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Online Disjoint Spanning Trees and Polymatroid Bases
Get PDFFinding the maximum number of disjoint spanning trees in a given graph is a well-studied problem with several applications and connections. The Tutte-Nash-Williams theorem provides a min-max relation for this problem which also extends to disjoint bases in a matroid and leads to efficient algorithms [Schrijver, 2003]. Several other packing problems such as element disjoint Steiner trees, disjoint set covers, and disjoint dominating sets are NP-Hard but admit an O(log n)-approximation [Feige et al., 2002; Cheriyan and Salavatipour, 2007]. Călinescu, Chekuri, and Vondrák [G. Călinescu et al., 2009] viewed all these packing problems as packing bases of a polymatroid and provided a unified perspective. Motivated by applications in wireless networks, recent works have studied the problem of packing set covers in the online model [Pananjady et al., 2015; Emek et al., 2019; Bienkowski et al., 2025]. The online model poses new challenges for packing problems. In particular, it is not clear how to pack a maximum number of disjoint spanning trees in a graph when edges arrive online. Motivated by these applications and theoretical considerations, we formulate an online model for packing bases of a polymatroid, and describe a randomized algorithm with a polylogarithmic competitive ratio. Our algorithm is based on interesting connections to the notion of quotients of a polymatroid that has recently seen applications in polymatroid sparsification [Quanrud, 2024]. We generalize the previously known result for the online disjoint set cover problem [Emek et al., 2019] and also address several other packing problems in a unified fashion. For the special case of packing disjoint spanning trees in a graph (or a hypergraph) whose edges arrive online, we provide an alternative to our general algorithm that is simpler and faster while achieving the same poly-logarithmic competitive ratio
Query Efficient Weighted Stochastic Matching
Get PDFIn this paper, we study the weighted stochastic matching problem. Let G = (V, E) be a given edge-weighted graph, and let its realization be a random subgraph of G that includes each edge e ∈ E independently with a known probability p_e. The goal in this problem is to pick a sparse subgraph Q of G without prior knowledge of , such that the maximum weight matching among the realized edges of Q (i.e., the subgraph Q ∩ ) in expectation approximates the maximum weight matching of the entire realization .
It is established by previous work that attaining any constant approximation ratio for this problem requires selecting a subgraph of max-degree Ω(1/p), where p = min_{e ∈ E} p_e. On the positive side, there exists a (1-ε)-approximation algorithm by Behnezhad and Derakhshan [FOCS'20], albeit at the cost of a max-degree having exponential dependence on 1/p. Within the O(1/p) query regime, however, the best-known algorithm achieves a 0.536 approximation ratio due to Dughmi, Kalayci, and Patel [ICALP'23], improving over the 0.501 approximation algorithm by Behnezhad, Farhadi, Hajiaghayi, and Reyhani [SODA'19].
In this work, we present a 0.68-approximation algorithm with the asymptotically optimal O(1/p) queries per vertex. Our result not only substantially improves the approximation ratio for weighted graphs, but also breaks the well-known 2/3 barrier with the optimal number of queries - even for unweighted graphs. Our analysis involves reducing the problem to designing a randomized matching algorithm on a given stochastic graph with some variance-bounding properties. To achieve these properties, we leverage a randomized algorithm by MacRury and Ma [STOC'24] for a variant of online stochastic matching
Tiling Random Regular Graphs Efficiently
Get PDFWe show that for every ε > 0 there exists a sufficiently large d₀ ∈ ℕ such that for every d ≥ d₀, whp the random d-regular graph G(n,d) contains a T-factor for every tree T on at most (1-ε)d/log d vertices. This is best possible since, for large enough integer d, whp G(n,d) does not contain a ((1+ε)d)/(log d)-star-factor. Our method gives a randomised algorithm which whp finds said T-factor and whose expected running time is O(n^{1+o(1)}), as well as an efficient deterministic counterpart
Minimizing Recourse in an Adaptive Balls and Bins Game
Get PDFWe consider a simple load-balancing game between an algorithm and an adaptive adversary. In a simplified version of this game, the adversary observes the assignment of jobs to machines and selects a machine to kill. The algorithm must then restart the jobs from the failed machine on other machines. The adversary repeats this process, observing the new assignment and eliminating another machine, and so on. The adversary aims to force the algorithm to perform many restarts, while we seek a robust algorithm that minimizes restarts regardless of the adversary’s strategy. This game was recently introduced by Bhattacharya et al. for designing a 3-spanner with low recourse against an adaptive adversary.
We prove that a simple algorithm, which assigns each job to a randomly chosen live bin, incurs O(n log n) recourse against an adaptive adversary. This enables us to construct a much simpler 3-spanner with a recourse that is smaller by a factor of O(log² n) compared to the previous construction, without increasing the update time or the size of the spanner.
