arXiv subjects

Ararat Harutyunyan

Publications and source records attributed to Ararat Harutyunyan.

At least 19 recordsLinked to original sources

arXiv2026

Tournaments not inducible by five voters

A tournament T is k-inducible if there are k linear orders on its vertex set such that, for every arc $i \to j$ of T, a majority of the orders rank i above j. For odd k, let N(k) be the least order at which some tournament is not k-inducible. Only N(3) = 8 is known exactly; for N(5) the best bounds were $12 \le N(5) \le 38$, from our previous paper [2], which also gave the first explicit example of moderate order, the Paley tournament $P_{43}$. Results. A bespoke search algorithm improves both ends: $13 \le N(5) \le 23$. The upper bound comes from proving that $P_{23}$ is not 5-inducible, the case Bachmeier et al. [1] reported they could not decide, their SAT solver not having terminated within a cumulative six weeks; ours takes 22 hours on one laptop. The lower bound comes from an analysis at order 12. We also show that $P_{31}$ is not 5-inducible, while $P_{19}$ is 5-inducible but not with unit margin, that is, not by a profile in which every arc is carried by exactly three voters against two. Both $P_{19}$ and $P_{23}$ are arc-critical for their respective properties, whereas $P_{31}$ and $P_{43}$ are not vertex-critical: deleting a vertex leaves a tournament that is still not 5-inducible. Method. The search places one vertex at a time, always choosing the vertex with the fewest options left, and propagates the consequences. Together with the automorphisms of the tournament, this decides on a single laptop instances that neither integer programming nor a general-purpose SAT solver can settle. The refutations for $P_{19}$ and $P_{23}$ are certified as well: the search is split into independent subproblems, a SAT solver emits a machine-checkable proof for each, and a separate program rechecks every proof. All results, subject to two human-checked lemmas, are reproducible from https://github.com/Leonardini/TournamentsBeyond5Voters.

cs.DM↗

arXiv2026

Tournaments determined by three and five voters

The Kemeny median problem asks for a linear order minimizing the total pairwise disagreement with $m$ given rankings of $n$ options; it is NP-hard for every even $m \ge 4$ and every odd $m \ge 7$, while $m = 3$ and $m = 5$ remain open. Weighting each arc of the majority tournament by its margin reduces the problem to minimum-weight feedback arc set (FAS). The fewest voters inducing a tournament is its McGarvey number, and its predictability $α^{*}(T)$ is the largest supermajority threshold at which $T$ is inducible. We refute three conjectures on inducibility. (i) In any tournament, every minimum FAS is a minimal hitting set of the directed 3-cycles, strengthening a theorem of Milosz, Hamel and Pierrot; both of their conjectures fail: the 3-cycle extension for all odd $m \ge 5$, and the equality $\mathrm{FAS} = \mathrm{HS}_3$ at $n = 11$. (ii) The threshold conjecture proposed by Shepardson and Tovey fails for $m = 3$, exactly on the boundary (predictability $= 2/3$). (iii) For $m = 5$ it fails strictly: the Paley tournament on 43 vertices, with predictability $181/301 > 3/5$, is not the majority of any 5 voters, making it the first explicit tournament of modest size beyond the reach of five voters.

cs.DM↗

arXiv2026

On the list version of a conjecture of Erdős and Neumann-Lara

The dichromatic number of a digraph $D$, denoted by $\vecχ(D)$, is the smallest number of colours required to colour the vertices of $D$ such that each colour class induces an acyclic digraph. A conjecture of Erdős and Neumann-Lara states that there exists a function $f(k)$ such that for every graph $G$ with $χ(G) \geq f(k)$ there is an orientation of $G$ such that the resulting digraph $D$ satisfies $\vecχ(D) \geq k$. We prove the list version of this conjecture: if $G$ has large list chromatic number then there is an orientation of $G$ such that the resulting digraph has large list dichromatic number. The main tool in our result is the following theorem, which is an extension of an analogous result of Alon for the chromatic number: every graph of minimum degree $d$ admits an orientation such that the resulting digraph has list dichromatic number of order at least $\ln d$.

math.CO↗

arXiv2026

Acyclic sets and colorings in digraphs under restrictions on degrees and cycle lengths

Given a digraph $D$, we denote by $\vecα(D)$ the maximum size of an acyclic set of $D$ (i.e. a set of vertices which induces a subdigraph with no directed cycles), and by $\vecχ(D)$ the minimum number of acyclic sets into which $V(D)$ can be partitioned. In this paper, we study $\vecα(D)$ and $\vecχ(D)$ from various perspectives, including restrictions on degrees and cycle lengths. A main result is that, if $D$ is a random $r$-regular digon-free simple digraph of order $n$, then $\vecα(D) = Θ(n \log r /r)$ with high probability. This corresponds to a result of Spencer and Subramanian on the Erdős--Rényi random digraph model. Along the way, we derive some related results and propose some conjectures. An example of this is an analogue of the theorem of Bondy which bounds the chromatic number of a graph by the circumference of any strong orientation.

