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1. Faster Algorithms for Rooted Connectivity in Directed Graphs

   Chekuri, Chandra ; Quanrud, Kent ( July 2021 , 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021))

   Bansal, Nikhil ; Merelli, Emanuela ; Worrell, James (Ed.)

   We consider the fundamental problems of determining the rooted and global edge and vertex connectivities (and computing the corresponding cuts) in directed graphs. For rooted (and hence also global) edge connectivity with small integer capacities we give a new randomized Monte Carlo algorithm that runs in time Õ(n²). For rooted edge connectivity this is the first algorithm to improve on the Ω(n³) time bound in the dense-graph high-connectivity regime. Our result relies on a simple combination of sampling coupled with sparsification that appears new, and could lead to further tradeoffs for directed graph connectivity problems. We extend the edge connectivity ideas to rooted and global vertex connectivity in directed graphs. We obtain a (1+ε)-approximation for rooted vertex connectivity in Õ(nW/ε) time where W is the total vertex weight (assuming integral vertex weights); in particular this yields an Õ(n²/ε) time randomized algorithm for unweighted graphs. This translates to a Õ(KnW) time exact algorithm where K is the rooted connectivity. We build on this to obtain similar bounds for global vertex connectivity. Our results complement the known results for these problems in the low connectivity regime due to work of Gabow [Harold N. Gabow, 1995] for edge connectivity from 1991, and the very recent work of Nanongkai et al. [Nanongkai et al., 2019] and Forster et al. [Sebastian Forster et al., 2020] for vertex connectivity. 

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2. Bilu-Linial Stability, Certified Algorithms and the Independent Set Problem

   Angelidakis, H ; Awasthi, P. ; Blum, A. ; Chatziafratis, V. ; Dan, C. ( September 2019 , 27th Annual European Symposium on Algorithms (ESA 2019))

   We study the classic Maximum Independent Set problem under the notion of stability introduced by Bilu and Linial (2010): a weighted instance of Independent Set is γ-stable if it has a unique optimal solution that remains the unique optimal solution under multiplicative perturbations of the weights by a factor of at most γ ≥ 1. The goal then is to efficiently recover this “pronounced” optimal solution exactly. In this work, we solve stable instances of Independent Set on several classes of graphs: we improve upon previous results by solving \tilde{O}(∆/sqrt(log ∆))-stable instances on graphs of maximum degree ∆, (k − 1)-stable instances on k-colorable graphs and (1 + ε)-stable instances on planar graphs (for any fixed ε > 0), using both combinatorial techniques as well as LPs and the Sherali-Adams hierarchy. For general graphs, we give an algorithm for (εn)-stable instances, for any fixed ε > 0, and lower bounds based on the planted clique conjecture. As a by-product of our techniques, we give algorithms as well as lower bounds for stable instances of Node Multiway Cut (a generalization of Edge Multiway Cut), by exploiting its connections to Vertex Cover. Furthermore, we prove a general structural result showing that the integrality gap of convex relaxations of several maximization problems reduces dramatically on stable instances. Moreover, we initiate the study of certified algorithms for Independent Set. The notion of a γ-certified algorithm was introduced very recently by Makarychev and Makarychev (2018) and it is a class of γ-approximation algorithms that satisfy one crucial property: the solution returned is optimal for a perturbation of the original instance, where perturbations are again multiplicative up to a factor of γ ≥ 1 (hence, such algorithms not only solve γ-stable instances optimally, but also have guarantees even on unstable instances). Here, we obtain ∆-certified algorithms for Independent Set on graphs of maximum degree ∆, and (1 + ε)-certified algorithms on planar graphs. Finally, we analyze the algorithm of Berman and Fürer (1994) and prove that it is a ((∆+1)/3 + ε)-certified algorithm for Independent Set on graphs of maximum degree ∆ where all weights are equal to 1. 

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3. A Distanced Matching Game, Decremental {APSP} in Expanders, and Faster Deterministic Algorithms for Graph Cut Problems

   Chuzhoy, Julia ( January 2023 , Proceedings of the 2023 {ACM-SIAM} Symposium on Discrete Algorithms,{SODA} 2023)

   Expander graphs play a central role in graph theory and algorithms. With a number of powerful algorithmic tools developed around them, such as the Cut-Matching game, expander pruning, expander decomposition, and algorithms for decremental All-Pairs Shortest Paths (APSP) in expanders, to name just a few, the use of expanders in the design of graph algorithms has become ubiquitous. Specific applications of interest to us are fast deterministic algorithms for cut problems in static graphs, and algorithms for dynamic distance-based graph problems, such as APSP. Unfortunately, the use of expanders in these settings incurs a number of drawbacks. For example, the best currently known algorithm for decremental APSP in constant-degree expanders can only achieve a (log n) O(1/ 2 ) -approximation with n 1+O( ) total update time for any . All currently known algorithms for the Cut Player in the Cut-Matching game are either randomized, or provide rather weak guarantees: expansion 1/(log n) 1/ with running time n 1+O( ) . This, in turn, leads to somewhat weak algorithmic guarantees for several central cut problems: the best current almost linear time deterministic algorithms for Sparsest Cut, Lowest Conductance Cut, and Balanced Cut can only achieve approximation factor (log n) ω(1). Lastly, when relying on expanders in distancebased problems, such as dynamic APSP, via current methods, it seems inevitable that one has to settle for approximation factors that are at least Ω(log n). In contrast, we do not have any negative results that rule out a factor-5 approximation with near-linear total update time. In this paper we propose the use of well-connected graphs, and introduce a new algorithmic toolkit for such graphs that, in a sense, mirrors the above mentioned algorithmic tools for expanders. One of these new tools is the Distanced Matching game, an analogue of the Cut-Matching game for well-connected graphs. We demonstrate the power of these new tools by obtaining better results for several of the problems mentioned above. First, we design an algorithm for decremental APSP in expanders with significantly better guarantees: in a constant-degree expander, the algorithm achieves (log n) 1+o(1)-approximation, with total update time n 1+o(1). We also obtain a deterministic algorithm for the Cut Player in the Cut-Matching game that achieves expansion 1 (log n) 5+o(1) in time n 1+o(1), deterministic almost linear-time algorithms for Sparsest Cut, Lowest-Conductance Cut, and Minimum Balanced Cut with approximation factors O(poly log n), as well as improved deterministic algorithm for Expander Decomposition. We believe that the use of well-connected graphs instead of expanders in various dynamic distance-based problems (such as APSP in general graphs) has the potential of providing much stronger guarantees, since we are no longer necessarily restricted to superlogarithmic approximation factors. 

