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1. A Simple Deterministic Near-Linear Time Approximation Scheme for Transshipment with Arbitrary Positive Edge Costs

   https://doi.org/10.4230/LIPIcs.ESA.2024.56  

   Fox, Emily (January 2024, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Chan, Timothy; Fischer, Johannes; Iacono, John; Herman, Grzegorz (Ed.)

   We describe a simple deterministic near-linear time approximation scheme for uncapacitated minimum cost flow in undirected graphs with positive real edge weights, a problem also known as transshipment. Specifically, our algorithm takes as input a (connected) undirected graph G = (V, E), vertex demands b ∈ R^V such that ∑_{v ∈ V} b(v) = 0, positive edge costs c ∈ R_{≥ 0}^E, and a parameter ε > 0. In O(ε^{-2} m log^{O(1)} n) time, it returns a flow f such that the net flow out of each vertex is equal to the vertex’s demand and the cost of the flow is within a (1 ± ε) factor of optimal. Our algorithm is combinatorial and has no running time dependency on the demands or edge costs. With the exception of a recent result presented at STOC 2022 for polynomially bounded edge weights, all almost- and near-linear time approximation schemes for transshipment relied on randomization to embed the problem instance into low-dimensional space. Our algorithm instead deterministically approximates the cost of routing decisions that would be made if the input were subject to a random tree embedding. To avoid computing the Ω(n²) vertex-vertex distances that an approximation of this kind suggests, we also take advantage of the clustering method used in the well-known Thorup-Zwick distance oracle. 

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2. Finding Most-Shattering Minimum Vertex Cuts of Polylogarithmic Size in Near-Linear Time

   https://doi.org/10.4230/LIPIcs.ICALP.2024.87  

   Hua, Kevin; Li, Daniel; Park, Jaewoo; Saranurak, Thatchaphol (January 2024, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Bringmann, Karl; Grohe, Martin; Puppis, Gabriele; Svensson, Ola (Ed.)

   We show the first near-linear time randomized algorithms for listing all minimum vertex cuts of polylogarithmic size that separate the graph into at least three connected components (also known as shredders) and for finding the most shattering one, i.e., the one maximizing the number of connected components. Our algorithms break the quadratic time bound by Cheriyan and Thurimella (STOC'96) for both problems that has been unimproved for more than two decades. Our work also removes an important bottleneck to near-linear time algorithms for the vertex connectivity augmentation problem (Jordan '95) and finding an even-length directed cycle in a graph, a problem shown to be equivalent to many other fundamental problems (Vazirani and Yannakakis '90, Robertson et al. '99). Note that it is necessary to list only minimum vertex cuts that separate the graph into at least three components because there can be an exponential number of minimum vertex cuts in general. To obtain a near-linear time algorithm, we have extended techniques in local flow algorithms developed by Forster et al. (SODA'20) to list shredders on a local scale. We also exploit fast queries to a pairwise vertex connectivity oracle subject to vertex failures (Long and Saranurak FOCS'22, Kosinas ESA'23). This is the first application of using connectivity oracles subject to vertex failures to speed up a static graph algorithm. 

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3. Parameterized Complexity of Fair Bisection: (FPT-Approximation meets Unbreakability)

   https://doi.org/10.4230/LIPIcs.ESA.2023.63  

   Inamdar, Tanmay; Lokshtanov, Daniel; Saurabh, Saket; Surianarayanan, Vaishali (January 2023, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Gørtz, Inge Li; Farach-Colton, Martin; Puglisi, Simon J; Herman, Grzegorz (Ed.)

