arXiv subjects

Joe Cheriyan

Publications and source records attributed to Joe Cheriyan.

8 recordsLinked to original sources

arXiv2026

A $5$-Approximation Analysis for the Cover Small Cuts Problem

In the Cover Small Cuts problem, we are given a capacitated (undirected) graph $G=(V,E,u)$ and a threshold value $\lambda$, as well as a set of links $L$ with end-nodes in $V$ and a non-negative cost for each link $\ell\in L$; the goal is to find a minimum-cost set of links such that each non-trivial cut of capacity less than $\lambda$ is covered by a link. Bansal, Cheriyan, Grout, and Ibrahimpur (arXiv:2209.11209, Algorithmica 2024) showed that the WGMV primal-dual algorithm, due to Williamson, Goemans, Mihail, and Vazirani (Combinatorica, 1995), achieves approximation ratio $16$ for the Cover Small Cuts problem; their analysis uses the notion of a pliable family of sets that satisfies a combinatorial property. Later, Bansal (arXiv:2308.15714v2, IPCO 2025) and then Nutov (arXiv:2504.03910, MFCS 2025) proved that the same algorithm achieves approximation ratio $6$. We show that the same algorithm achieves approximation ratio $5$, by using a stronger notion, namely, a pliable family of sets that satisfies symmetry and structural submodularity.

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arXiv2025

Symmetric Submodular Functions, Uncrossable Functions, and Structural Submodularity

Diestel, et al. (see Order 35 (2017), JCT-A 167 (2019), arXiv:1805.01439) introduced the notion of abstract separation systems that satisfy a submodularity property, and they call this structural submodularity. Williamson, Goemans, Mihail, and Vazirani (Combinatorica 15 (1995)) call a family of sets $\mathcal{F}$ uncrossable if the following holds: for any pair of sets $A,B\in\mathcal{F}$, both $A\cap{B},A\cup{B}$ are in $\mathcal{F}$, or both $A-B,B-A$ are in $\mathcal{F}$. Bansal, Cheriyan, Grout, and Ibrahimpur (Algorithmica 86 (2024), arXiv:2209.11209) call a family of sets $\mathcal{F}$ pliable if the following holds: for any pair of sets $A,B\in\mathcal{F}$, at least two of the sets $A\cap{B},A\cup{B},A-B,B-A$ are in $\mathcal{F}$. We say that a pliable family of sets $\mathcal{F}$ satisfies structural submodularity if the following holds: for any pair of crossing sets $A,B\in\mathcal{F}$, at least one of the sets $A\cap{B},A\cup{B}$ is in $\mathcal{F}$, and at least one of the sets $A-B,B-A$ is in $\mathcal{F}$. For any positive integer $d\geq2$, we construct a pliable family of sets $\mathcal{F}$ that satisfies structural submodularity such that (a) there do not exist a symmetric submodular function $g$ and $\lambda\in{\mathbb Q}$ such that $\mathcal{F} = \{ S \,:\, g(S)<\lambda \}$, and (b) $\mathcal{F}$ cannot be partitioned into $d$ (or fewer) uncrossable families.

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arXiv2025

A Bad Example for Jain's Iterative Rounding Theorem for the Cover Small Cuts Problem

Jain's iterative rounding theorem is a well-known result in the area of approximation algorithms and, more broadly, in combinatorial optimization. The theorem asserts that LP relaxations of several problems in network design and combinatorial optimization have the following key property: for every basic solution $x$ there exists a variable $x_e$ that has value at least a constant (e.g., $x_e\geq\frac12$). We construct an example showing that this property fails to hold for the Cover Small Cuts problem. In this problem, we are given an undirected, capacitated graph $G=(V,E),u$ and a threshold value $\lambda$, as well as a set of links $L$ with end-nodes in $V$ and a non-negative cost for each link $\ell\in L$; the goal is to find a minimum-cost set of links such that each non-trivial cut of capacity less than $\lambda$ is covered by a link. This indicates that the polyhedron of feasible solutions to the LP relaxation (of Cover Small Cuts) differs in an essential way from the polyhedrons associated with several problems in combinatorial optimization. Moreover, our example shows that a direct application of Jain's iterative rounding algorithm does not give an $O(1)$ approximation algorithm for Cover Small Cuts. We mention that Bansal et al. (Algorithmica 2024) present an $O(1)$ approximation algorithm for Cover Small Cuts based on the primal-dual method of Williamson et al. (Combinatorica 1995).

