7,478 research outputs found

    Lower Bounds for the Graph Homomorphism Problem

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    The graph homomorphism problem (HOM) asks whether the vertices of a given nn-vertex graph GG can be mapped to the vertices of a given hh-vertex graph HH such that each edge of GG is mapped to an edge of HH. The problem generalizes the graph coloring problem and at the same time can be viewed as a special case of the 22-CSP problem. In this paper, we prove several lower bound for HOM under the Exponential Time Hypothesis (ETH) assumption. The main result is a lower bound 2Ω(nloghloglogh)2^{\Omega\left( \frac{n \log h}{\log \log h}\right)}. This rules out the existence of a single-exponential algorithm and shows that the trivial upper bound 2O(nlogh)2^{{\mathcal O}(n\log{h})} is almost asymptotically tight. We also investigate what properties of graphs GG and HH make it difficult to solve HOM(G,H)(G,H). An easy observation is that an O(hn){\mathcal O}(h^n) upper bound can be improved to O(hvc(G)){\mathcal O}(h^{\operatorname{vc}(G)}) where vc(G)\operatorname{vc}(G) is the minimum size of a vertex cover of GG. The second lower bound hΩ(vc(G))h^{\Omega(\operatorname{vc}(G))} shows that the upper bound is asymptotically tight. As to the properties of the "right-hand side" graph HH, it is known that HOM(G,H)(G,H) can be solved in time (f(Δ(H)))n(f(\Delta(H)))^n and (f(tw(H)))n(f(\operatorname{tw}(H)))^n where Δ(H)\Delta(H) is the maximum degree of HH and tw(H)\operatorname{tw}(H) is the treewidth of HH. This gives single-exponential algorithms for graphs of bounded maximum degree or bounded treewidth. Since the chromatic number χ(H)\chi(H) does not exceed tw(H)\operatorname{tw}(H) and Δ(H)+1\Delta(H)+1, it is natural to ask whether similar upper bounds with respect to χ(H)\chi(H) can be obtained. We provide a negative answer to this question by establishing a lower bound (f(χ(H)))n(f(\chi(H)))^n for any function ff. We also observe that similar lower bounds can be obtained for locally injective homomorphisms.Comment: 19 page

    On Index Coding and Graph Homomorphism

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    In this work, we study the problem of index coding from graph homomorphism perspective. We show that the minimum broadcast rate of an index coding problem for different variations of the problem such as non-linear, scalar, and vector index code, can be upper bounded by the minimum broadcast rate of another index coding problem when there exists a homomorphism from the complement of the side information graph of the first problem to that of the second problem. As a result, we show that several upper bounds on scalar and vector index code problem are special cases of one of our main theorems. For the linear scalar index coding problem, it has been shown in [1] that the binary linear index of a graph is equal to a graph theoretical parameter called minrank of the graph. For undirected graphs, in [2] it is shown that minrank(G)=k\mathrm{minrank}(G) = k if and only if there exists a homomorphism from Gˉ\bar{G} to a predefined graph Gˉk\bar{G}_k. Combining these two results, it follows that for undirected graphs, all the digraphs with linear index of at most k coincide with the graphs GG for which there exists a homomorphism from Gˉ\bar{G} to Gˉk\bar{G}_k. In this paper, we give a direct proof to this result that works for digraphs as well. We show how to use this classification result to generate lower bounds on scalar and vector index. In particular, we provide a lower bound for the scalar index of a digraph in terms of the chromatic number of its complement. Using our framework, we show that by changing the field size, linear index of a digraph can be at most increased by a factor that is independent from the number of the nodes.Comment: 5 pages, to appear in "IEEE Information Theory Workshop", 201

    Full Complexity Classification of the List Homomorphism Problem for Bounded-Treewidth Graphs

