arXiv subjects

Omer Wasim

Publications and source records attributed to Omer Wasim.

3 recordsLinked to original sources

arXiv2024

Fully Dynamic (\Delta+1) Coloring Against Adaptive Adversaries

Over the years, there has been extensive work on fully dynamic algorithms for classic graph problems that admit greedy solutions. Examples include $(\Delta+1)$ vertex coloring, maximal independent set, and maximal matching. For all three problems, there are randomized algorithms that maintain a valid solution after each edge insertion or deletion to the $n$-vertex graph by spending $\polylog n$ time, provided that the adversary is oblivious. However, none of these algorithms work against adaptive adversaries whose updates may depend on the output of the algorithm. In fact, even breaking the trivial bound of $O(n)$ against adaptive adversaries remains open for all three problems. For instance, in the case of $(\Delta+1)$ vertex coloring, the main challenge is that an adaptive adversary can keep inserting edges between vertices of the same color, necessitating a recoloring of one of the endpoints. The trivial algorithm would simply scan all neighbors of one endpoint to find a new available color (which always exists) in $O(n)$ time. In this paper, we break this linear barrier for the $(\Delta+1)$ vertex coloring problem. Our algorithm is randomized, and maintains a valid $(\Delta+1)$ vertex coloring after each edge update by spending $\widetilde{O}(n^{8/9})$ time with high probability.

cs.DS↗

arXiv2024

Competitive Capacitated Online Recoloring

In this paper, we revisit the online recoloring problem introduced recently by Azar et al. In online recoloring, there is a fixed set $V$ of $n$ vertices and an initial coloring $c_0: V\rightarrow [k]$ for some $k\in \mathbb{Z}^{>0}$. Under an online sequence $\sigma$ of requests where each request is an edge $(u_t,v_t)$, a proper vertex coloring $c$ of the graph $G_t$ induced by requests until time $t$ needs to be maintained for all $t$; i.e., for any $(u,v)\in G_t$, $c(u)\neq c(v)$. The objective is to minimize the total weight of vertices recolored for the sequence $\sigma$. We obtain the first competitive algorithms for capacitated online recoloring and fully dynamic recoloring. Our first set of results is for $2$-recoloring using algorithms that are $(1+\varepsilon)$-resource augmented where $\varepsilon\in (0,1)$ is an arbitrarily small constant. Our main result is an $O(\log n)$-competitive deterministic algorithm for weighted bipartite graphs, which is asymptotically optimal in light of an $\Omega(\log n)$ lower bound that holds for an unbounded amount of augmentation. We also present an $O(n\log n)$-competitive deterministic algorithm for fully dynamic recoloring, which is optimal within an $O(\log n)$ factor in light of a $\Omega(n)$ lower bound that holds for an unbounded amount of augmentation. Our second set of results is for $\Delta$-recoloring in an $(1+\varepsilon)$-overprovisioned setting where the maximum degree of $G_t$ is bounded by $(1-\varepsilon)\Delta$ for all $t$, and each color assigned to at most $(1+\varepsilon)\frac{n}{\Delta}$ vertices, for an arbitrary $\varepsilon > 0$. Our main result is an $O(1)$-competitive randomized algorithm for $\Delta = O(\sqrt{n/\log n})$. We also present an $O(\Delta)$-competitive deterministic algorithm for $\Delta \le \varepsilon n/2$. Both results are asymptotically optimal.

cs.DS↗

arXiv2023

Sample Complexity of Linear Regression Models for Opinion Formation in Networks

Consider public health officials aiming to spread awareness about a new vaccine in a community interconnected by a social network. How can they distribute information with minimal resources, so as to avoid polarization and ensure community-wide convergence of opinion? To tackle such challenges, we initiate the study of sample complexity of opinion convergence in networks. Our framework is built on the recognized opinion formation game, where we regard the opinion of each agent as a data-derived model, unlike previous works that treat opinions as data-independent scalars. The opinion model for every agent is initially learned from its local samples and evolves game-theoretically as all agents communicate with neighbors and revise their models towards an equilibrium. Our focus is on the sample complexity needed to ensure that the opinions converge to an equilibrium such that the final model of every agent has low generalization error. Our paper has two main technical results. First, we present a novel polynomial time optimization framework to quantify the total sample complexity for arbitrary networks, when the underlying learning problem is (generalized) linear regression. Second, we leverage this optimization to study the network gain which measures the improvement of sample complexity when learning over a network compared to that in isolation. Towards this end, we derive network gain bounds for various network classes including cliques, star graphs, and random regular graphs. Additionally, our framework provides a method to study sample distribution within the network, suggesting that it is sufficient to allocate samples inversely to the degree. Empirical results on both synthetic and real-world networks strongly support our theoretical findings.

cs.GT↗