arXiv subjects

Ishay Haviv

Publications and source records attributed to Ishay Haviv.

At least 19 recordsLinked to original sources

arXiv2026

Setwise Distinguishable Permutations

A family of permutations of $[n]$ is called setwise distinguishable if for every permutation in the family there exists a subset of $[n]$ whose image under this permutation differs from its image under any other permutation in the family. We prove that there exists a setwise distinguishable family of $2^{(2-o(1)) \cdot n}$ permutations of $[n]$. The result is optimal up to the $o(1)$ term in the exponent and is achieved through an explicit construction. As an application, we obtain nearly tight conditional lower bounds on the kernelization complexity of graph coloring problems parameterized by the vertex-deletion distance to split graphs. This improves a result of Jansen and Kratsch (Inf. Comput., 2013).

cs.DM↗

arXiv2026

Fair Allocation under Conflict Constraints via Strong Colorability

In the fair allocation problem under conflict constraints, the goal is to partition the vertices of a graph among agents in a fair manner, such that no two adjacent vertices are assigned to the same agent. We study this problem for agents with common preferences through the lens of three fairness criteria: stochastic-dominance envy-freeness up to one item for preference orders (SD-EF1), envy-freeness up to one item for monotone additive valuations (EF1), and envy-freeness up to one item from each side for general additive valuations (EF[1,1]). To do so, we introduce a hierarchy of variants of the strong chromatic number, a graph quantity introduced independently by Alon and Fellows in the early nineties. Our results reveal a close connection between fair allocation under conflict constraints and the first two levels of this hierarchy, providing a unified route to both existential and algorithmic results. For SD-EF1, we fully characterize the number of agents needed to guarantee a fair allocation of a given graph for every common preference order. For EF1 and EF[1,1], we provide analogous sufficient conditions, extending a result on path graphs due to Equbal, Gurjar, Igarashi, Kumar, Manurangsi, Nath, Saxena, Vaish, and Yoneda. We also show that, unlike in the SD-EF1 setting, the sufficient conditions for EF1 and EF[1,1] are not necessary in general. Our framework yields existential and algorithmic consequences in terms of the maximum degree. We obtain that every graph with maximum degree $Δ$ admits SD-EF1, EF1, and EF[1,1] allocations for common preferences whenever the number of agents is at least $3Δ-1$. We further provide, for any $\varepsilon>0$, deterministic polynomial-time algorithms that find such allocations whenever the number of agents is at least $(3+\varepsilon)Δ$. These guarantees strengthen earlier work by Barman and Viswanathan on equitable colorings.

cs.GT↗

arXiv2026

Kernelization Bounds for Constrained Coloring

We study the kernel complexity of constraint satisfaction problems over a finite domain, parameterized by the number of variables, whose constraint language consists of two relations: the non-equality relation and an additional permutation-invariant relation $R$. We establish a conditional lower bound on the kernel size in terms of the largest arity of an OR relation definable from $R$. Building on this, we investigate the kernel complexity of uniformly rainbow free coloring problems. In these problems, for fixed positive integers $d$, $\ell$, and $q \geq d$, we are given a graph $G$ on $n$ vertices and a collection $\cal F$ of $\ell$-tuples of $d$-subsets of its vertex set, and the goal is to decide whether there exists a proper coloring of $G$ with $q$ colors such that no $\ell$-tuple in $\cal F$ is uniformly rainbow, that is, no tuple has all its sets colored with the same $d$ distinct colors. We determine, for all admissible values of $d$, $\ell$, and $q$, the infimum over all values $η$ for which the problem admits a kernel of size $O(n^η)$, under the assumption $\mathsf{NP} \nsubseteq \mathsf{coNP/poly}$. As applications, we obtain nearly tight bounds on the kernel complexity of various coloring problems under diverse settings and parameterizations. This includes graph coloring problems parameterized by the vertex-deletion distance to a disjoint union of cliques, resolving a question of Schalken (2020), as well as uniform hypergraph coloring problems parameterized by the number of vertices, extending results of Jansen and Pieterse (2019) and Beukers (2021).

