# Knapsack Problem

Every on-chain trader in crypto have the end goal on trading their digital assets; Buying or selling with maximum amount of profit with minimum cost on their limited balance. The goal leads to questions below:

• How does a trader buy a cryptocurrency in lowest price in orders in AMM?

• How do I avoid slippage in my order, if not, how do I minimize it?

• How can I be sure that MEVs don't steal my coins to trade on AMM?

• How do I provide liquidity in AMM so that I can have most effective position to trade assets in profiting position?

The concept of buying or selling assets with the objective of maximizing profit while minimizing costs within the constraints of a limited balance is akin to a mathematical analogy to the classic knapsack problem. In this analogous scenario, traders aim to select specific asset transactions, each characterized by its profit potential (analogous to the value of items) and associated costs (analogous to the weight of items). The challenge lies in optimizing these selections to achieve the highest possible total profit while adhering to the constraints imposed by their available balance, much like the way the knapsack problem seeks to maximize the value of chosen items within the confines of a fixed weight limit. This comparison underscores the mathematical nature of trading strategies aimed at achieving financial gains while efficiently managing resources.

To resolve this, DEX aggregators, off-chain settlement, Liquidity provision-based option projects claim to be brokers, market makers, or wholesalers of the traders' orders. However, this approach is not only computationally inefficient, shady, centralized, and illegalized practice which could harm fair market competition. By integrating discrete orders in different DEXes, they eventually have to solve an NP-hard problem, 0/1 knapsack problem.

Standard exchange integrates continuous orders within its platform. Market maker is an automated computer algorithm to provide best deals in the market, comparing order by order. All orders are made peer-to-peer, competing against each other to give users the best price by solving fractional knapsack problem, in which it always returns the best outcome in polynomial time.

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