   In the Minimum Bisection problem input is a graph G and the goal is to partition the vertex set into two parts A and B, such that ||A|-|B|| ≤ 1 and the number k of edges between A and B is minimized. The problem is known to be NP-hard, and assuming the Unique Games Conjecture even NP-hard to approximate within a constant factor [Khot and Vishnoi, J.ACM'15]. On the other hand, a 𝒪(log n)-approximation algorithm [Räcke, STOC'08] and a parameterized algorithm [Cygan et al., ACM Transactions on Algorithms'20] running in time k^𝒪(k) n^𝒪(1) is known. The Minimum Bisection problem can be viewed as a clustering problem where edges represent similarity and the task is to partition the vertices into two equally sized clusters while minimizing the number of pairs of similar objects that end up in different clusters. Motivated by a number of egregious examples of unfair bias in AI systems, many fundamental clustering problems have been revisited and re-formulated to incorporate fairness constraints. In this paper we initiate the study of the Minimum Bisection problem with fairness constraints. Here the input is a graph G, positive integers c and k, a function χ:V(G) → {1, …, c} that assigns a color χ(v) to each vertex v in G, and c integers r_1,r_2,⋯,r_c. The goal is to partition the vertex set of G into two almost-equal sized parts A and B with at most k edges between them, such that for each color i ∈ {1, …, c}, A has exactly r_i vertices of color i. Each color class corresponds to a group which we require the partition (A, B) to treat fairly, and the constraints that A has exactly r_i vertices of color i can be used to encode that no group is over- or under-represented in either of the two clusters. We first show that introducing fairness constraints appears to make the Minimum Bisection problem qualitatively harder. Specifically we show that unless FPT=W[1] the problem admits no f(c)n^𝒪(1) time algorithm even when k = 0. On the other hand, our main technical contribution shows that is that this hardness result is simply a consequence of the very strict requirement that each color class i has exactly r_i vertices in A. In particular we give an f(k,c,ε)n^𝒪(1) time algorithm that finds a balanced partition (A, B) with at most k edges between them, such that for each color i ∈ [c], there are at most (1±ε)r_i vertices of color i in A. Our approximation algorithm is best viewed as a proof of concept that the technique introduced by [Lampis, ICALP'18] for obtaining FPT-approximation algorithms for problems of bounded tree-width or clique-width can be efficiently exploited even on graphs of unbounded width. The key insight is that the technique of Lampis is applicable on tree decompositions with unbreakable bags (as introduced in [Cygan et al., SIAM Journal on Computing'14]). An important ingredient of our approximation scheme is a combinatorial result that may be of independent interest, namely that for every k, every graph G admits a tree decomposition with adhesions of size at most 𝒪(k), unbreakable bags, and logarithmic depth. 

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4. A Weighted Approach to the Maximum Cardinality Bipartite Matching Problem with Applications in Geometric Settings

   Lahn, Nathaniel; Raghvendra, Sharath (June 2019, Leibniz international proceedings in informatics)

   We present a weighted approach to compute a maximum cardinality matching in an arbitrary bipartite graph. Our main result is a new algorithm that takes as input a weighted bipartite graph G(A cup B,E) with edge weights of 0 or 1. Let w <= n be an upper bound on the weight of any matching in G. Consider the subgraph induced by all the edges of G with a weight 0. Suppose every connected component in this subgraph has O(r) vertices and O(mr/n) edges. We present an algorithm to compute a maximum cardinality matching in G in O~(m(sqrt{w} + sqrt{r} + wr/n)) time. When all the edge weights are 1 (symmetrically when all weights are 0), our algorithm will be identical to the well-known Hopcroft-Karp (HK) algorithm, which runs in O(m sqrt{n}) time. However, if we can carefully assign weights of 0 and 1 on its edges such that both w and r are sub-linear in n and wr=O(n^{gamma}) for gamma < 3/2, then we can compute maximum cardinality matching in G in o(m sqrt{n}) time. Using our algorithm, we obtain a new O~(n^{4/3}/epsilon^4) time algorithm to compute an epsilon-approximate bottleneck matching of A,B subsetR^2 and an 1/(epsilon^{O(d)}}n^{1+(d-1)/(2d-1)}) poly log n time algorithm for computing epsilon-approximate bottleneck matching in d-dimensions. All previous algorithms take Omega(n^{3/2}) time. Given any graph G(A cup B,E) that has an easily computable balanced vertex separator for every subgraph G'(V',E') of size |V'|^{delta}, for delta in [1/2,1), we can apply our algorithm to compute a maximum matching in O~(mn^{delta/1+delta}) time improving upon the O(m sqrt{n}) time taken by the HK-Algorithm. 

