Periodic Reporting for period 1 - DynOpt (Towards a New Theory of Optimal Dynamic Graph Algorithms)
Période du rapport: 2022-10-01 au 2025-03-31
Modern real-life networks often rapidly change over time, triggering the need for efficient dynamic graph algorithms. Given a sequence of update operations, where
each update is an insertion or deletion of a single edge or node, a common goal is to design algorithms
maintaining some key functionality of the graph with low update time.
Not only are dynamic graph algorithms important in their own right, they have also been instrumental in achieving numerous breakthroughs in
static graph algorithms. The field of dynamic algorithms is also related to other fields of computer science, such as distributed computing.
The holy grail in the field has been to achieve update time polylogarithmic in the input size.
In this project we are interested in achieving a constant (independent of the input size) update time.
This difference is not just of theoretical appeal, but could also be of practical importance.
Further, constant update time can be achieved only for problems that are statically solvable via linear time algorithms.
The update time of an algorithm cannot be better than the normalized static runtime for the problem, defined as the ratio of the best static time complexity of the problem to
the graph size. An algorithm is called intrinsically optimal if its update time matches the normalized static runtime for the problem. The main question underlying this project is:
Which graph problems admit intrinsically optimal update time?
The unique goal of this project is to establish a theory of intrinsically optimal algorithms.
There is a strong connection between static and dynamic algorithms, also captured in the definition of intrinsically optimal algorithms.
Our goal is thus to advance both fields of dynamic and static algorithms.
To achieve this goal, we must go far beyond the current state-of-the-art, and confront several major open problems in dynamic and static algorithms.
each update is an insertion or deletion of a single edge or node, a common goal is to design algorithms
maintaining some key functionality of the graph with low update time.
Not only are dynamic graph algorithms important in their own right, they have also been instrumental in achieving numerous breakthroughs in
static graph algorithms. The field of dynamic algorithms is also related to other fields of computer science, such as distributed computing.
The holy grail in the field has been to achieve update time polylogarithmic in the input size.
In this project we are interested in achieving a constant (independent of the input size) update time.
This difference is not just of theoretical appeal, but could also be of practical importance.
Further, constant update time can be achieved only for problems that are statically solvable via linear time algorithms.
The update time of an algorithm cannot be better than the normalized static runtime for the problem, defined as the ratio of the best static time complexity of the problem to
the graph size. An algorithm is called intrinsically optimal if its update time matches the normalized static runtime for the problem. The main question underlying this project is:
Which graph problems admit intrinsically optimal update time?
The unique goal of this project is to establish a theory of intrinsically optimal algorithms.
There is a strong connection between static and dynamic algorithms, also captured in the definition of intrinsically optimal algorithms.
Our goal is thus to advance both fields of dynamic and static algorithms.
To achieve this goal, we must go far beyond the current state-of-the-art, and confront several major open problems in dynamic and static algorithms.
I have been working with my group to achieve a better understanding of intrinsically optimal algorithms, and to advance the state-of-the-art of dynamic and static graph algorithms.
Some of the main achievements so far can be summarized as follows.
Vizing's Theorem from 1964 states that any n-vertex m-edge graph of maximum degree Δ can be edge colored with Δ+1 colors; moreover, it implies that such a coloring can be found in O(m n) time. This runtime was improved to ~O(m sqrt{n}) by Arjomandi (1982) and Gabow et al. (1985). In SODA'24, my co-authors and I presented a dynamic ((1+ϵ)Δ)-edge coloring algorithm with poly(1/ε) amortized update time, assuming Δ is a sufficiently large polylog. As a corollary, we got the first linear time (for constant ε) static algorithm for ((1+ϵ)Δ)-coloring. This gives another evidence that dynamic algorithms can contribute to static algorithm design.
In my FOCS’24 paper with co-authors we presented the first polynomial time improvement for (Δ+1)-edge coloring in over 40 years (concurrently and independently to Assadi's SODA'25 result). Recently we got a near-linear time algorithm, using different techniques (accepted at STOC'25).
