Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2207.07363

Exact Flow Sparsification Requires Unbounded Size

Robert Krauthgamer·Ron Mosenzon
View original source ↗Read source document ↗Export metadata ↓

Abstract

Given a large edge-capacitated network $G$ and a subset of $k$ vertices called terminals, an (exact) flow sparsifier is a small network $G'$ that preserves (exactly) all multicommodity flows that can be routed between the terminals. Flow sparsifiers were introduced by Leighton and Moitra [STOC 2010], and have been studied and used in many algorithmic contexts. A fundamental question that remained open for over a decade, asks whether every $k$-terminal network admits an exact flow sparsifier whose size is bounded by some function $f(k)$ (regardless of the size of $G$ or its capacities). We resolve this question in the negative by proving that there exist $6$-terminal networks $G$ whose flow sparsifiers $G'$ must have arbitrarily large size. This unboundedness is perhaps surprising, since the analogous sparsification that preserves all terminal cuts (called exact cut sparsifier or mimicking network) admits sparsifiers of size $f_0(k)\leq 2^{2^k}$ [Hagerup, Katajainen, Nishimura, and Ragde, JCSS 1998]. We prove our results by analyzing the set of all feasible demands in the network, known as the demand polytope. We identify an invariant of this polytope, essentially the slope of certain facets, that can be made arbitrarily large even for $k=6$, and implies an explicit lower bound on the size of the network. We further use this technique to answer, again in the negative, an open question of Seymour [JCTB 2015] regarding flow-sparsification that uses only contractions and preserves the infeasibility of one demand vector.

Explore related subjects

cs.DS

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Robert Krauthgamer, Ron Mosenzon. 2022-08-03. Exact Flow Sparsification Requires Unbounded Size. https://arxiv.org/abs/2207.07363

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Beyond Kruskal: Polynomial-Time Tensor Decomposition under the Lovitz-Petrov Condition

Identifiability criteria certify that a given tensor decomposition is a unique rank decomposition. Kruskal's classical condition is one of the best-known deterministic criteria for identifiability. However, no polynomial-time decomposition algorithm is known under the Kruskal condition, and verifying the condition itself is NP-hard. Lovitz and Petrov introduced a strictly more general identifiability condition which, in contrast, is polynomial-time verifiable, but no polynomial-time decomposition algorithm was previously known under this condition. We give a polynomial-time algorithm for tensor decomposition under the Lovitz--Petrov condition. Moreover, combining our algorithm with polynomial-time verification of the Lovitz--Petrov condition yields an efficient end-to-end certification procedure: after computing a decomposition, one can deterministically certify in polynomial time that it is unique and therefore of minimum rank. This contrasts with an arbitrary tensor decomposition, which certifies only an upper bound on the tensor rank, while determining tensor rank is NP-hard in general.

cs.DS↗
arXiv2026

Poisson Exchange Beyond Submodularity: Effective Approximation Algorithms for Offline and Online Subset Selection over Matroids

Over the past decade, a growing body of research has shown that $γ$-weak submodularity broadly arises in numerous subset selection tasks, including feature selection, neural network pruning, and video summarization. Despite its prevalence, maximizing a $γ$-weakly submodular function subject to a general matroid constraint remains challenging. To date, the only known approximation guarantee is the conservative $(1+1/γ)^{-2}$ factor established by \citet{chen2018weakly}. To improve upon this result, this paper proposes a novel algorithm called \MGPE, which repeatedly performs maximum-gain local exchanges through careful control of a non-homogeneous Poisson clock, and proves that this \MGPE\ can attain an approximation ratio arbitrarily close to $ρ_γ=1-\left(γ/(2-γ)\right)^{ \frac{γ^2}{2(1-γ)} }$. In sharp contrast to the previous guarantee, our obtained factor $ρ_γ$ not only strictly improves upon $(1+1/γ)^{-2}$ for every $γ\in(0,1]$, but also can asymptotically approach the optimal $(1-1/e)$-approximation for submodular maximization as $γ\to1$. Furthermore, we surprisingly find that when the matroid constraint reduces to a cardinality or the objective satisfies the stronger notion of $α$-weak DR-submodularity, \MGPE\ can automatically recover the tight approximation ratios of $1-e^{-γ}$ and $1-e^{-α}$, respectively. Here, $α\in(0,1]$ denotes the DR ratio.

cs.DS↗
arXiv2026

Approximating Prize-Collecting TSP below 1.556

The prize-collecting traveling salesperson problem is a variant of the metric traveling salesperson problem in which vertices may be left unvisited by paying their associated penalties. The objective is to minimize the length of the tour plus the total penalty of the unvisited vertices. Blauth, Klein, and Nägele gave the previously best-known LP-relative $1.599$-approximation. We show that a simpler version of their algorithm, obtained by omitting the splitting-off preprocessing before the tree decomposition, has an LP-relative approximation ratio of $1.555761$. The improvement comes entirely from a new analysis of the parity-correction step: a simple analysis already gives $1.56$, and the stated factor follows from a numerical parameter search with exact verification.

cs.DS↗