math.CO↗

arXiv2025

$(Δ-1)$-dicolouring of digraphs

In 1977, Borodin and Kostochka conjectured that every graph with maximum degree $Δ\geq 9$ is $(Δ-1)$-colourable, unless it contains a clique of size $Δ$. In 1999, Reed confirmed the conjecture when $Δ\geq 10^{14}$. We propose different generalisations of this conjecture for digraphs, and prove the analogue of Reed's result for each of them. The chromatic number and clique number are replaced respectively by the dichromatic number and the biclique number of digraphs. If $D$ is a digraph such that $\min(\tildeΔ(D),Δ^+(D)) = Δ\geq 9$, we conjecture that $D$ has dichromatic number at most $Δ-1$, unless either (i) $D$ contains a biclique of size $Δ$, or (ii) $D$ contains a biclique $K$ of size $Δ-2$, a directed $3$-cycle $\vec{C_3}$ disjoint from $K$, and all possible arcs in both directions between $\vec{C_3}$ and $K$. If true, this implies the conjecture of Borodin and Kostochka. We prove it when $Δ$ is large enough, thereby generalising the result of Reed. We finally give a sufficient condition for a digraph $D$ to have dichromatic number at most $Δ_{\min}(D)-1$, assuming that $Δ_{\min}(D)$ is large enough. In particular, this holds when the underlying graph of $D$ has no clique of size $Δ_{\min}(D)$, thus yielding a third independent generalisation of Reed's result. We further give a hardness result witnessing that our sufficient condition is best possible. To obtain these new upper bounds on the dichromatic number, we prove a dense decomposition lemma for digraphs having large maximum degree, which generalises to the directed setting the so-called dense decomposition of graphs due to Molloy and Reed. We believe this may be of independent interest, especially as a tool in various applications.

math.CO↗

arXiv2025

ACCORD: Autoregressive Constraint-satisfying Generation for COmbinatorial Optimization with Routing and Dynamic attention

Large Language Models (LLMs) have demonstrated impressive reasoning capabilities, yet their direct application to NP-hard combinatorial problems (CPs) remains underexplored. In this work, we systematically investigate the reasoning abilities of LLMs on a variety of NP-hard combinatorial optimization tasks and introduce ACCORD: Autoregressive Constraint-satisfying generation for COmbinatorial optimization with Routing and Dynamic attention. ACCORD features a novel dataset representation and model architecture that leverage the autoregressive nature of LLMs to dynamically enforce feasibility constraints, coupled with attention-based routing to activate problem-specific LoRA modules. We also present the ACCORD-90k supervised dataset, covering six NP-hard combinatorial problems: TSP, VRP, Knapsack, FlowShop, JSSP, and BinPacking. Extensive experiments demonstrate that our ACCORD model, built on an 8B-parameter Llama backbone, consistently outperforms standard prompting and input-output methods, even when compared to much larger LLMs, such as gpt-4. Ablation studies further show that our output structure enhances solution feasibility. To the best of our knowledge, this is the first large-scale, end-to-end framework for exploring the applications of LLMs to a broad spectrum of combinatorial optimization problems. The codes are publicly available at https://github.com/starjob42/ACCORD

cs.LG↗

arXiv2025

Colouring Complete Multipartite and Kneser-type Digraphs

The dichromatic number of a digraph $D$ is the smallest $k$ such that $D$ can be partitioned into $k$ acyclic subdigraphs, and the dichromatic number of an undirected graph is the maximum dichromatic number over all its orientations. Extending a well-known result of Lovász, we show that the dichromatic number of the Kneser graph $KG(n,k)$ is $Θ(n-2k+2)$ and that the dichromatic number of the Borsuk graph $BG(n+1,a)$ is $n+2$ if $a$ is large enough. We then study the list version of the dichromatic number. We show that, for any $\varepsilon>0$ and $2\leq k\leq n^{1/2-\varepsilon}$, the list dichromatic number of $KG(n,k)$ is $Θ(n\ln n)$. This extends a recent result of Bulankina and Kupavskii on the list chromatic number of $KG(n,k)$, where the same behaviour was observed. We also show that for any $ρ>3$, $r\geq 2$ and $m\geq\max\{\ln^ρr,2\}$, the list dichromatic number of the complete $r$-partite graph with $m$ vertices in each part is $Θ(r\ln m)$, extending a classical result of Alon. Finally, we give a directed analogue of Sabidussi's theorem on the chromatic number of graph products.