The minimum set cover problem is a central combinatorial optimization problem. The best possible approximation factors for this problem in the static setting are either ln(n) or f, where n is the universe size and f is the maximum frequency. In my STOC'23 paper with Uzrad, we presented a dynamic ((1+ε)ln(n))-approximation-algorithm for maintaining a set cover with amortized update time O(f log n poly(1/ε)), for any ε > 0, providing an improvement over the previous state-of-the-art, by Gupta et al. from STOC'17, which had the same update time but a higher approximation. In my SODA'25 paper with Bukov and Zhang, we presented dynamic ((1+ ε)f)-approximation algorithms that are faster than the previous state-of-the-art.
In my STOC'23 paper with Le, where we designed a unified framework for constructing light spanners that is applicable to a variety of graph classes and to all stretch regimes, we had to go significantly beyond the state-of-the art. Our framework is built on top of a new and general technique.
The main highlight of my SODA’24 paper with co-authors was to resolve in the affirmative a major open question that was open for two decades, called the Steiner point removal problem (SPR). We answered this question affirmatively in any minor-free graph, yet the question was open even in planar graphs.
Some of the main achievements so far can be summarized as follows.
Vizing's Theorem from 1964 states that any n-vertex m-edge graph of maximum degree Δ can be edge colored with Δ+1 colors; moreover, it implies that such a coloring can be found in O(m n) time. This runtime was improved to ~O(m sqrt{n}) by Arjomandi (1982) and Gabow et al. (1985). In SODA'24, my co-authors and I presented a dynamic ((1+ϵ)Δ)-edge coloring algorithm with poly(1/ε) amortized update time, assuming Δ is a sufficiently large polylog. As a corollary, we got the first linear time (for constant ε) static algorithm for ((1+ϵ)Δ)-coloring. This gives another evidence that dynamic algorithms can contribute to static algorithm design.
In my FOCS’24 paper with co-authors we presented the first polynomial time improvement for (Δ+1)-edge coloring in over 40 years (concurrently and independently to Assadi's SODA'25 result). Recently we got a near-linear time algorithm, using different techniques (accepted at STOC'25).
The minimum set cover problem is a central combinatorial optimization problem. The best possible approximation factors for this problem in the static setting are either ln(n) or f, where n is the universe size and f is the maximum frequency. In my STOC'23 paper with Uzrad, we presented a dynamic ((1+ε)ln(n))-approximation-algorithm for maintaining a set cover with amortized update time O(f log n poly(1/ε)), for any ε > 0, providing an improvement over the previous state-of-the-art, by Gupta et al. from STOC'17, which had the same update time but a higher approximation. In my SODA'25 paper with Bukov and Zhang, we presented dynamic ((1+ ε)f)-approximation algorithms that are faster than the previous state-of-the-art.
In my STOC'23 paper with Le, where we designed a unified framework for constructing light spanners that is applicable to a variety of graph classes and to all stretch regimes, we had to go significantly beyond the state-of-the art. Our framework is built on top of a new and general technique.
The main highlight of my SODA’24 paper with co-authors was to resolve in the affirmative a major open question that was open for two decades, called the Steiner point removal problem (SPR). We answered this question affirmatively in any minor-free graph, yet the question was open even in planar graphs.
Some of the progress made so far beyond the state-of-the-art results in the literature, as part of the current ERC project, can be summarized as follows:
My FOCS’24 paper with co-authors provided the first polynomial time improvements for (Δ+1)-edge coloring in over 40 years (concurrently to Assadi's SODA'25 result).
In my STOC'23 paper with Le, where we designed a unified framework for constructing light spanners that is applicable to a variety of graph classes and to all stretch regimes, we had to go significantly beyond the state-of-the art.