cs.CC↗

arXiv2025

Kernelization for $H$-Coloring

For a fixed graph $H$, the $H$-Coloring problem asks whether a given graph admits an edge-preserving function from its vertex set to that of $H$. A seminal theorem of Hell and Nešetřil asserts that the $H$-Coloring problem is NP-hard whenever $H$ is loopless and non-bipartite. A result of Jansen and Pieterse implies that for every graph $H$, the $H$-Coloring problem parameterized by the vertex cover number $k$ admits a kernel with $O(k^{Δ(H)})$ vertices and bit-size bounded by $O(k^{Δ(H)} \cdot \log k)$, where $Δ(H)$ denotes the maximum degree in $H$. For the case where $H$ is a complete graph on at least three vertices, this kernel size nearly matches conditional lower bounds established by Jansen and Kratsch and by Jansen and Pieterse. This paper presents new upper and lower bounds on the kernel size of $H$-Coloring problems parameterized by the vertex cover number. The upper bounds arise from two kernelization algorithms. The first is purely combinatorial, and its size is governed by a structural quantity of the graph $H$, called the non-adjacency witness number. As applications, we obtain kernels whose size is bounded by a fixed polynomial for natural classes of graphs $H$ with unbounded maximum degree. More strikingly, we show that for almost every graph $H$, the degree of the polynomial that bounds the size of our combinatorial kernel grows only logarithmically in $Δ(H)$. Our second kernel leverages linear-algebraic tools and involves the notion of faithful independent representations of graphs. It strengthens the general bound from prior work and, among other applications, yields near-optimal kernels for problems concerning the dimension of orthogonal graph representations over finite fields. We complement these results with conditional lower bounds, thereby nearly settling the kernel complexity of the problem for various target graphs $H$.

cs.DS↗

arXiv2025

New Hardness Results for Low-Rank Matrix Completion

The low-rank matrix completion problem asks whether a given real matrix with missing values can be completed so that the resulting matrix has low rank or is close to a low-rank matrix. The completed matrix is often required to satisfy additional structural constraints, such as positive semi-definiteness or a bounded infinity norm. The problem arises in various research fields, including machine learning, statistics, and theoretical computer science, and has broad real-world applications. This paper presents new $\mathsf{NP} $-hardness results for low-rank matrix completion problems. We show that for every sufficiently large integer $d$ and any real number $\varepsilon \in [ 2^{-O(d)},\frac{1}{7}]$, given a partial matrix $A$ with exposed values of magnitude at most $1$ that admits a positive semi-definite completion of rank $d$, it is $\mathsf{NP}$-hard to find a positive semi-definite matrix that agrees with each given value of $A$ up to an additive error of at most $\varepsilon$, even when the rank is allowed to exceed $d$ by a multiplicative factor of $O (\frac{1}{\varepsilon ^2 \cdot \log(1/\varepsilon)} )$. This strengthens a result of Hardt, Meka, Raghavendra, and Weitz (COLT, 2014), which applies to multiplicative factors smaller than $2$ and to $\varepsilon $ that decays polynomially in $d$. We establish similar $\mathsf{NP}$-hardness results for the case where the completed matrix is constrained to have a bounded infinity norm (rather than be positive semi-definite), for which all previous hardness results rely on complexity assumptions related to the Unique Games Conjecture. Our proofs involve a novel notion of nearly orthonormal representations of graphs, the concept of line digraphs, and bounds on the rank of perturbed identity matrices.

cs.CC↗

arXiv2025

Kernels for Storage Capacity and Dual Index Coding

The storage capacity of a graph measures the maximum amount of information that can be stored across its vertices, such that the information at any vertex can be recovered from the information stored at its neighborhood. The study of this graph quantity is motivated by applications in distributed storage and by its intimate relations to the index coding problem from the area of network information theory. In the latter, one wishes to minimize the amount of information that has to be transmitted to a collection of receivers, in a way that enables each of them to discover its required data using some prior side information. In this paper, we initiate the study of the Storage Capacity and Index Coding problems from the perspective of parameterized complexity. We prove that the Storage Capacity problem parameterized by the solution size admits a kernelization algorithm producing kernels of linear size. We also provide such a result for the Index Coding problem, in the linear and non-linear settings, where it is parameterized by the dual value of the solution, i.e., the length of the transmission that can be saved using the side information. A key ingredient in the proofs is the crown decomposition technique due to Chor, Fellows, and Juedes (WG 2003, WG 2004). As an application, we significantly extend an algorithmic result of Dau, Skachek, and Chee (IEEE Trans. Inform. Theory, 2014).

