arXiv subjects

Morteza Alimi

Publications and source records attributed to Morteza Alimi.

9 recordsLinked to original sources

arXiv2026

NP-Hardness and a PTAS for Maximum Interval Multicover, with Applications to Dense Subgraphs

Given intervals and points on a line, each point with a demand, and a budget $K$, Maximum Interval Multicover asks for at most $K$ intervals that satisfy as many points as possible, a point being satisfied if it lies in at least as many chosen intervals as its demand. One application is committee elections in which each candidate is approved by consecutive voters and voters have individual demands. Xu, Ran and Zhang introduced the problem and gave a $1/2$-approximation; the objective is not submodular when demands exceed one. Li, Tu, Zhang and Cao gave a polynomial-time approximation scheme (PTAS) for the minimization counterpart, unweighted partial interval multicover, and asked for its complexity. We prove that both problems are NP-hard, even for intervals of equal length; as their optima are integers bounded by the input size, neither has a fully polynomial-time approximation scheme unless $\mathrm{P}=\mathrm{NP}$. We then give a PTAS for Maximum Interval Multicover, also for arbitrary nonnegative nondecreasing rewards for the coverage of each point. Building on a decomposition from Nonner's PTAS for Densest $k$-Subgraph on interval graphs, our scheme cuts the line into pieces at a small loss and solves every piece exactly. The key dynamic program handles intervals crossing a common point by choosing an interleaving of their left and right endpoints rather than sweeping along the line; it needs only two counters to track these intervals. The same algorithm yields PTASes on convex bipartite graphs for Bipartite Dense Subgraph, improving on the ratio $2/3$ of Backer and Keil for the subclass of bipartite permutation graphs, and on interval graphs for Maximum $k$-Vertex Cover and weighted clique counts.

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arXiv2026

A $59/33$ Cut-LP Guarantee for Matching Augmentation

The Matching Augmentation Problem (MAP) asks for a minimum-cardinality set of unit-cost edges that, together with a zero-cost matching, forms a 2-edge-connected spanning multigraph. We study the standard cut relaxation. Bamas, Drygala, and Svensson proposed a particularly simple LP-guided algorithm: compute an extreme optimum, run a depth-first search that prioritizes large LP coordinates, and augment the resulting DFS tree optimally. We give a new structural analysis of the Bamas--Drygala--Svensson LP-guided DFS algorithm. The analysis combines an exact primal--dual identity for the residual uplink problem with a rank bound that measures fractional support relative to the unit-valued skeleton. The result is that for every root and every deterministic tie-breaking order consistent with the LP priorities, the algorithm returns a solution of cost at most $\frac{59}{33}c(x^*)-\frac{25}{33}=\left(2-\frac7{33}\right)c(x^*)-\frac{25}{33}\approx1.788c(x^*)-0.758$, where $x^*$ is an optimum of the cut LP. Consequently, the integrality gap of the relaxation is at most $59/33\approx1.788$. No new algorithmic step is required; the improvement is analytical. The exact packing certificate for the residual uplink problem yields a cost identity with a packing-slack term, while a rank theorem bounds fractional support relative to the unit-valued skeleton. A regional classification accounts for the non-tree edges, and a two-cut identity handles self-holes. As a direct corollary, the same $59/33\approx1.788$ bound holds for Forest Augmentation in the minimum-value regime. The proof is self-contained apart from one theorem on the dimension of minimum-cut vectors.

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arXiv2026

Remote Matching: Exact-Cardinality Approximation and Tight UGC Hardness

In the unrestricted max--min metric $T$-join problem, one seeks an even terminal set $T$ maximizing the cost of a minimum $T$-join. Iwata and Ravi gave a factor-$3/2$ approximation for this problem. We show that this guarantee is tight under the Unique Games Conjecture: no polynomial-time approximation with factor strictly smaller than $3/2$ exists under UGC. We then consider the exact-cardinality variant, which prescribes an even number $k$ of terminals. Writing $p:=k/n$, we give a deterministic polynomial-time $ρ(p)$-approximation for every feasible cardinality, where \[ ρ(p)= \begin{cases} 7/2, & \makebox[1.5em][r]{$0$}<p\le2/7,\\ 1/p, & 2/7\le p\le2/3,\\ 1/[2(1-p)], & 2/3\le p\le7/8,\\ 4, & 7/8\le p<1. \end{cases} \] In particular, a factor-$4$ approximation holds throughout the entire feasible cardinality range, the factor is at most $7/2$ whenever $0<k\le 6n/7$, and equals $3/2$ at $k=2n/3$. The algorithmic framework is based on optimal laminar cut packings, their weighted tree representations, exact-cardinality rounding, and tree dynamic programming.