The main highlight of my SODA’24 paper was to resolve in the affirmative the aforementioned SPR problem, which was open for two decades. We answered this question affirmatively in any minor-free graph, yet the question was open even in planar graphs. Our main insight was that shortcut partition, a new type of graph partition into low-diameter clusters that we developed in our FOCS’23 paper for planar metrics, and which is much more relaxed than similar previous types of graph partitions, is strong enough to resolve the SPR problem. To resolve the SPR problem in minor-free graphs we had to then extend our FOCS’23 construction of shortcut partition from planar metrics to minor-free metrics; breaking the “planarity barrier” was a major technical challenge, and we did so by devising a modified cop decomposition construction, first introduced in the context of the cops-and-robbers game by Andreae in 1986.
In my aforementioned SODA’25 paper with Bukov and Zhang, we presented two dynamic ((1+ ε)f)-approximation algorithms for maintaining a set cover: a deterministic one with amortized update time O(f log f poly(1/ε)) and a randomized one with amortized update time O(f log* f poly(1/ε)); the previous best time bounds were O(f log n poly(1/ε)) and O(f^2 poly(1/ε)), which coincide at f = log n. We first broke this time barrier using a deterministic algorithm, by refining techniques in previous work. Then we employed this deterministic algorithm recursively, by a new technique, which ultimately led to reducing the log f factor to a factor of log* f. We are not aware of any nontrivial graph problem for which the state-of-the-art dynamic algorithm admits a ``log-star’’ slack from optimality.
Looking forward, we anticipate progress in all fields described in the proposal: dynamic graph algorithms, static graph algorithms, and distributed computing.
We will continue to develop both randomized and deterministic algorithms and constructions as well as strive toward a better understanding of the gaps between randomized and deterministic solutions. We will continue working on fast edge coloring dynamic and static algorithms, on efficient constructions of graph spanners and related structures, and on fast matching and set cover algorithms for the dynamic and static settings.
My FOCS’24 paper with co-authors provided the first polynomial time improvements for (Δ+1)-edge coloring in over 40 years (concurrently to Assadi's SODA'25 result).
In my STOC'23 paper with Le, where we designed a unified framework for constructing light spanners that is applicable to a variety of graph classes and to all stretch regimes, we had to go significantly beyond the state-of-the art.
The main highlight of my SODA’24 paper was to resolve in the affirmative the aforementioned SPR problem, which was open for two decades. We answered this question affirmatively in any minor-free graph, yet the question was open even in planar graphs. Our main insight was that shortcut partition, a new type of graph partition into low-diameter clusters that we developed in our FOCS’23 paper for planar metrics, and which is much more relaxed than similar previous types of graph partitions, is strong enough to resolve the SPR problem. To resolve the SPR problem in minor-free graphs we had to then extend our FOCS’23 construction of shortcut partition from planar metrics to minor-free metrics; breaking the “planarity barrier” was a major technical challenge, and we did so by devising a modified cop decomposition construction, first introduced in the context of the cops-and-robbers game by Andreae in 1986.
In my aforementioned SODA’25 paper with Bukov and Zhang, we presented two dynamic ((1+ ε)f)-approximation algorithms for maintaining a set cover: a deterministic one with amortized update time O(f log f poly(1/ε)) and a randomized one with amortized update time O(f log* f poly(1/ε)); the previous best time bounds were O(f log n poly(1/ε)) and O(f^2 poly(1/ε)), which coincide at f = log n. We first broke this time barrier using a deterministic algorithm, by refining techniques in previous work. Then we employed this deterministic algorithm recursively, by a new technique, which ultimately led to reducing the log f factor to a factor of log* f. We are not aware of any nontrivial graph problem for which the state-of-the-art dynamic algorithm admits a ``log-star’’ slack from optimality.
Looking forward, we anticipate progress in all fields described in the proposal: dynamic graph algorithms, static graph algorithms, and distributed computing.
We will continue to develop both randomized and deterministic algorithms and constructions as well as strive toward a better understanding of the gaps between randomized and deterministic solutions. We will continue working on fast edge coloring dynamic and static algorithms, on efficient constructions of graph spanners and related structures, and on fast matching and set cover algorithms for the dynamic and static settings.