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arXiv2022

Algorithms for 2-connected network design and flexible Steiner trees with a constant number of terminals

The $k$-Steiner-2NCS problem is as follows: Given a constant $k$, and an undirected connected graph $G = (V,E)$, non-negative costs $c$ on $E$, and a partition $(T, V-T)$ of $V$ into a set of terminals, $T$, and a set of non-terminals (or, Steiner nodes), where $|T|=k$, find a minimum-cost two-node connected subgraph that contains the terminals. We present a randomized polynomial-time algorithm for the unweighted problem, and a randomized PTAS for the weighted problem. We obtain similar results for the $k$-Steiner-2ECS problem, where the input is the same, and the algorithmic goal is to find a minimum-cost two-edge connected subgraph that contains the terminals. Our methods build on results by Björklund, Husfeldt, and Taslaman (ACM-SIAM SODA 2012) that give a randomized polynomial-time algorithm for the unweighted $k$-Steiner-cycle problem; this problem has the same inputs as the unweighted $k$-Steiner-2NCS problem, and the algorithmic goal is to find a minimum-size simple cycle $C$ that contains the terminals ($C$ may contain any number of Steiner nodes).

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arXiv2018

The Matching Augmentation Problem: A $\frac74$-Approximation Algorithm

We present a $\frac74$ approximation algorithm for the matching augmentation problem (MAP): given a multi-graph with edges of cost either zero or one such that the edges of cost zero form a matching, find a 2-edge connected spanning subgraph (2-ECSS) of minimum cost. We first present a reduction of any given MAP instance to a collection of well-structured MAP instances such that the approximation guarantee is preserved. Then we present a $\frac74$ approximation algorithm for a well-structured MAP instance. The algorithm starts with a min-cost 2-edge cover and then applies ear-augmentation steps. We analyze the cost of the ear-augmentations using an approach similar to the one proposed by Vempala and Vetta for the (unweighted) min-size 2-ECSS problem (`Factor 4/3 approximations for minimum 2-connected subgraphs,' APPROX 2000, LNCS 1913, pp.262-273).

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arXiv2016

Approximating (Unweighted) Tree Augmentation via Lift-and-Project (Part 0: $1.8+ε$ approximation for (Unweighted) TAP)

We study the unweighted Tree Augmentation Problem (TAP) via the Lasserre (Sum of Squares) system. We prove an approximation guarantee of ($1.8+ε$) relative to an SDP relaxation, which matches the combinatorial approximation guarantee of Even, Feldman, Kortsarz and Nutov in ACM TALG (2009), where $ε>0$ is a constant. We generalize the combinatorial analysis of integral solutions of Even, et al., to fractional solutions by identifying some properties of fractional solutions of the Lasserre system via the decomposition result of Rothvoß (arXiv:1111.5473, 2011) and Karlin, Mathieu and Nguyen (IPCO 2011).

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arXiv2015

Approximating (Unweighted) Tree Augmentation via Lift-and-Project, Part II

In Part II, we study the unweighted Tree Augmentation Problem (TAP) via the Lasserre (Sum~of~Squares) system. We prove that the integrality ratio of an SDP relaxation (the Lasserre tightening of an LP relaxation) is $\leq \frac{3}{2}+ε$, where $ε>0$ can be any small constant. We obtain this result by designing a polynomial-time algorithm for TAP that achieves an approximation guarantee of ($\frac32+ε$) relative to the SDP relaxation. The algorithm is combinatorial and does not solve the SDP relaxation, but our analysis relies on the SDP relaxation. We generalize the combinatorial analysis of integral solutions from the previous literature to fractional solutions by identifying some properties of fractional solutions of the Lasserre system via the decomposition result of Karlin, Mathieu and Nguyen (IPCO 2011).

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arXiv2015

Approximating (Unweighted) Tree Augmentation via Lift-and-Project, Part I: Stemless TAP

In Part I, we study a special case of the unweighted Tree Augmentation Problem (TAP) via the Lasserre (Sum of Squares) system. In the special case, we forbid so-called stems; these are a particular type of subtree configuration. For stemless TAP, we prove that the integrality ratio of an SDP relaxation (the Lasserre tightening of an LP relaxation) is $\leq \frac{3}{2}+ε$, where $ε>0$ can be any small constant. We obtain this result by designing a polynomial-time algorithm for stemless TAP that achieves an approximation guarantee of ($\frac32+ε$) relative to the SDP relaxation. The algorithm is combinatorial and does not solve the SDP relaxation, but our analysis relies on the SDP relaxation. We generalize the combinatorial analysis of integral solutions from the previous literature to fractional solutions by identifying some properties of fractional solutions of the Lasserre system via the decomposition result of Karlin, Mathieu and Nguyen (IPCO 2011). Also, we present an example of stemless TAP such that the approximation guarantee of $\frac32$ is tight for the algorithm. In Part II of this paper, we extend the methods of Part I to prove the same results relative to the same SDP relaxation for TAP.

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