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arXiv2026

An 8/5 Rounding for Half-Integral Forest-BCR via Root Supports and Circuit Rank

We study the rounding of a supplied half-integral feasible solution of the root-assignment bidirected cut relaxation for Steiner Forest (Forest-BCR). Byrka, Grandoni, and Traub [IPCO 2025] proved a $16/9$ guarantee for a recursive framework that normalizes the LP point, selects a vertex set of maximum projected LP density, buys a minimum spanning tree on that set, contracts it, and recurses. We prove that the same framework has guarantee $8/5$. The new analysis keeps the orientation and the root label of each projected half-unit of LP mass. In a simple projection, the cut constraints at a terminal of degree two determine the root-assignment vector of every demand incident with it, and half-integrality leaves only two possibilities: a unit assignment to one root, which forces excess outdegree inside that root's support, or a split assignment to two roots, which forces overlap between their supports. For every connected component $C$ of the split-root graph this yields $ β_C\ \ge\ \frac{L_C}{2}$, where $β_C$ is the circuit rank of the union of the root supports in $C$ and $L_C$ is the number of its vertices of degree two in the full projection. Balancing the density certificate obtained from this inequality against the ordinary degree sum gives a vertex set of density at least $5/8$, and the inherited contraction lemma turns that into the $8/5$ rounding. For every $q\ge3$ we also construct a normalized half-integral point whose maximum projected density is exactly $5q/[2(4q-1)]$, so the universal projected-density bound is asymptotically tight.

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arXiv2026

An Entropy Potential for Type-Composition Games

Potential functions are a key tool in theoretical computer science with applications ranging from the runtime analysis of algorithms and data structures, through the analysis of the expected behavior of random processes and search heuristics, to proving the existence of equilibrium states in strategic games. Typically, proofs that employ potential functions are short, elegant, and easy to verify, yet very powerful. Moreover, potential functions are essential ingredients for constructive proofs, in particular in algorithmic game theory. There, a key question is the existence of equilibrium states, but the most powerful theorem in the field -- Nash's theorem -- is unfortunately non-constructive. For many strategic games, potential functions come to the rescue by enabling constructive proofs that sometimes even yield efficient algorithms for finding equilibria. We add to this by providing a novel class of entropy-inspired log-multinomial potential functions for natural game-theoretic settings where rational agents of different types strategically choose actions to maximize their utility. In particular, we consider utility functions that are based on the fraction of same- and other-type agents taking the same action. We demonstrate the versatility of the new potential function class by presenting simple equilibrium existence proofs for two recent game-theoretic models, for which only involved technical proofs were previously known. Even better, the new potential function class yields efficient algorithms for constructing equilibria for much more general models. Thereby, we positively resolve several open problems.

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arXiv2025

Efficient Dynamic Rank Aggregation

The rank aggregation problem, which has many real-world applications, refers to the process of combining multiple input rankings into a single aggregated ranking. In dynamic settings, where new rankings arrive over time, efficiently updating the aggregated ranking is essential. This paper develops a fast, theoretically and practically efficient dynamic rank aggregation algorithm. First, we develop the LR-Aggregation algorithm, built on top of the LR-tree data structure, which is itself modeled on the LR-distance, a novel and equivalent take on the classical Spearman's footrule distance. We then analyze the theoretical efficiency of the Pick-A-Perm algorithm, and show how it can be combined with the LR-aggregation algorithm using another data structure that we develop. We demonstrate through experimental evaluations that LR-Aggregation produces close to optimal solutions in practice. We show that Pick-A-Perm has a theoretical worst case approximation guarantee of 2. We also show that both the LR-Aggregation and Pick-A-Perm algorithms, as well as the methodology for combining them can be run in $O(n \log n)$ time. To the best of our knowledge, this is the first fast, near linear time rank aggregation algorithm in the dynamic setting, having both a theoretical approximation guarantee, and excellent practical performance (much better than the theoretical guarantee).