math.CO↗

arXiv2025

Starjob: Dataset for LLM-Driven Job Shop Scheduling

Large Language Models (LLMs) have shown remarkable capabilities across various domains, but their potential for solving combinatorial optimization problems remains largely unexplored. In this paper, we investigate the applicability of LLMs to the Job Shop Scheduling Problem (JSSP), a classic challenge in combinatorial optimization that requires efficient job allocation to machines to minimize makespan. To this end, we introduce Starjob, the first supervised dataset for JSSP, comprising 130k instances specifically designed for training LLMs. Leveraging this dataset, we fine-tune the LLaMA 8B 4-bit quantized model with the LoRA method to develop an end-to-end scheduling approach. Our evaluation on standard benchmarks demonstrates that the proposed LLM-based method not only surpasses traditional Priority Dispatching Rules (PDRs) but also achieves notable improvements over state-of-the-art neural approaches like L2D, with an average improvement of 15.36% on DMU and 7.85% on Taillard benchmarks. These results highlight the untapped potential of LLMs in tackling combinatorial optimization problems, paving the way for future advancements in this area.

cs.LG↗

arXiv2024

LLMs can Schedule

The job shop scheduling problem (JSSP) remains a significant hurdle in optimizing production processes. This challenge involves efficiently allocating jobs to a limited number of machines while minimizing factors like total processing time or job delays. While recent advancements in artificial intelligence have yielded promising solutions, such as reinforcement learning and graph neural networks, this paper explores the potential of Large Language Models (LLMs) for JSSP. We introduce the very first supervised 120k dataset specifically designed to train LLMs for JSSP. Surprisingly, our findings demonstrate that LLM-based scheduling can achieve performance comparable to other neural approaches. Furthermore, we propose a sampling method that enhances the effectiveness of LLMs in tackling JSSP.

cs.AI↗

arXiv2023

Odd Chromatic Number of Graph Classes

A graph is called odd (respectively, even) if every vertex has odd (respectively, even) degree. Gallai proved that every graph can be partitioned into two even induced subgraphs, or into an odd and an even induced subgraph. We refer to a partition into odd subgraphs as an odd colouring of G. Scott [Graphs and Combinatorics, 2001] proved that a graph admits an odd colouring if and only if it has an even number of vertices. We say that a graph G is k-odd colourable if it can be partitioned into at most k odd induced subgraphs. We initiate the systematic study of odd colouring and odd chromatic number of graph classes. In particular, we consider for a number of classes whether they have bounded odd chromatic number. Our main results are that interval graphs, graphs of bounded modular-width and graphs of bounded maximum degree all have bounded odd chromatic number.

cs.DM↗

arXiv2022

Digraph Coloring and Distance to Acyclicity

In $k$-Digraph Coloring we are given a digraph and are asked to partition its vertices into at most $k$ sets, so that each set induces a DAG. This well-known problem is NP-hard, as it generalizes (undirected) $k$-Coloring, but becomes trivial if the input digraph is acyclic. This poses the natural parameterized complexity question what happens when the input is "almost" acyclic. In this paper we study this question using parameters that measure the input's distance to acyclicity in either the directed or the undirected sense. It is already known that, for all $k\ge 2$, $k$-Digraph Coloring is NP-hard on digraphs of DFVS at most $k+4$. We strengthen this result to show that, for all $k\ge 2$, $k$-Digraph Coloring is NP-hard for DFVS $k$. Refining our reduction we obtain two further consequences: (i) for all $k\ge 2$, $k$-Digraph Coloring is NP-hard for graphs of feedback arc set (FAS) at most $k^2$; interestingly, this leads to a dichotomy, as we show that the problem is FPT by $k$ if FAS is at most $k^2-1$; (ii) $k$-Digraph Coloring is NP-hard for graphs of DFVS $k$, even if the maximum degree $Δ$ is at most $4k-1$; we show that this is also almost tight, as the problem becomes FPT for DFVS $k$ and $Δ\le 4k-3$. We then consider parameters that measure the distance from acyclicity of the underlying graph. We show that $k$-Digraph Coloring admits an FPT algorithm parameterized by treewidth, whose parameter dependence is $(tw!)k^{tw}$. Then, we pose the question of whether the $tw!$ factor can be eliminated. Our main contribution in this part is to settle this question in the negative and show that our algorithm is essentially optimal, even for the much more restricted parameter treedepth and for $k=2$. Specifically, we show that an FPT algorithm solving $2$-Digraph Coloring with dependence $td^{o(td)}$ would contradict the ETH.