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    A homomorphism from a graph G to a graph H is an edge-preserving mapping from V(G) to V(H). Let H be a fixed graph with possible loops. In the list homomorphism problem, denoted by LHom(H), we are given a graph G, whose every vertex v is assigned with a list L(v) of vertices of H. We ask whether there exists a homomorphism h from G to H, which respects lists L, i.e., for every v ? V(G) it holds that h(v) ? L(v). The complexity dichotomy for LHom(H) was proven by Feder, Hell, and Huang [JGT 2003]. The authors showed that the problem is polynomial-time solvable if H belongs to the class called bi-arc graphs, and for all other graphs H it is NP-complete. We are interested in the complexity of the LHom(H) problem, parameterized by the treewidth of the input graph. This problem was investigated by Egri, Marx, and Rz??ewski [STACS 2018], who obtained tight complexity bounds for the special case of reflexive graphs H, i.e., if every vertex has a loop. In this paper we extend and generalize their results for all relevant graphs H, i.e., those, for which the LHom(H) problem is NP-hard. For every such H we find a constant k = k(H), such that the LHom(H) problem on instances G with n vertices and treewidth t - can be solved in time k^t ? n^?(1), provided that G is given along with a tree decomposition of width t, - cannot be solved in time (k-?)^t ? n^?(1), for any ? > 0, unless the SETH fails. For some graphs H the value of k(H) is much smaller than the trivial upper bound, i.e., |V(H)|. Obtaining matching upper and lower bounds shows that the set of algorithmic tools that we have discovered cannot be extended in order to obtain faster algorithms for LHom(H) in bounded-treewidth graphs. Furthermore, neither the algorithm, nor the proof of the lower bound, is very specific to treewidth. We believe that they can be used for other variants of the LHom(H) problem, e.g. with different parameterizations

    Sparsification Lower Bounds for List H-Coloring

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    We investigate the List H-Coloring problem, the generalization of graph coloring that asks whether an input graph G admits a homomorphism to the undirected graph H (possibly with loops), such that each vertex v ? V(G) is mapped to a vertex on its list L(v) ? V(H). An important result by Feder, Hell, and Huang [JGT 2003] states that List H-Coloring is polynomial-time solvable if H is a so-called bi-arc graph, and NP-complete otherwise. We investigate the NP-complete cases of the problem from the perspective of polynomial-time sparsification: can an n-vertex instance be efficiently reduced to an equivalent instance of bitsize ?(n^(2-?)) for some ? > 0? We prove that if H is not a bi-arc graph, then List H-Coloring does not admit such a sparsification algorithm unless NP ? coNP/poly. Our proofs combine techniques from kernelization lower bounds with a study of the structure of graphs H which are not bi-arc graphs

    Quantum Query Complexity of Subgraph Isomorphism and Homomorphism

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    Let HH be a fixed graph on nn vertices. Let fH(G)=1f_H(G) = 1 iff the input graph GG on nn vertices contains HH as a (not necessarily induced) subgraph. Let αH\alpha_H denote the cardinality of a maximum independent set of HH. In this paper we show: Q(fH)=Ω(αHn),Q(f_H) = \Omega\left(\sqrt{\alpha_H \cdot n}\right), where Q(fH)Q(f_H) denotes the quantum query complexity of fHf_H. As a consequence we obtain a lower bounds for Q(fH)Q(f_H) in terms of several other parameters of HH such as the average degree, minimum vertex cover, chromatic number, and the critical probability. We also use the above bound to show that Q(fH)=Ω(n3/4)Q(f_H) = \Omega(n^{3/4}) for any HH, improving on the previously best known bound of Ω(n2/3)\Omega(n^{2/3}). Until very recently, it was believed that the quantum query complexity is at least square root of the randomized one. Our Ω(n3/4)\Omega(n^{3/4}) bound for Q(fH)Q(f_H) matches the square root of the current best known bound for the randomized query complexity of fHf_H, which is Ω(n3/2)\Omega(n^{3/2}) due to Gr\"oger. Interestingly, the randomized bound of Ω(αHn)\Omega(\alpha_H \cdot n) for fHf_H still remains open. We also study the Subgraph Homomorphism Problem, denoted by f[H]f_{[H]}, and show that Q(f[H])=Ω(n)Q(f_{[H]}) = \Omega(n). Finally we extend our results to the 33-uniform hypergraphs. In particular, we show an Ω(n4/5)\Omega(n^{4/5}) bound for quantum query complexity of the Subgraph Isomorphism, improving on the previously known Ω(n3/4)\Omega(n^{3/4}) bound. For the Subgraph Homomorphism, we obtain an Ω(n3/2)\Omega(n^{3/2}) bound for the same.Comment: 16 pages, 2 figure
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