cs.DS↗

arXiv2025

A Near-Optimal Kernel for a Coloring Problem

For a fixed integer $q$, the $q$-Coloring problem asks to decide if a given graph has a vertex coloring with $q$ colors such that no two adjacent vertices receive the same color. In a series of papers, it has been shown that for every $q \geq 3$, the $q$-Coloring problem parameterized by the vertex cover number $k$ admits a kernel of bit-size $\widetilde{O}(k^{q-1})$, but admits no kernel of bit-size $O(k^{q-1-\varepsilon})$ for $\varepsilon >0$ unless $\mathsf{NP} \subseteq \mathsf{coNP/poly}$ (Jansen and Kratsch, 2013; Jansen and Pieterse, 2019). In 2020, Schalken proposed the question of the kernelizability of the $q$-Coloring problem parameterized by the number $k$ of vertices whose removal results in a disjoint union of edges and isolated vertices. He proved that for every $q \geq 3$, the problem admits a kernel of bit-size $\widetilde{O}(k^{2q-2})$, but admits no kernel of bit-size $O(k^{2q-3-\varepsilon})$ for $\varepsilon >0$ unless $\mathsf{NP} \subseteq \mathsf{coNP/poly}$. He further proved that for $q \in \{3,4\}$ the problem admits a near-optimal kernel of bit-size $\widetilde{O}(k^{2q-3})$ and asked whether such a kernel is achievable for all integers $q \geq 3$. In this short paper, we settle this question in the affirmative.

cs.DS↗

arXiv2024

Larger Nearly Orthogonal Sets over Finite Fields

For a field $\mathbb{F}$ and integers $d$ and $k$, a set ${\cal A} \subseteq \mathbb{F}^d$ is called $k$-nearly orthogonal if its members are non-self-orthogonal and every $k+1$ vectors of ${\cal A}$ include an orthogonal pair. We prove that for every prime $p$ there exists some $δ= δ(p)>0$, such that for every field $\mathbb{F}$ of characteristic $p$ and for all integers $k \geq 2$ and $d \geq k$, there exists a $k$-nearly orthogonal set of at least $d^{δ\cdot k/\log k}$ vectors of $\mathbb{F}^d$. The size of the set is optimal up to the $\log k$ term in the exponent. We further prove two extensions of this result. In the first, we provide a large set ${\cal A}$ of non-self-orthogonal vectors of $\mathbb{F}^d$ such that for every two subsets of ${\cal A}$ of size $k+1$ each, some vector of one of the subsets is orthogonal to some vector of the other. In the second extension, every $k+1$ vectors of the produced set ${\cal A}$ include $\ell+1$ pairwise orthogonal vectors for an arbitrary fixed integer $1 \leq \ell \leq k$. The proofs involve probabilistic and spectral arguments and the hypergraph container method.

math.CO↗

arXiv2024

A Fixed-Parameter Algorithm for the Kneser Problem

The Kneser graph $K(n,k)$ is defined for integers $n$ and $k$ with $n \geq 2k$ as the graph whose vertices are all the $k$-subsets of $\{1,2,\ldots,n\}$ where two such sets are adjacent if they are disjoint. A classical result of Lovász asserts that the chromatic number of $K(n,k)$ is $n-2k+2$. In the computational Kneser problem, we are given an oracle access to a coloring of the vertices of $K(n,k)$ with $n-2k+1$ colors, and the goal is to find a monochromatic edge. We present a randomized algorithm for the Kneser problem with running time $n^{O(1)} \cdot k^{O(k)}$. This shows that the problem is fixed-parameter tractable with respect to the parameter $k$. The analysis involves structural results on intersecting families and on induced subgraphs of Kneser graphs. We also study the Agreeable-Set problem of assigning a small subset of a set of $m$ items to a group of $\ell$ agents, so that all agents value the subset at least as much as its complement. As an application of our algorithm for the Kneser problem, we obtain a randomized polynomial-time algorithm for the Agreeable-Set problem for instances that satisfy $\ell \geq m - O(\frac{\log m}{\log \log m})$. We further show that the Agreeable-Set problem is at least as hard as a variant of the Kneser problem with an extended access to the input coloring.

