Tigran Tonoyan’s research while affiliated with Technion – Israel Institute of Technology and other places

We present a new technique to efficiently sample and communicate a large number of elements from a distributed sampling space. When used in the context of a recent LOCAL algorithm for $(\operatorname{degree}+1)$-list-coloring (D1LC), this allows us to solve D1LC in $O(\log^5 \log n)$ CONGEST rounds, and in only $O(\log^* n)$ rounds when the graph has minimum degree $\Omega(\log^7 n)$, w.h.p. The technique also has immediate applications in testing some graph properties locally, and for estimating the sparsity/density of local subgraphs in O(1) CONGEST rounds, w.h.p.

Linial’s famous color reduction algorithm reduces a given m -coloring of a graph with maximum degree $\varDelta$ Δ to an $O(\varDelta ^2\log m)$ O ( Δ 2 log m ) -coloring, in a single round in the LOCAL model. We give a similar result when nodes are restricted to choose their color from a list of allowed colors: given an m -coloring in a directed graph of maximum outdegree $\beta$ β , if every node has a list of size $\varOmega (\beta ^2 (\log \beta +\log \log m + \log \log |{\mathcal {C}}|))$ Ω ( β 2 ( log β + log log m + log log | C | ) ) from a color space ${\mathcal {C}}$ C then they can select a color in two rounds in the LOCAL model. Moreover, the communication of a node essentially consists of sending its list to the neighbors. This is obtained as part of a framework that also contains Linial’s color reduction (with an alternative proof) as a special case. Our result also leads to a defective list coloring algorithm. As a corollary, we improve the state-of-the-art truly local $({\text {deg}}+1)$ ( deg + 1 ) -list coloring algorithm from Barenboim et al. (PODC, pp 437–446, 2018) by slightly reducing the runtime to $O(\sqrt{\varDelta \log \varDelta })+\log ^* n$ O ( Δ log Δ ) + log ∗ n and significantly reducing the message size (from $\varDelta ^{O(\log ^* \varDelta )}$ Δ O ( log ∗ Δ ) to roughly $\varDelta$ Δ ). Our techniques are inspired by the local conflict coloring framework of Fraigniaud et al. (in: FOCS, pp 625–634, 2016).

We present a new approach to randomized distributed graph coloring that is simpler and more efficient than previous ones. In particular, it allows us to tackle the $(\operatorname{deg}+1)$-list-coloring (D1LC) problem, where each node v of degree $d_v$ is assigned a palette of $d_v+1$ colors, and the objective is to find a proper coloring using these palettes. While for $(\Delta+1)$-coloring (where $\Delta$ is the maximum degree), there is a fast randomized distributed $O(\log^3\log n)$-round algorithm (Chang, Li, and Pettie [SIAM J. Comp. 2020]), no $o(\log n)$-round algorithms are known for the D1LC problem. We give a randomized distributed algorithm for D1LC that is optimal under plausible assumptions about the deterministic complexity of the problem. Using the recent deterministic algorithm of Ghaffari and Kuhn [FOCS2021], our algorithm runs in $O(\log^3 \log n)$ time, matching the best bound known for $(\Delta+1)$-coloring. In addition, it colors all nodes of degree $\Omega(\log^7 n)$ in $O(\log^* n)$ rounds. A key contribution is a subroutine to generate slack for D1LC. When placed into the framework of Assadi, Chen, and Khanna [SODA2019] and Alon and Assadi [APPROX/RANDOM2020], this almost immediately leads to a palette sparsification theorem for D1LC, generalizing previous results. That gives fast algorithms for D1LC in three different models: an O(1)-round algorithm in the MPC model with $\tilde{O}(n)$ memory per machine; a single-pass semi-streaming algorithm in dynamic streams; and an $\tilde{O}(n\sqrt{n})$-time algorithm in the standard query model.

We introduce the problem of finding a spanning tree along with a partition of the tree edges into the fewest number of feasible sets, where constraints on the edges define feasibility. The motivation comes from wireless networking, where we seek to model the irregularities seen in actual wireless environments. Not all node pairs may be able to communicate, even if geographically close—thus, the available pairs are specified with a link graph ${{\mathcal {G}}}=(V,E)$. Also, signal attenuation need not follow a nice geometric formula—hence, interference is modeled by a conflict (hyper)graph ${{\mathcal {C}}}=(E,F)$ on the links. The objective is to maximize the efficiency of the communication, or equivalently, to minimize the length of a schedule of the tree edges in the form of a coloring. We find that in spite of all this generality, the problem can be approximated linearly in terms of a versatile parameter, the inductive independence of the conflict graph. Specifically, we give a simple algorithm that attains a $O(\rho \log n)$-approximation, where n is the number of nodes and $\rho$ is the inductive independence. For an extension to Steiner trees, modeling multicasting, we obtain a $O(\rho \log ^2 n)$-approximation. We also consider a natural geometric setting when only links longer than a threshold can be unavailable, and analyze the performance of a geometric minimum spanning tree.

Reconfiguration schedules, i.e., sequences that gradually transform one solution of a problem to another while always maintaining feasibility, have been extensively studied. Most research has dealt with the decision problem of whether a reconfiguration schedule exists, and the complexity of finding one. A prime example is the reconfiguration of vertex covers. We initiate the study of batched vertex cover reconfiguration, which allows to reconfigure multiple vertices concurrently while requiring that any adversarial reconfiguration order within a batch maintains feasibility. The latter provides robustness, e.g., if the simultaneous reconfiguration of a batch cannot be guaranteed. The quality of a schedule is measured by the number of batches until all nodes are reconfigured, and its cost, i.e., the maximum size of an intermediate vertex cover. To set a baseline for batch reconfiguration, we show that for graphs belonging to one of the classes $\{\mathsf{cycles, trees, forests, chordal, cactus, even\text{-}hole\text{-}free, claw\text{-}free}\}$, there are schedules that use $O(\varepsilon^{-1})$ batches and incur only a $1+\varepsilon$ multiplicative increase in cost over the best sequential schedules. Our main contribution is to compute such batch schedules in $O(\varepsilon^{-1}\log^* n)$ distributed time, which we also show to be tight. Further, we show that once we step out of these graph classes we face a very different situation. There are graph classes on which no efficient distributed algorithm can obtain the best (or almost best) existing schedule. Moreover, there are classes of bounded degree graphs which do not admit any reconfiguration schedules without incurring a large multiplicative increase in the cost at all.

We study problems with stochastic uncertainty information on intervals for which the precise value can be queried by paying a cost. The goal is to devise an adaptive decision tree to find a correct solution to the problem in consideration while minimizing the expected total query cost. We show that, for the sorting problem, such a decision tree can be found in polynomial time. For the problem of finding the data item with minimum value, we have some evidence for hardness. This contradicts intuition, since the minimum problem is easier both in the online setting with adversarial inputs and in the offline verification setting. However, the stochastic assumption can be leveraged to beat both deterministic and randomized approximation lower bounds for the online setting.

A graph G is a Generalized Disk Graph if for some dimension $\eta \ge 1$, a non-decreasing sub-linear function f and natural number t, each vertex $v_i$ can be assigned a length $l_i$ and set $P_i \subseteq \mathbb {R}^\eta$ of t points such that $v_iv_j$ is an edge of G if and only if $l_i \le l_j$ and $d(P_i, P_j) \le l_if(l_j/l_i)\,+\,l_if(1)$, where $d(\cdot , \cdot )$ is the least distance between points in either set. Generalized disk graphs were introduced as a model of wireless network interference and have been shown to be dramatically more accurate than disk graphs or other previously known graph classes. However, their properties have not been studied extensively before.