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arXiv2025

Approximating Graphic Multi-Path TSP and Graphic Ordered TSP

The path version of the Traveling Salesman Problem is one of the most well-studied variants of the ubiquitous TSP. Its generalization, the Multi-Path TSP, has recently been used in the best known algorithm for path TSP by Traub and Vygen [Cambridge University Press, 2024]. The best known approximation factor for this problem is $2.214$ by Böhm, Friggstad, Mömke and Spoerhase [SODA 2025]. In this paper we show that for the case of graphic metrics, a significantly better approximation guarantee of $2$ can be attained. Our algorithm is based on sampling paths from a decomposition of the flow corresponding to the optimal solution to the LP for the problem, and connecting the left-out vertices with doubled edges. The cost of the latter is twice the optimum in the worst case; we show how the cost of the sampled paths can be absorbed into it without increasing the approximation factor. Furthermore, we prove that any below-$2$ approximation algorithm for the special case of the problem where each source is the same as the corresponding sink yields a below-$2$ approximation algorithm for Graphic Multi-Path TSP. We also show that our ideas can be utilized to give a factor $1.791$-approximation algorithm for Ordered TSP in graphic metrics, for which the aforementioned paper [SODA 2025] and Armbruster, Mnich and Nägele [APPROX 2024] give a $1.868$-approximation algorithm in general metrics.

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arXiv2024

Approximating Prize-Collecting Variants of TSP

We present an approximation algorithm for the Prize-collecting Ordered Traveling Salesman Problem (PCOTSP), which simultaneously generalizes the Prize-collecting TSP and the Ordered TSP. The Prize-collecting TSP is well-studied and has a long history, with the current best approximation factor slightly below $1.6$, shown by Blauth, Klein and Nägele [IPCO 2024]. The best approximation ratio for Ordered TSP is $\frac{3}{2}+\frac{1}{e}$, presented by Böhm, Friggstad, Mömke, Spoerhase [SODA 2025] and Armbruster, Mnich, Nägele [Approx 2024]. The former also present a factor 2.2131 approximation algorithm for Multi-Path-TSP. By carefully tuning the techniques of the latest results on the aforementioned problems and leveraging the unique properties of our problem, we present a 2.097-approximation algorithm for PCOTSP. A key idea in our result is to first sample a set of trees, and then probabilistically pick up some vertices, while using the pruning ideas of Blauth, Klein, Nägele [IPCO 2024] on other vertices to get cheaper parity correction; the sampling probability and the penalty paid by the LP playing a crucial part in both cases. A straightforward adaptation of the aforementioned pruning ideas would only give minuscule improvements over standard parity correction methods. Instead, we use the specific characteristics of our problem together with properties gained from running a simple combinatorial algorithm to bring the approximation factor below 2.1. Our techniques extend to Prize-collecting Multi-Path TSP, building on results from Böhm, Friggstad, Mömke, Spoerhase [SODA 2025], leading to a 2.41-approximation.

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arXiv2021

Mean Isoperimetry with Control on Outliers: Exact and Approximation Algorithms

Given a weighted graph $G=(V,E)$ with weight functions $c:E\to \mathbb{R}_+$ and $π:V\to \mathbb{R}_+$, and a subset $U\subseteq V$, the normalized cut value for $U$ is defined as the sum of the weights of edges exiting $U$ divided by the weight of vertices in $U$. The {\it mean isoperimetry problem}, $\mathsf{ISO}^1(G,k)$, for a weighted graph $G$ is a generalization of the classical uniform sparsest cut problem in which, given a parameter $k$, the objective is to find $k$ disjoint nonempty subsets of $V$ minimizing the average normalized cut value of the parts. The robust version of the problem seeks an optimizer where the number of vertices that fall out of the subpartition is bounded by some given integer $0 \leq ρ\leq |V|$. Our main result states that $\mathsf{ISO}^1(G,k)$, as well as its robust version, $\mathsf{CRISO}^1(G,k,ρ)$, subjected to the condition that each part of the subpartition induces a connected subgraph, are solvable in time $O(k^2 ρ^2\ π(V(T)^3)$ on any weighted tree $T$, in which $π(V(T))$ is the sum of the vertex-weights. This result implies that $\mathsf{ISO}^1(G,k)$ is strongly polynomial-time solvable on weighted trees when the vertex-weights are polynomially bounded and may be compared to the fact that the problem is NP-Hard for weighted trees in general. Also, using this, we show that both mentioned problems, $\mathsf{ISO}^1(G,k)$ and $\mathsf{CRISO}^1(G,k,ρ)$ as well as the ordinary robust mean isoperimetry problem $\mathsf{RISO}^1(G,k,ρ)$, admit polynomial-time $O(\log^{1.5}|V| \log\log |V|)$-approximation algorithms for weighted graphs with polynomially bounded weights, using the R{ä}cke-Shah tree cut sparsifier.

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