cs.DS↗

arXiv2024

The Chromatic Number of Kneser Hypergraphs via Consensus Division

We show that the Consensus Division theorem implies lower bounds on the chromatic number of Kneser hypergraphs, offering a novel proof for a result of Alon, Frankl, and Lovász (Trans. Amer. Math. Soc., 1986) and for its generalization by Kř\'ıž (Trans. Amer. Math. Soc., 1992). Our approach is applied to study the computational complexity of the total search problem Kneser$^p$, which given a succinct representation of a coloring of a $p$-uniform Kneser hypergraph with fewer colors than its chromatic number, asks to find a monochromatic hyperedge. We prove that for every prime $p$, the Kneser$^p$ problem with an extended access to the input coloring is efficiently reducible to a quite weak approximation of the Consensus Division problem with $p$ shares. In particular, for $p=2$, the problem is efficiently reducible to any non-trivial approximation of the Consensus Halving problem on normalized monotone functions. We further show that for every prime $p$, the Kneser$^p$ problem lies in the complexity class $\mathsf{PPA}$-$p$. As an application, we establish limitations on the complexity of the Kneser$^p$ problem, restricted to colorings with a bounded number of colors.

cs.CC↗

arXiv2024

Kernelization for Orthogonality Dimension

The orthogonality dimension of a graph over $\mathbb{R}$ is the smallest integer $d$ for which one can assign to every vertex a nonzero vector in $\mathbb{R}^d$ such that every two adjacent vertices receive orthogonal vectors. For an integer $d$, the $d$-Ortho-Dim$_\mathbb{R}$ problem asks to decide whether the orthogonality dimension of a given graph over $\mathbb{R}$ is at most $d$. We prove that for every integer $d \geq 3$, the $d$-Ortho-Dim$_\mathbb{R}$ problem parameterized by the vertex cover number $k$ admits a kernel with $O(k^{d-1})$ vertices and bit-size $O(k^{d-1} \cdot \log k)$. We complement this result by a nearly matching lower bound, showing that for any $\varepsilon > 0$, the problem admits no kernel of bit-size $O(k^{d-1-\varepsilon})$ unless $\mathsf{NP} \subseteq \mathsf{coNP/poly}$. We further study the kernelizability of orthogonality dimension problems in additional settings, including over general fields and under various structural parameterizations.

cs.DS↗

arXiv2024

Improved Approximation Algorithms for Index Coding

The index coding problem is concerned with broadcasting encoded information to a collection of receivers in a way that enables each receiver to discover its required data based on its side information, which comprises the data required by some of the others. Given the side information map, represented by a graph in the symmetric case and by a digraph otherwise, the goal is to devise a coding scheme of minimum broadcast length. We present a general method for developing efficient algorithms for approximating the index coding rate for prescribed families of instances. As applications, we obtain polynomial-time algorithms that approximate the index coding rate of graphs and digraphs on $n$ vertices to within factors of $O(n/\log^2 n)$ and $O(n/\log n)$ respectively. This improves on the approximation factors of $O(n/\log n)$ for graphs and $O(n \cdot \log \log n/\log n)$ for digraphs achieved by Blasiak, Kleinberg, and Lubetzky (IEEE Trans. Inform. Theory, 2013). For the family of quasi-line graphs, we exhibit a polynomial-time algorithm that approximates the index coding rate to within a factor of $2$. This improves on the approximation factor of $O(n^{2/3})$ achieved by Arbabjolfaei and Kim (ISIT, 2016) for graphs on $n$ vertices taken from certain sub-families of quasi-line graphs. Our approach is applicable for approximating a variety of additional graph and digraph quantities to within the same approximation factors. Specifically, it captures every graph quantity sandwiched between the independence number and the clique cover number and every digraph quantity sandwiched between the maximum size of an acyclic induced sub-digraph and the directed clique cover number.