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4. Minimum Cuts in Directed Graphs via Partial Sparsification

   https://doi.org/10.1109/FOCS52979.2021.00113  

   Cen, Ruoxu ; Li, Jason ; Nanongkai, Danupon ; Panigrahi, Debmalya ; Saranurak, Thatchaphol ; Quanrud, Kent ( February 2022 , 62nd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2021, Denver, CO, USA, February 7-10, 2022)

   We give an algorithm to find a minimum cut in an edge-weighted directed graph with n vertices and m edges in O ̃(n · max{m^{2/3}, n}) time. This improves on the 30 year old bound of O ̃(nm) obtained by Hao and Orlin for this problem. Using similar techniques, we also obtain O ̃ (n^2 /ε^2 )-time (1+ε)-approximation algorithms for both the minimum edge and minimum vertex cuts in directed graphs, for any fixed ε. Before our work, no (1+ε)-approximation algorithm better than the exact runtime of O ̃(nm) is known for either problem. Our algorithms follow a two-step template. In the first step, we employ a partial sparsification of the input graph to preserve a critical subset of cut values approximately. In the second step, we design algorithms to find the (edge/vertex) mincut among the preserved cuts from the first step. For edge mincut, we give a new reduction to O ̃ (min{n/m^{1/3} , √n}) calls of any maxflow subroutine, via packing arborescences in the sparsifier. For vertex mincut, we develop new local flow algorithms to identify small unbalanced cuts in the sparsified graph. 

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5. Can We Break Symmetry with o(m) Communication?

   https://doi.org/10.1145/3465084  

   Pai, Shreyas ; Pandurangan, Gopal ; Pemmaraju, Sriram ; Robinson, Peter ( July 2021 , PODC'21: Proceedings of the 2021 ACM Symposium on Principles of Distributed Computing)

   null (Ed.)

   We study the communication cost (or message complexity) of fundamental distributed symmetry breaking problems, namely, coloring and MIS. While significant progress has been made in understanding and improving the running time of such problems, much less is known about the message complexity of these problems. In fact, all known algorithms need at least Ω(m) communication for these problems, where m is the number of edges in the graph. We addressthe following question in this paper: can we solve problems such as coloring and MIS using sublinear, i.e., o(m) communication, and if sounder what conditions? In a classical result, Awerbuch, Goldreich, Peleg, and Vainish [JACM 1990] showed that fundamental global problems such asbroadcast and spanning tree construction require at least o(m) messages in the KT-1 Congest model (i.e., Congest model in which nodes have initial knowledge of the neighbors' ID's) when algorithms are restricted to be comparison-based (i.e., algorithms inwhich node ID's can only be compared). Thirty five years after this result, King, Kutten, and Thorup [PODC 2015] showed that onecan solve the above problems using Õ(n) messages (n is the number of nodes in the graph) in Õ(n) rounds in the KT-1 Congest model if non-comparison-based algorithms are permitted. An important implication of this result is that one can use the synchronous nature of the KT-1 Congest model, using silence to convey information,and solve any graph problem using non-comparison-based algorithms with Õ(n) messages, but this takes an exponential number of rounds. In the asynchronous model, even this is not possible. In contrast, much less is known about the message complexity of local symmetry breaking problems such as coloring and MIS. Our paper fills this gap by presenting the following results. Lower bounds: In the KT-1 CONGEST model, we show that any comparison-based algorithm, even a randomized Monte Carlo algorithm with constant success probability, requires Ω(n 2) messages in the worst case to solve either (△ + 1)-coloring or MIS, regardless of the number of rounds. We also show that Ω(n) is a lower bound on the number ofmessages for any (△ + 1)-coloring or MIS algorithm, even non-comparison-based, and even with nodes having initial knowledge of up to a constant radius. Upper bounds: In the KT-1 CONGEST model, we present the following randomized non-comparison-based algorithms for coloring that, with high probability, use o(m) messages and run in polynomially many rounds.(a) A (△ + 1)-coloring algorithm that uses Õ(n1.5) messages, while running in Õ(D + √ n) rounds, where D is the graph diameter. Our result also implies an asynchronous algorithm for (△ + 1)-coloring with the same message bound but running in Õ(n) rounds. (b) For any constantε > 0, a (1+ε)△-coloring algorithm that uses Õ(n/ε 2 ) messages, while running in Õ(n) rounds. If we increase our input knowledge slightly to radius 2, i.e.,in the KT-2 CONGEST model, we obtain:(c) A randomized comparison-based MIS algorithm that uses Õ(n 1.5) messages. while running in Õ( √n) rounds. While our lower bound results can be viewed as counterparts to the classical result of Awerbuch, Goldreich, Peleg, and Vainish [JACM 90], but for local problems, our algorithms are the first-known algorithms for coloring and MIS that take o(m) messages and run in polynomially many rounds. 

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