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Ali Çivril

Publications and source records attributed to Ali Çivril.

14 recordsLinked to original sources
arXiv2026

Improved Inapproximability Results for Steiner Tree via Long Code Based Reductions

The best algorithm for approximating Steiner tree has performance ratio $\ln(4)+ε\approx 1.386$ [J. Byrka et al., \textit{Proceedings of the 42th Annual ACM Symposium on Theory of Computing (STOC)}, 2010, pp. 583-592], whereas the inapproximability result stays at the factor $\frac{96}{95} \approx 1.0105$ [M. Chlebík and J. Chlebíková, \textit{Proceedings of the 8th Scandinavian Workshop on Algorithm Theory (SWAT)}, 2002, pp. 170-179]. In this article, we take a step forward to bridge this gap and show that there is no polynomial time algorithm approximating Steiner tree with constant ratio better than $\frac{19}{18} \approx 1.0555$ unless \textsf{P = NP}. We also relate the problem to the Unique Games Conjecture by showing that it is \textsf{UG}-hard to find a constant approximation ratio better than $\frac{17}{16} = 1.0625$. In the special case of quasi-bipartite graphs, we prove an inapproximability factor of $\frac{25}{24} \approx 1.0416$ unless \textsf{P = NP}, which improves upon the previous bound of $\frac{128}{127} \approx 1.0078$. The reductions that we present for all the cases are of the same spirit with appropriate modifications. Our main technical contribution is an adaptation of a Set-Cover type reduction in which the Long Code is used to the geometric setting of the problems we consider.

cs.CC↗
arXiv2026

Sparse Approximation by Semidefinite Programming

The problem of sparse approximation and the closely related compressed sensing have received tremendous attention in the past decade. Primarily studied from the viewpoint of applied harmonic analysis and signal processing, there have been two dominant algorithmic approaches to this problem: Greedy methods called the matching pursuit (MP) and the linear programming based approaches called the basis pursuit (BP). The aim of the current paper is to bring a fresh perspective to sparse approximation by treating it as a combinatorial optimization problem and providing an algorithm based on the powerful optimization technique semidefinite programming (SDP). In particular, we show that there is a randomized algorithm based on a semidefinite relaxation of the problem with performance guarantees depending on the coherence and the restricted isometry constant of the dictionary used. We then show a derandomization of the algorithm based on the method of conditional probabilities.

cs.IT↗
arXiv2026

An Improved Integrality Gap for Steiner Tree

A promising approach for obtaining improved approximation algorithms for Steiner tree is to use the bidirected cut relaxation (BCR). The integrality gap of this relaxation is at least $36/31$, and it has long been conjectured that its true value is very close to this lower bound. However, the best upper bound for general graphs was an almost trivial $2$. We improve this bound to $3/2$ by a combinatorial algorithm based on the primal-dual schema.

cs.DS↗
arXiv2026

A New Approximation Algorithm for the Minimum 2-Edge-Connected Spanning Subgraph Problem

We present a new approximation algorithm for the minimum 2-edge-connected spanning subgraph problem. Its approximation ratio is $\frac{4}{3}$, which matches the current best ratio. The approximation ratio of the algorithm is $\frac{6}{5}$ on subcubic graphs, which is an improvement upon the previous best ratio of $\frac{5}{4}$. The algorithm is a novel extension of the primal-dual schema, which consists of two distinct phases. Both the algorithm and the analysis are much simpler than those of the previous approaches.

cs.DS↗
arXiv2026

Scheme-theoretic Approach to Computational Complexity I. The Separation of P and NP

We lay the foundations of a new theory for algorithms and computational complexity by parameterizing the instances of a computational problem as a moduli scheme. Considering the geometry of the scheme associated to 3-SAT, we separate P and NP. In particular, we show that no deterministic algorithm can solve \textsf{3-SAT} in time less than $1.296839^n$ in the worst case.

cs.CC↗
arXiv2026

4/3-Approximation of Graphic TSP

We describe a $\frac{4}{3}$-approximation algorithm for the traveling salesman problem in which the distances between points are induced by graph-theoretical distances in an unweighted graph. The algorithm is based on finding a minimum cost perfect matching on the odd degree vertices of a carefully computed 2-edge-connected spanning subgraph.