cs.DS↗

arXiv2024

Testing Intersectingness of Uniform Families

A set family ${\cal F}$ is called intersecting if every two members of ${\cal F}$ intersect, and it is called uniform if all members of ${\cal F}$ share a common size. A uniform family ${\cal F} \subseteq \binom{[n]}{k}$ of $k$-subsets of $[n]$ is $\varepsilon$-far from intersecting if one has to remove more than $\varepsilon \cdot \binom{n}{k}$ of the sets of ${\cal F}$ to make it intersecting. We study the property testing problem that given query access to a uniform family ${\cal F} \subseteq \binom{[n]}{k}$, asks to distinguish between the case that ${\cal F}$ is intersecting and the case that it is $\varepsilon$-far from intersecting. We prove that for every fixed integer $r$, the problem admits a non-adaptive two-sided error tester with query complexity $O(\frac{\ln n}{\varepsilon})$ for $\varepsilon \geq Ω( (\frac{k}{n})^r)$ and a non-adaptive one-sided error tester with query complexity $O(\frac{\ln k}{\varepsilon})$ for $\varepsilon \geq Ω( (\frac{k^2}{n})^r)$. The query complexities are optimal up to the logarithmic terms. For $\varepsilon \geq Ω( (\frac{k^2}{n})^2)$, we further provide a non-adaptive one-sided error tester with optimal query complexity of $O(\frac{1}{\varepsilon})$. Our findings show that the query complexity of the problem behaves differently from that of testing intersectingness of non-uniform families, studied recently by Chen, De, Li, Nadimpalli, and Servedio (ITCS, 2024).

cs.DS↗

arXiv2024

Nearly Orthogonal Sets over Finite Fields

For a field $\mathbb{F}$ and integers $d$ and $k$, a set of vectors of $\mathbb{F}^d$ is called $k$-nearly orthogonal if its members are non-self-orthogonal and every $k+1$ of them include an orthogonal pair. We prove that for every prime $p$ there exists a positive constant $δ= δ(p)$, such that for every field $\mathbb{F}$ of characteristic $p$ and for all integers $k \geq 2$ and $d \geq k^{1/(p-1)}$, there exists a $k$-nearly orthogonal set of at least $d^{δ\cdot k^{1/(p-1)}/ \log k}$ vectors of $\mathbb{F}^d$. In particular, for the binary field we obtain a set of $d^{Ω( k /\log k)}$ vectors, and this is tight up to the $\log k$ term in the exponent. For comparison, the best known lower bound over the reals is $d^{Ω( \log k / \log \log k)}$ (Alon and Szegedy, Graphs and Combin., 1999). The proof combines probabilistic and spectral arguments.

cs.CG↗

arXiv2024

Fixed-Parameter Algorithms for the Kneser and Schrijver Problems

The Kneser graph $K(n,k)$ is defined for integers $n$ and $k$ with $n \geq 2k$ as the graph whose vertices are all the $k$-subsets of $[n]=\{1,2,\ldots,n\}$ where two such sets are adjacent if they are disjoint. The Schrijver graph $S(n,k)$ is defined as the subgraph of $K(n,k)$ induced by the collection of all $k$-subsets of $[n]$ that do not include two consecutive elements modulo $n$. It is known that the chromatic number of both $K(n,k)$ and $S(n,k)$ is $n-2k+2$. In the computational Kneser and Schrijver problems, we are given an access to a coloring with $n-2k+1$ colors of the vertices of $K(n,k)$ and $S(n,k)$ respectively, and the goal is to find a monochromatic edge. We prove that the problems admit randomized algorithms with running time $n^{O(1)} \cdot k^{O(k)}$, hence they are fixed-parameter tractable with respect to the parameter $k$. The analysis involves structural results on intersecting families and on induced subgraphs of Kneser and Schrijver graphs. We also study the Agreeable-Set problem of assigning a small subset of a set of $m$ items to a group of $\ell$ agents, so that all agents value the subset at least as much as its complement. As an application of our algorithm for the Kneser problem, we obtain a randomized polynomial-time algorithm for the Agreeable-Set problem for instances with $\ell \geq m - O(\frac{\log m}{\log \log m})$. We further show that the Agreeable-Set problem is at least as hard as a variant of the Kneser problem with an extended access to the input coloring.

cs.DS↗

arXiv2023

Improved NP-Hardness of Approximation for Orthogonality Dimension and Minrank

The orthogonality dimension of a graph $G$ over $\mathbb{R}$ is the smallest integer $k$ for which one can assign a nonzero $k$-dimensional real vector to each vertex of $G$, such that every two adjacent vertices receive orthogonal vectors. We prove that for every sufficiently large integer $k$, it is $\mathsf{NP}$-hard to decide whether the orthogonality dimension of a given graph over $\mathbb{R}$ is at most $k$ or at least $2^{(1-o(1)) \cdot k/2}$. We further prove such hardness results for the orthogonality dimension over finite fields as well as for the closely related minrank parameter, which is motivated by the index coding problem in information theory. This in particular implies that it is $\mathsf{NP}$-hard to approximate these graph quantities to within any constant factor. Previously, the hardness of approximation was known to hold either assuming certain variants of the Unique Games Conjecture or for approximation factors smaller than $3/2$. The proofs involve the concept of line digraphs and bounds on their orthogonality dimension and on the minrank of their complement.

