---
title: Lift-One Algorithm in Binary Knapsack Lifting
url: https://www.emergentmind.com/topics/lift-one-algorithm
type: topic
---

# Lift-One Algorithm in Binary Knapsack Lifting

The **Lift-One Algorithm** usually denotes the **standard sequential lifting method** used in practical mixed-integer programming for binary knapsack inequalities: coefficients are determined **one-by-one**, typically by solving knapsack subproblems with **dynamic programming on capacities** or equivalent exact subproblems [2602.22640]. In the material summarized here, the term is practical rather than canonical: the 2026 paper on binary knapsack lifting does **not** name the classical array-based procedure “Lift-One Algorithm” explicitly in its main text, but it identifies that procedure as exactly the implementation commonly associated with **Lift-One-style** exact sequential lifting [2602.22640]. The same corpus also makes clear that “lifting” is heavily overloaded across optimization, program synthesis, MCMC, and semidefinite relaxation; those uses are related only by analogy, not by a shared algorithmic definition [2504.19705].

## 1. Binary knapsack setting and lifted inequalities

The principal setting is the binary knapsack set
\[
\mathcal{X}=\left\{\boldsymbol{x}\in\{0,1\}^n:\sum_{i=1}^n a_i x_i\le b\right\},
\]
with \(a_i\in\mathbb{Q}_+\), \(b\in\mathbb{Q}_+\), \(0<a_i\le b\), and \(\sum_i a_i>b\). Its convex hull is the binary knapsack polytope
\[
\mathcal{P}=\operatorname{conv}(\mathcal{X}).
\]
Sequential lifting begins from a lower-dimensional valid inequality defined on a fixed set of variables. The index set \(N=\{1,\dots,n\}\) is partitioned as \((C,N_0,N_1)\), where \(C\) contains variables kept in the seed inequality, \(N_0\) contains variables fixed to \(0\), and \(N_1\) contains variables fixed to \(1\) [2602.22640].

The corresponding fixed knapsack set is
\[
\mathcal{X}(N_0,N_1)=\mathcal{X}\cap\{x_i=0\ \forall i\in N_0,\ x_i=1\ \forall i\in N_1\},
\]
equivalently
\[
\sum_{i\in C} a_i x_i \le b-\sum_{i\in N_1} a_i.
\]
A seed inequality is assumed in the form
\[
\sum_{i\in C}\alpha_i x_i\le \beta^\ell,
\]
valid for \(\operatorname{conv}(\mathcal{X}(N_0,N_1))\), where \(C=\{1,\dots,\ell\}\). After lifting, the resulting inequality on the original knapsack set has the form
\[
\sum_{i\in C}\alpha_i x_i+\sum_{i\in N_0}\alpha_i x_i+\sum_{i\in N_1}\alpha_i x_i \le \beta^\ell+\sum_{i\in N_1}\alpha_i.
\]
This formulation is the core object to which Lift-One-style procedures apply [2602.22640].

A central structural property is that **sequential lifting means lifting projected or fixed variables one at a time in a chosen order**. The order is not incidental: the result depends on the lifting order, and each new coefficient is obtained from a knapsack optimization problem. The same source states that if the seed inequality is facet-defining for the lower-dimensional fixed set, then **exact sequential lifting preserves facetness in the original space** [2602.22640].

## 2. Exact sequential lifting: one coefficient at a time

Let the lifting order be \(\{\ell+1,\dots,n\}\). Suppose variables up to \(k\) have already been lifted, and the current valid inequality is
\[
\sum_{i=1}^k \alpha_i x_i \le \beta^k
\]
for the current fixed set \(\mathcal{X}(N_0^k,N_1^k)\), with
\[
b^k = b-\sum_{i\in N_1^k} a_i,\qquad \beta^k = \beta^\ell + \sum_{i\in N_1\setminus N_1^k}\alpha_i.
\]
The knapsack value function is
\[
F_k(z)=\max_{\boldsymbol{x}\in\{0,1\}^k}\left\{\sum_{i=1}^k \alpha_i x_i:\sum_{i=1}^k a_i x_i\le z\right\}.
\]
Lift-One-style exact sequential lifting computes each new coefficient through this value function [2602.22640].