This motivates a careful examination of the range of attacks an adaptive adversary can deploy against simple algorithms before resorting to more complex ones. As our case study demonstrates, this attack space may not be as large as it initially appears, enabling the development of robust algorithms that are both simpler and easier to analyze
Faster Dynamic (Δ+1)-Coloring Against Adaptive Adversaries
Get PDFWe consider the problem of maintaining a proper (Δ + 1)-vertex coloring in a graph on n-vertices and maximum degree Δ undergoing edge insertions and deletions. We give a randomized algorithm with amortized update time Õ(n^{2/3}) against adaptive adversaries, meaning that updates may depend on past decisions by the algorithm. This improves on the very recent Õ(n^{8/9})-update-time algorithm by Behnezhad, Rajaraman, and Wasim (SODA 2025) and matches a natural barrier for dynamic (Δ+1)-coloring algorithms. The main improvements are on the densest regions of the graph, where we use structural hints from the study of distributed graph algorithms
A Nearly Optimal Deterministic Algorithm for Online Transportation Problem
Get PDFFor the online transportation problem with m server sites, it has long been known that the competitive ratio of any deterministic algorithm is at least 2m-1. Kalyanasundaram and Pruhs conjectured in 1998 that a deterministic (2m-1)-competitive algorithm exists for this problem, a conjecture that has remained open for over two decades.
In this paper, we propose a new deterministic algorithm for the online transportation problem and show that it achieves a competitive ratio of at most 8m-5. This is the first O(m)-competitive deterministic algorithm, coming close to the lower bound of 2m-1 within a constant factor
Fourier Analysis of Iterative Algorithms
Get PDFWe study a general class of nonlinear iterative algorithms which includes power iteration, belief propagation and approximate message passing, and many forms of gradient descent. When the input is a random matrix with i.i.d. entries, we use Boolean Fourier analysis to analyze these algorithms as low-degree polynomials in the entries of the input matrix. Each symmetrized Fourier character represents all monomials with a certain shape as specified by a small graph, which we call a Fourier diagram.
We prove fundamental asymptotic properties of the Fourier diagrams: over the randomness of the input, all diagrams with cycles are negligible; the tree-shaped diagrams form a basis of asymptotically independent Gaussian vectors; and, when restricted to the trees, iterative algorithms exactly follow an idealized Gaussian dynamic. We use this to prove a state evolution formula, giving a "complete" asymptotic description of the algorithm’s trajectory.
The restriction to tree-shaped monomials mirrors the assumption of the cavity method, a 40-year-old non-rigorous technique in statistical physics which has served as one of the most important techniques in the field. We demonstrate how to implement cavity method derivations by 1) restricting the iteration to its tree approximation, and 2) observing that heuristic cavity method-type arguments hold rigorously on the simplified iteration. Our proofs use combinatorial arguments similar to the trace method from random matrix theory.
Finally, we push the diagram analysis to a number of iterations that scales with the dimension n of the input matrix, proving that the tree approximation still holds for a simple variant of power iteration all the way up to n^{Ω(1)} iterations
Maximum Bipartite vs. Triangle-Free Subgraph
Get PDFGiven a (multi)graph G which contains a bipartite subgraph with ρ edges, what is the largest triangle-free subgraph of G that can be found efficiently? We present an SDP-based algorithm that finds one with at least 0.8823 ρ edges, thus improving on the subgraph with 0.878 ρ edges obtained by the classic Max-Cut algorithm of Goemans and Williamson. On the other hand, by a reduction from Håstad’s 3-bit PCP we show that it is NP-hard to find a triangle-free subgraph with (25 / 26 + ε) ρ ≈ (0.961 + ε) ρ edges.
As an application, we classify the Maximum Promise Constraint Satisfaction Problem, denoted byMaxPCSP(G, H), for all bipartite G: Given an input (multi)graph X which admits a G-colouring satisfying ρ edges, find an H-colouring of X that satisfies ρ edges. This problem is solvable in polynomial time, apart from trivial cases, if H contains a triangle, and is NP-hard otherwise
New and Improved Bounds for Markov Paging
Get PDFIn the Markov paging model, one assumes that page requests are drawn from a Markov chain over the pages in memory, and the goal is to maintain a fast cache that suffers few page faults in expectation. While computing the optimal online algorithm (OPT) for this problem naively takes time exponential in the size of the cache, the best-known polynomial-time approximation algorithm is the dominating distribution algorithm due to Lund, Phillips and Reingold (FOCS 1994), who showed that the algorithm is 4-competitive against OPT. We substantially improve their analysis and show that the dominating distribution algorithm is in fact 2-competitive against OPT. We also show a lower bound of 1.5907-competitiveness for this algorithm - to the best of our knowledge, no such lower bound was previously known
An Upper Bound on the Weisfeiler-Leman Dimension
Get PDFThe Weisfeiler-Leman (WL) algorithms form a family of incomplete approaches to the graph isomorphism problem. They recently found various applications in algorithmic group theory and machine learning. In fact, the algorithms form a parameterized family: for each k ∈ ℕ there is a corresponding k-dimensional algorithm WLk. The algorithms become increasingly powerful with increasing dimension, but at the same time the running time increases. The WL-dimension of a graph G is the smallest k ∈ ℕ for which WLk correctly decides isomorphism between G and every other graph. In some sense, the WL-dimension measures how difficult it is to test isomorphism of one graph to others using a fairly general class of combinatorial algorithms. Nowadays, it is a standard measure in descriptive complexity theory for the structural complexity of a graph.
We prove that the WL-dimension of a graph on n vertices is at most 3/20 ⋅ n + o(n) = 0.15 ⋅ n + o(n).
Reducing the question to coherent configurations, the proof develops various techniques to analyze their structure. This includes sufficient conditions under which a fiber can be restored uniquely up to isomorphism if it is removed, a recursive proof exploiting a degree reduction and treewidth bounds, as well as an exhaustive analysis of interspaces involving small fibers.
As a base case, we also analyze the dimension of coherent configurations with small fiber size and thereby graphs with small color class size