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5. Parallel, Distributed, and Quantum Exact Single-Source Shortest Paths with Negative Edge Weights

   https://doi.org/10.4230/LIPIcs.ESA.2024.13  

   Ashvinkumar, Vikrant; Bernstein, Aaron; Cao, Nairen; Grunau, Christoph; Haeupler, Bernhard; Jiang, Yonggang; Nanongkai, Danupon; Su, Hsin-Hao (January 2024, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Chan, Timothy; Fischer, Johannes; Iacono, John; Herman, Grzegorz (Ed.)

   This paper presents parallel, distributed, and quantum algorithms for single-source shortest paths when edges can have negative integer weights (negative-weight SSSP). We show a framework that reduces negative-weight SSSP in all these settings to n^{o(1)} calls to any SSSP algorithm that works on inputs with non-negative integer edge weights (non-negative-weight SSSP) with a virtual source. More specifically, for a directed graph with m edges, n vertices, undirected hop-diameter D, and polynomially bounded integer edge weights, we show randomized algorithms for negative-weight SSSP with - W_{SSSP}(m,n)n^{o(1)} work and S_{SSSP}(m,n)n^{o(1)} span, given access to a non-negative-weight SSSP algorithm with W_{SSSP}(m,n) work and S_{SSSP}(m,n) span in the parallel model, and - T_{SSSP}(n,D)n^{o(1)} rounds, given access to a non-negative-weight SSSP algorithm that takes T_{SSSP}(n,D) rounds in CONGEST, and - Q_{SSSP}(m,n)n^{o(1)} quantum edge queries, given access to a non-negative-weight SSSP algorithm that takes Q_{SSSP}(m,n) queries in the quantum edge query model. This work builds off the recent result of Bernstein, Nanongkai, Wulff-Nilsen [Bernstein et al., 2022], which gives a near-linear time algorithm for negative-weight SSSP in the sequential setting. Using current state-of-the-art non-negative-weight SSSP algorithms yields randomized algorithms for negative-weight SSSP with - m^{1+o(1)} work and n^{1/2+o(1)} span in the parallel model, and - (n^{2/5}D^{2/5} + √n + D)n^{o(1)} rounds in CONGEST, and - m^{1/2}n^{1/2+o(1)} quantum queries to the adjacency list or n^{1.5+o(1)} quantum queries to the adjacency matrix. Up to a n^{o(1)} factor, the parallel and distributed results match the current best upper bounds for reachability [Jambulapati et al., 2019; Cao et al., 2021]. Consequently, any improvement to negative-weight SSSP in these models beyond the n^{o(1)} factor necessitates an improvement to the current best bounds for reachability. The quantum result matches the lower bound up to an n^{o(1)} factor [Aija Berzina et al., 2004]. Our main technical contribution is an efficient reduction from computing a low-diameter decomposition (LDD) of directed graphs to computations of non-negative-weight SSSP with a virtual source. Efficiently computing an LDD has heretofore only been known for undirected graphs in both the parallel and distributed models, and been rather unstudied in quantum models. The directed LDD is a crucial step of the sequential algorithm in [Bernstein et al., 2022], and we think that its applications to other problems in parallel and distributed models are far from being exhausted. Other ingredients of our results include altering the recursion structure of the scaling algorithm in [Bernstein et al., 2022] to surmount difficulties that arise in these models, and also an efficient reduction from computing strongly connected components to computations of SSSP with a virtual source in CONGEST. The latter result answers a question posed in [Bernstein and Nanongkai, 2019] in the negative. 

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