cs.DS↗
arXiv2026

Scheme-Theoretic Approach to Computational Complexity. III. SETH

We show that there exist infinitely many $n \in \mathbb{Z}^+$ such that for any constant $ε> 0$, any deterministic algorithm to solve $k$-\textsf{SAT} for $k \geq 3$ must perform at least $(2^{k-\frac{3}{2}-ε})^{\frac{n}{k+1}}$ operations, where $n$ is the number of variables in the $k$\textsf{-SAT} instance.

cs.CC↗
arXiv2026

Scheme-Theoretic Approach to Computational Complexity. IV. A New Perspective on Hardness of Approximation

We provide a new approach for establishing hardness of approximation results, based on the theory recently introduced by the author. It allows one to directly show that approximating a problem beyond a certain threshold requires super-polynomial time. To exhibit the framework, we revisit two famous problems in this paper. The particular results we prove are: MAX-3-SAT$(1,\frac{7}{8}+ε)$ requires exponential time for any constant $ε$ satisfying $\frac{1}{8} \geq ε> 0$. In particular, the gap exponential time hypothesis (Gap-ETH) holds. MAX-3-LIN-2$(1-ε, \frac{1}{2}+ε)$ requires exponential time for any constant $ε$ satisfying $\frac{1}{4} \geq ε> 0$.

cs.CC↗
arXiv2026

9/7-Approximation for Two-Edge-Connectivity and Two-Vertex-Connectivity

We provide algorithms for the minimum 2-edge-connected spanning subgraph problem and the minimum 2-vertex-connected spanning subgraph problem with approximation ratio $\frac{9}{7}$. This improves upon a recent algorithm with ratio slightly smaller than $\frac{4}{3}$ for 2-edge-connectivity, and another one with ratio $\frac{4}{3}$ for 2-vertex-connectivity.

cs.DS↗
arXiv2026

3/2-Approximation for the Forest Augmentation Problem

We describe a $\frac{3}{2}$-approximation algorithm for the Forest Augmentation Problem (\textsf{FAP}), which is a special case of the Weighted 2-Edge-Connected Spanning Subgraph Problem (\textsf{Weighted 2-ECSS}). This significantly improves upon the previous best ratio $1.9973$, and proceeds toward the goal of a $\frac{3}{2}$-approximation algorithm for \textsf{Weighted 2-ECSS}.

cs.DS↗
arXiv2019

Approximation of Steiner Forest via the Bidirected Cut Relaxation

The classical algorithm of Agrawal, Klein and Ravi [SIAM J. Comput., 24 (1995), pp. 440-456], stated in the setting of the primal-dual schema by Goemans and Williamson [SIAM J. Comput., 24 (1995), pp. 296-317] uses the undirected cut relaxation for the Steiner forest problem. Its approximation ratio is $2-\frac{1}{k}$, where $k$ is the number of terminal pairs. A variant of this algorithm more recently proposed by Könemann et al. [SIAM J. Comput., 37 (2008), pp. 1319-1341] is based on the lifted cut relaxation. In this paper, we continue this line of work and consider the bidirected cut relaxation for the Steiner forest problem, which lends itself to a novel algorithmic idea yielding the same approximation ratio as the classical algorithm. In doing so, we introduce an extension of the primal-dual schema in which we run two different phases to satisfy connectivity requirements in both directions. This reveals more about the combinatorial structure of the problem. In particular, there are examples on which the classical algorithm fails to give a good approximation, but the new algorithm finds a near-optimal solution.

cs.DS↗
arXiv2017

Sparse Approximation is Provably Hard under Coherent Dictionaries

It is well known that sparse approximation problem is \textsf{NP}-hard under general dictionaries. Several algorithms have been devised and analyzed in the past decade under various assumptions on the \emph{coherence} $μ$ of the dictionary represented by an $M \times N$ matrix from which a subset of $k$ column vectors is selected. All these results assume $μ=O(k^{-1})$. This article is an attempt to bridge the big gap between the negative result of \textsf{NP}-hardness under general dictionaries and the positive results under this restrictive assumption. In particular, it suggests that the aforementioned assumption might be asymptotically the best one can make to arrive at any efficient algorithmic result under well-known conjectures of complexity theory. In establishing the results, we make use of a new simple multilayered PCP which is tailored to give a matrix with small coherence combined with our reduction.

cs.CC↗