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A hash function is a mathematical process that transforms input data, often called a 'key', into a condensed form known as a hash value or hash code. This process is comparable to creating a unique fingerprint for any piece of data. However, unlike fingerprints that are universally unique, hash functions deal with a limited range of possible outputs, which leads to the inevitability of hash collisions. Although effectively designed hash functions aim to minimize collision occurrences by distributing hash values as evenly as possible across the output space, the reality of 'pigeonhole principle' implies that collisions can still happen, especially as the amount of keys grows large.

When a hash collision occurs, it's crucial to manage it gracefully to maintain the integrity and performance of the hash table. Two popular strategies are: Chaining and Open Addressing.

• Chaining involves creating a linked list for every slot in the hash table. When a collision takes place, the colliding element is added to the list corresponding to that slot. The downside is that data lookup times can increase if a list becomes too long.
• Open Addressing involves finding another empty slot within the hash table for the colliding entry. Techniques like linear probing, quadratic probing, and double hashing are employed to determine the sequence of probes. Though this avoids pointers and linked lists, clustering can occur, where continuous slots get filled, leading to increased probing and reduced efficiency.

Another approach includes Double Hashing, which uses a secondary hash function in case of a collision. It's essential to evaluate the requirements of the system, such as the expected number of entries and frequency of access, to decide on the most appropriate collision resolution strategy.

The performance of a hash table is often measured by its 'load factor', which is the ratio of the number of stored entries to the total number of slots in the table. A lower load factor means fewer collisions and faster operations, while a higher load factor may increase collision frequency, prompting the need for collision resolution mechanisms to work harder. Balancing the load factor is hence key in maintaining the efficiency of hash table operations.

In terms of computational complexity, the ideal scenario is O(1), or constant time, for insertion, deletion, and lookup operations. However, when collisions become frequent, these operations can degrade to O(n), where n is the number of entries within a particular slot in the case of chaining, or the probing sequence length with open addressing. Periodic resizing of the hash table and rehashing of entries can help maintain optimal performance levels but must be done judiciously to avoid large processing overheads.