Unifying and Strengthening Hardness for Dynamic Problems via the Online Matrix-Vector Multiplication Conjecture (1511.06773v1)

Abstract: Consider the following Online Boolean Matrix-Vector Multiplication problem: We are given an $n\times n$ matrix $M$ and will receive $n$ column-vectors of size $n$, denoted by $v_1,\ldots,v_n$, one by one. After seeing each vector $v_i$, we have to output the product $Mv_i$ before we can see the next vector. A naive algorithm can solve this problem using $O(n^3)$ time in total, and its running time can be slightly improved to $O(n^3/\log^2 n)$ [Williams SODA'07]. We show that a conjecture that there is no truly subcubic ($O(n^{3-\epsilon})$) time algorithm for this problem can be used to exhibit the underlying polynomial time hardness shared by many dynamic problems. For a number of problems, such as subgraph connectivity, Pagh's problem, $d$-failure connectivity, decremental single-source shortest paths, and decremental transitive closure, this conjecture implies tight hardness results. Thus, proving or disproving this conjecture will be very interesting as it will either imply several tight unconditional lower bounds or break through a common barrier that blocks progress with these problems. This conjecture might also be considered as strong evidence against any further improvement for these problems since refuting it will imply a major breakthrough for combinatorial Boolean matrix multiplication and other long-standing problems if the term "combinatorial algorithms" is interpreted as "non-Strassen-like algorithms" [Ballard et al. SPAA'11]. The conjecture also leads to hardness results for problems that were previously based on diverse problems and conjectures, such as 3SUM, combinatorial Boolean matrix multiplication, triangle detection, and multiphase, thus providing a uniform way to prove polynomial hardness results for dynamic algorithms; some of the new proofs are also simpler or even become trivial. The conjecture also leads to stronger and new, non-trivial, hardness results.

• The paper unifies and strengthens computational hardness results for various dynamic problems by using the Online Matrix-Vector Multiplication (OMv) conjecture as a single hypothesis.
• Assuming the OMv conjecture, the authors derive stronger and simpler hardness proofs for problems like dynamic shortest paths and connectivity, replacing reliance on disparate conjectures.
• The research implies that significant efficiency improvements for many dynamic algorithms may be limited unless the OMv conjecture is disproven.

Unifying and Strengthening Hardness for Dynamic Problems via the Online Matrix-Vector Multiplication Conjecture

The paper "Unifying and Strengthening Hardness for Dynamic Problems via the Online Matrix-Vector Multiplication Conjecture" by Monika Henzinger, Sebastian Krinninger, Danupon Nanongkai, and Thatchaphol Saranurak, proposes an intriguing exploration into the theoretical limits of dynamic algorithms, leveraging the Online Matrix-Vector Multiplication (OMv) conjecture as a foundational tool. This research extends and unifies existing hardness results for a wide array of dynamic problems, replacing a diverse set of conjectures with a single coherent hypothesis.

Key Insights and Contributions

1. OMv Conjecture as a Central Hypothesis: The authors elevate the OMv conjecture, which posits the absence of truly subcubic time algorithms for the OMv problem, to a position of central importance in establishing computational hardness across numerous dynamic problems. This conjecture serves as a uniform basis to understand and articulate the computational limits of dynamic algorithms.
2. Enhanced Hardness Results: By assuming the OMv conjecture, the paper derives more robust and often simpler hardness proofs compared to prior results based on other conjectures (such as 3SUM and APSP). Notably, these new proofs consolidate previously disparate conjectures into a single framework, greatly simplifying the landscape of computational hardness for dynamic problems.
3. Broad Range of Affected Problems: The OMv conjecture is utilized to demonstrate strong hardness results for a variety of dynamic problems, including subgraph connectivity, decremental single-source shortest paths, and dynamic densest subgraph. Each of these problems is shown to have significant computational barriers when tackled with dynamic algorithms, unless the OMv conjecture is disproved.
4. Implications for Algorithm Design: The implications of this research are significant for the design of dynamic algorithms. The demonstrated hardness results suggest that for many problems, unless the OMv conjecture is broken, substantial improvements in algorithmic efficiency are unlikely without changing the underlying computational assumptions.
5. Connection to Combinatorial Algorithms: Interestingly, the paper establishes that many hardness results, previously shown only for combinatorial algorithms, now extend to all algorithms under the OMv conjecture. This broader application implies that certain combinatorial challenges are inherent to the problems themselves, not just their specific algorithmic implementations.

Practical and Theoretical Implications

The practical implications of this research are predominantly found in the guidance it offers to algorithm designers about which avenues might yield fruitful results and which might be fundamentally limited by the problem’s computational hardness. This direction helps prioritize efforts towards problem domains where tangible improvements are feasible given current computational assumptions.

Theoretically, this paper suggests that further understanding and possibly resolving the OMv conjecture could be pivotal in breaking through existing computational barriers. Such breakthroughs could potentially pave the way for new algorithmic strategies that efficiently address what are currently considered computationally hard problems. Furthermore, this work could inspire similar unifying conjectures in other areas of computational science, leading to a cleaner and more intuitive theoretical framework for understanding problem complexity.

Speculation on Future Developments

Should future work corroborate the OMv conjecture, the algorithmic landscape for dynamic problems could see significant shifts. Researchers may find themselves compelled to innovate beyond traditional algorithmic frameworks, possibly exploring novel computational paradigms or leveraging emerging technologies like quantum computing to overcome the barriers outlined in this work.

Moreover, disproving the OMv conjecture could herald a new era where previously intractable problems become solvable within reasonable computational bounds, mirroring the significant algorithmic advances witnessed in other fields following the resolution of foundational theoretical questions.

Overall, the paper by Henzinger et al. represents a potent contribution to the field of computer science, highlighting the profound impact that theoretical conjectures can have on practical algorithmic advancements.