a new variant called the Unique Projection Subset Sum (u – PSSUM) problem: Given $\left(a_{1}, \ldots, a_{n}\right) \in \mathbb{Z}_{\geq 0}^{n}$, such that for every $t, \sum_{i \in[n]} c_{i} a_{i}=t$, for $c_{i} \in\{0,1\}$ has unique solution if exists, u-PSSUM asks for a vector $\vec{x}=\left(x_{1}, \ldots, x_{n}\right)$, such that there exists an $S \subseteq[n]$, where $\sum_{i \in S} x_{i}=t$, and $\|\vec{a}-\vec{x}\|_{p}$ is minimized, where $\|\cdot\|_{p}$ denotes the $\ell_{p}$ norm. We present a deterministic $O(n t)$-time algorithm and a randomized $\widetilde{O}(n+t)$-time algorithm for u-PSSUM, in $\ell_{1}$-norm. The second one is already a known variant called Un-bounded Subset Sum (UBSSUM) problem, which takes a tuple of non-negative integers $\left(a_{1}, \ldots, a_{n}, t\right)$, as an input, and asks whether there exists non-negative integers $\beta_{1}, \ldots, \beta_{n}$, such that $\sum_{i=1}^{n} \beta_{i} a_{i}=t$. We present the first polynomial time reductions from UBSSUM to Closest Vector Problem (CVP) in $\ell_{1}$ and $\ell_{\infty}$ norms. We also give new algorithms for two variants of UBSSUM problem.