cs.CC↗

arXiv2021

Filling Crosswords is Very Hard

We revisit a classical crossword filling puzzle which already appeared in Garey\&Jonhson's book. We are given a grid with $n$ vertical and horizontal slots and a dictionary with $m$ words and are asked to place words from the dictionary in the slots so that shared cells are consistent. We attempt to pinpoint the source of intractability of this problem by taking into account the structure of the grid graph, which contains a vertex for each slot and an edge if two slots intersect. Our main approach is to consider the case where this graph has a tree-like structure. Unfortunately, if we impose the common rule that words cannot be reused, we show that the problem remains NP-hard even under very severe structural restrictions. The problem becomes slightly more tractable if word reuse is allowed, as we obtain an $m^{tw}$ algorithm in this case, where $tw$ is the treewidth of the grid graph. However, even in this case, we show that our algorithm cannot be improved. More strongly, we show that under the ETH the problem cannot be solved in time $m^{o(k)}$, where $k$ is the number of horizontal slots of the instance. Motivated by these mostly negative results, we consider the much more restricted case where the problem is parameterized by the number of slots $n$. Here, we show that the problem becomes FPT, but the parameter dependence is exponential in $n^2$. We show that this dependence is also justified: the existence of an algorithm with running time $2^{o(n^2)}$ would contradict the randomized ETH. Finally, we consider an optimization version of the problem, where we seek to place as many words on the grid as possible. Here it is easy to obtain a $\frac{1}{2}$-approximation, even on weighted instances. We show that this algorithm is also likely to be optimal, as obtaining a better approximation ratio in polynomial time would contradict the Unique Games Conjecture.

cs.CC↗

arXiv2020

Disproving the normal graph conjecture

A graph $G$ is called normal if there exist two coverings, $\mathbb{C}$ and $\mathbb{S}$ of its vertex set such that every member of $\mathbb{C}$ induces a clique in $G$, every member of $\mathbb{S}$ induces an independent set in $G$ and $C \cap S \neq \emptyset$ for every $C \in \mathbb{C}$ and $S \in \mathbb{S}$. It has been conjectured by De Simone and Körner in 1999 that a graph $G$ is normal if $G$ does not contain $C_5$, $C_7$ and $\overline{C_7}$ as an induced subgraph. We disprove this conjecture.

math.CO↗

arXiv2019

Coloring dense digraphs

The chromatic number of a digraph $D$ is the minimum number of acyclic subgraphs covering the vertex set of $D$. A tournament $H$ is a hero if every $H$-free tournament $T$ has chromatic number bounded by a function of $H$. Inspired by the celebrated Erdős--Hajnal conjecture, Berger et al. fully characterized the class of heroes in 2013. We extend this framework to dense digraphs: A digraph $H$ is a superhero if every $H$-free digraph $D$ has chromatic number bounded by a function of $H$ and $α(D)$, the independence number of the underlying graph of $D$. We prove here that a digraph is a superhero if and only if it is a hero, and hence characterize all superheroes. This answers a question of Aboulker, Charbit and Naserasr.

math.CO↗

arXiv2019

Average-case complexity of a branch-and-bound algorithm for min dominating set

The average-case complexity of a branch-and-bound algorithms for Minimum Dominating Set problem in random graphs in the G(n,p) model is studied. We identify phase transitions between subexponential and exponential average-case complexities, depending on the growth of the probability p with respect to the number n of nodes.

cs.DS↗

arXiv2018

Maximum Independent Sets in Subcubic Graphs: New Results

The maximum independent set problem is known to be NP-hard in the class of subcubic graphs, i.e. graphs of vertex degree at most 3. We present a polynomial-time solution in a subclass of subcubic graphs generalizing several previously known results.

cs.DS↗

arXiv2018

Exact distance coloring in trees

For an integer $q\ge 2$ and an even integer $d$, consider the graph obtained from a large complete $q$-ary tree by connecting with an edge any two vertices at distance exactly $d$ in the tree. This graph has clique number $q+1$, and the purpose of this short note is to prove that its chromatic number is $Θ\big(\tfrac{d \log q}{\log d}\big)$. It was not known that the chromatic number of this graph grows with $d$. As a simple corollary of our result, we give a negative answer to a problem of van den Heuvel and Naserasr, asking whether there is a constant $C$ such that for any odd integer $d$, any planar graph can be colored with at most $C$ colors such that any pair of vertices at distance exactly $d$ have distinct colors. Finally, we study interval coloring of trees (where vertices at distance at least $d$ and at most $cd$, for some real $c>1$, must be assigned distinct colors), giving a sharp upper bound in the case of bounded degree trees.

math.CO↗

arXiv2018

Domination and fractional domination in digraphs

In this paper, we investigate the relation between the (fractional) domination number of a digraph $G$ and the independence number of its underlying graph, denoted by $α(G)$. More precisely, we prove that every digraph $G$ has fractional domination number at most $2α(G)$, and every directed triangle-free digraph $G$ has domination number at most $α(G)\cdot α(G)!$. The first bound is sharp.

math.CO↗