If \(k+1\in N_0^k\), the procedure performs **up lifting**:
\[
\alpha_{k+1} = \min_{\boldsymbol{x}\in\{0,1\}^k} \left\{ \beta^k-\sum_{i=1}^k \alpha_i x_i:\sum_{i=1}^k a_i x_i\le b^k-a_{k+1} \right\}
= \beta^k-F_k(b^k-a_{k+1}).
\]
If \(b^k<a_{k+1}\), the problem is infeasible and the coefficient is set to \(\alpha_{k+1}=0\). The update is
\[
N_0^{k+1}=N_0^k\setminus\{k+1\},\quad N_1^{k+1}=N_1^k,\quad b^{k+1}=b^k,\quad \beta^{k+1}=\beta^k.
\]

If \(k+1\in N_1^k\), the procedure performs **down lifting**, writing the inequality as
\[
\sum_{i=1}^k \alpha_i x_i+\alpha_{k+1}(x_{k+1}-1)\le \beta^k,
\]
with coefficient
\[
\alpha_{k+1} = \max_{\boldsymbol{x}\in\{0,1\}^k} \left\{ \sum_{i=1}^k \alpha_i x_i-\beta^k: \sum_{i=1}^k a_i x_i\le b^k+a_{k+1} \right\}
= F_k(b^k+a_{k+1})-\beta^k.
\]
The associated update is
\[
N_0^{k+1}=N_0^k,\quad N_1^{k+1}=N_1^k\setminus\{k+1\},\quad b^{k+1}=b^k+a_{k+1},\quad \beta^{k+1}=\beta^k+\alpha_{k+1}.
\]
These formulas make precise the “lift one variable at a time” interpretation: each step solves a small exact subproblem and then updates the current inequality and reduced knapsack parameters [2602.22640].

## 3. Classical Lift-One implementation via dynamic programming

In the 2026 treatment, the **classical exact lifting approach** computes \(F_k(z)\) by dynamic programming on capacities:
\[
F_0(z)=0,\qquad \forall z\in[0,b],
\]
and
\[
F_{k+1}(z)=
\begin{cases}
F_k(z), & z<a_{k+1},\\[2mm]
\max\{F_k(z),\,F_k(z-a_{k+1})+\alpha_{k+1}\}, & z\ge a_{k+1}.
\end{cases}
\]
This is explicitly identified as the Bellman recursion [2602.22640].

In this context, **Lift-One / classical exact sequential lifting** is the method that computes one coefficient at a time using full dynamic-programming arrays over capacities. The paper characterizes the approach as exact, but also emphasizes its limitations. When all weights and capacity are integers, the classical array-based DP has complexity
\[
\mathcal{O}(nb)
\]
time and
\[
\mathcal{O}(b)
\]
space. The same discussion states that this implementation stores many redundant DP states and is tied to integer capacities and weights [2602.22640].

This distinction is important because “Lift-One” is sometimes used as though it denoted an abstract lifting principle. In the knapsack literature summarized here, however, it denotes a very concrete computational pattern: compute a coefficient, update the DP representation, then compute the next coefficient. The procedure is exact, sequential, order-dependent, and closely bound to the representation chosen for \(F_k\) [2602.22640].

## 4. Dominance-list exact lifting as a refinement of classical Lift-One

The main refinement proposed in the 2026 paper replaces the full DP array by a **dominance-list representation**. A state is a pair \((w,p)\), where
\[
w=\sum_{i=1}^k a_i x_i,\qquad p=\sum_{i=1}^k \alpha_i x_i
\]
for some \(x\in\{0,1\}^k\). The full state list is
\[
\bar L_k=\{(w,p): w=\sum_{i=1}^k a_ix_i,\ p=\sum_{i=1}^k \alpha_ix_i,\ \forall x\in\{0,1\}^k\}.
\]
A state \((w',p')\) dominates \((w'',p'')\) if
\[
w'<w''\ \text{and}\ p'\ge p'', \quad\text{or}\quad w'\le w''\ \text{and}\ p'>p''.
\]
Dominated states can never be optimal for any capacity, so the **dominance list**
\[
L_k=\bar L_k\setminus\{\text{dominated states}\}
\]
stores only the nondominated breakpoints [2602.22640].

A structural lemma stated in the summary is that any two distinct states in \(L_k\) satisfy
\[
w^j<w^{j+1}\quad\text{and}\quad p^j<p^{j+1}.
\]
Thus \(L_k\) forms an increasing staircase. If
\[
L_k=\{(w^1,p^1),\dots,(w^m,p^m)\},
\]
then the entire DP value function is recovered by
\[
F_k(\bar z)=
\begin{cases}
p^j, & \bar z\in[w^j,w^{j+1}),\quad j=1,\dots,m-1,\\
p^m, & \bar z\in[w^m,b].
\end{cases}
\]
The refinement therefore changes only the representation, not the exact lifting coefficients: it computes the same \(F_k\), but stores only its changes [2602.22640].

The list update is based on a shifted copy
\[
L'_{k-1}=L_{k-1}\oplus (a_k,\alpha_k)
= \{(w^1+a_k,p^1+\alpha_k),\dots,(w^m+a_k,p^m+\alpha_k)\},
\]
and \(L_k\) is obtained by merging \(L_{k-1}\) and \(L'_{k-1}\), keeping only feasible nondominated states with \(w\le b\). The merge is implemented by a two-pointer scan and runs in time linear in the list length [2602.22640].