cs.CC↗

arXiv2023

On Finding Constrained Independent Sets in Cycles

A subset of $[n] = \{1,2,\ldots,n\}$ is called stable if it forms an independent set in the cycle on the vertex set $[n]$. In 1978, Schrijver proved via a topological argument that for all integers $n$ and $k$ with $n \geq 2k$, the family of stable $k$-subsets of $[n]$ cannot be covered by $n-2k+1$ intersecting families. We study two total search problems whose totality relies on this result. In the first problem, denoted by $\mathsf{Schrijve}r(n,k,m)$, we are given an access to a coloring of the stable $k$-subsets of $[n]$ with $m = m(n,k)$ colors, where $m \leq n-2k+1$, and the goal is to find a pair of disjoint subsets that are assigned the same color. While for $m = n-2k+1$ the problem is known to be $\mathsf{PPA}$-complete, we prove that for $m < d \cdot \lfloor \frac{n}{2k+d-2} \rfloor$, with $d$ being any fixed constant, the problem admits an efficient algorithm. For $m = \lfloor n/2 \rfloor-2k+1$, we prove that the problem is efficiently reducible to the $\mathsf{Kneser}$ problem. Motivated by the relation between the problems, we investigate the family of unstable $k$-subsets of $[n]$, which might be of independent interest. In the second problem, called Unfair Independent Set in Cycle, we are given $\ell$ subsets $V_1, \ldots, V_\ell$ of $[n]$, where $\ell \leq n-2k+1$ and $|V_i| \geq 2$ for all $i \in [\ell]$, and the goal is to find a stable $k$-subset $S$ of $[n]$ satisfying the constraints $|S \cap V_i| \leq |V_i|/2$ for $i \in [\ell]$. We prove that the problem is $\mathsf{PPA}$-complete and that its restriction to instances with $n=3k$ is at least as hard as the Cycle plus Triangles problem, for which no efficient algorithm is known. On the contrary, we prove that there exists a constant $c$ for which the restriction of the problem to instances with $n \geq c \cdot k$ can be solved in polynomial time.

cs.DS↗

arXiv2023

Local Orthogonality Dimension

An orthogonal representation of a graph $G$ over a field $\mathbb{F}$ is an assignment of a vector $u_v \in \mathbb{F}^t$ to every vertex $v$ of $G$, such that $\langle u_v,u_v \rangle \neq 0$ for every vertex $v$ and $\langle u_v,u_{v'} \rangle = 0$ whenever $v$ and $v'$ are adjacent in $G$. The locality of the orthogonal representation is the largest dimension of a subspace spanned by the vectors associated with a closed neighborhood in the graph. We introduce a novel graph parameter, called the local orthogonality dimension, defined for a given graph $G$ and a given field $\mathbb{F}$, as the smallest possible locality of an orthogonal representation of $G$ over $\mathbb{F}$. We investigate the usefulness of topological methods for proving lower bounds on the local orthogonality dimension. We prove that graphs for which topological methods imply a lower bound of $t$ on their chromatic number have local orthogonality dimension at least $\lceil t/2 \rceil +1$ over every field, strengthening a result of Simonyi and Tardos on the local chromatic number. We show that for certain graphs this lower bound is tight, whereas for others, the local orthogonality dimension over the reals is equal to the chromatic number. More generally, we prove that for every complement of a line graph, the local orthogonality dimension over $\mathbb{R}$ coincides with the chromatic number. This strengthens a recent result by Daneshpajouh, Meunier, and Mizrahi, who proved that the local and standard chromatic numbers of these graphs are equal. As another extension of their result, we prove that the local and standard chromatic numbers are equal for some additional graphs, from the family of Kneser graphs. We also show an $\mathsf{NP}$-hardness result for the local orthogonality dimension and present an application of this graph parameter to the index coding problem from information theory.

math.CO↗