The same paper also introduces a reduction rule. For lifting step \(k+1\), only certain capacities are needed, and it proves that if \(F_k(z)\) is known for all
\[
z\in\left[(b-\sum_{i=k+1}^n a_i)^+,\, b\right],
\]
then all future lifting coefficients \(\alpha_{k+1},\dots,\alpha_n\) can be computed exactly. The restricted lower bound is
\[
r=(b-\sum_{i=k+1}^n a_i)^+.
\]
This leads to reduced variants of both the dominance-list method and the DP-array method [2602.22640].

The experiments compare four implementations: **DL**, **DP**, **DL-R**, and **DP-R**. The reported findings are that **DL beats DP strongly on large capacities**, that **reduction helps both methods**, that **DL is more scale invariant**, and that when integrated into **HiGHS**, the exact lifting algorithm solved **4 more instances** than default approximate lifting and **slightly reduced average time** on solved instances [2602.22640].

## 5. Conceptual relatives and common confusions

The phrase **Lift-One Algorithm** is easily confused with other algorithmic uses of “lifting,” but the sources separate these meanings sharply. In one direction, the 2020 paper on **Climbing LP Algorithms** presents the **Dual Matrix Algorithm (DMA)** as a climbing-style LP method that is “very much in the same family of ideas” as Lift-One-style LP methods, because it maintains a meaningful intermediate certificate, improves a monotone valuation, and uses local rank-one-like updates rather than recomputing everything from scratch [2101.00101].

That LP framework considers feasibility of
\[
Ax>0
\]
for an integer matrix \(A\), and maintains a matrix structure \(B,C,V,d\). The core update is a single-entry increment of \(B\):
\[
C' \df \left(B+\frac{e_j e_i}{t d_j}\right)A = (V')^{-1},
\]
with corresponding determinant growth and a guaranteed progress per step of at least
\[
> \frac{1}{2n^2},
\]
yielding an \(O(n^3L)\)-step algorithm [2101.00101]. The paper explicitly describes this as a **conceptual cousin** of Lift-One rather than the Lift-One algorithm itself. The relationship is therefore one of style—localized updates, monotone progress, incremental linear-algebraic maintenance—not one of identity.

In other directions, the summarized papers use “lifting” in entirely different senses. **Guided Tensor Lifting** concerns synthesis of TACO programs from C code via LLM-guided probabilistic grammars and weighted \(A^*\)-style enumeration; the source explicitly states that it is **not about a Lift-One algorithm in the classical sense** [2504.19705]. The MCMC paper uses lifting to mean enlarging a state space to create nonreversible Markov chains [1412.8762]. The POCS paper uses a one-dimensional lift from \(\mathbb{R}^N\) to \(\mathbb{R}^{N+1}\) for optimization [1306.2516]. The sparse-recovery paper introduces auxiliary weights in a lifted \(\ell_1\) model [2203.05125]. The STRIDE framework uses lift-round-descend steps between polynomial variables and rank-one moment matrices in semidefinite relaxation [2105.14033]. These are not variants of the knapsack Lift-One procedure; they are separate algorithmic constructions that happen to reuse the word “lifting.”

## 6. Exactness, limitations, and present significance

Within binary knapsack lifting, the essential properties of Lift-One-style methods are **exactness**, **sequentiality**, and **order dependence**. The method computes one coefficient at a time from a knapsack subproblem, and if the seed inequality is facet-defining, exact sequential lifting preserves facetness in the original knapsack polytope [2602.22640]. This gives the method a precise polyhedral role: it is not merely heuristic coefficient adjustment, but an exact procedure for extending a valid inequality from a restricted face back to the full variable space.

Its main historical limitation, in the formulation summarized here, is computational. The classical DP-array implementation is exact but stores many redundant states and is tied to integer capacities and weights. This is why modern solver practice often relies on approximate lifting: exact DP-based lifting can be too expensive on real-scale instances with large capacities or non-integer coefficients [2602.22640]. The dominance-list implementation directly addresses those limitations. The paper emphasizes that the dominance-list method is **scale invariant**, can handle **non-integer coefficients**, is often much faster for **large capacities**, and is **better suited for modern solver implementations** [2602.22640].

A plausible implication is that the current significance of the Lift-One idea lies less in the old array-based dynamic program than in the underlying exact sequential-lifting principle. The 2026 work preserves that principle while replacing the storage and update mechanism. In that sense, the modern form of Lift-One is best understood as an **exact sequential lifting framework** whose computational realization may vary, with the dominance-list representation emerging in the cited material as the most effective exact implementation for binary knapsack sets [2602.22640].

Source: https://www.emergentmind.com/topics/lift-one-algorithm