7,605 research outputs found

    Optimal Vertex Fault Tolerant Spanners (for fixed stretch)

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    A kk-spanner of a graph GG is a sparse subgraph HH whose shortest path distances match those of GG up to a multiplicative error kk. In this paper we study spanners that are resistant to faults. A subgraph HGH \subseteq G is an ff vertex fault tolerant (VFT) kk-spanner if HFH \setminus F is a kk-spanner of GFG \setminus F for any small set FF of ff vertices that might "fail." One of the main questions in the area is: what is the minimum size of an ff fault tolerant kk-spanner that holds for all nn node graphs (as a function of ff, kk and nn)? This question was first studied in the context of geometric graphs [Levcopoulos et al. STOC '98, Czumaj and Zhao SoCG '03] and has more recently been considered in general undirected graphs [Chechik et al. STOC '09, Dinitz and Krauthgamer PODC '11]. In this paper, we settle the question of the optimal size of a VFT spanner, in the setting where the stretch factor kk is fixed. Specifically, we prove that every (undirected, possibly weighted) nn-node graph GG has a (2k1)(2k-1)-spanner resilient to ff vertex faults with Ok(f11/kn1+1/k)O_k(f^{1 - 1/k} n^{1 + 1/k}) edges, and this is fully optimal (unless the famous Erdos Girth Conjecture is false). Our lower bound even generalizes to imply that no data structure capable of approximating distGF(s,t)dist_{G \setminus F}(s, t) similarly can beat the space usage of our spanner in the worst case. We also consider the edge fault tolerant (EFT) model, defined analogously with edge failures rather than vertex failures. We show that the same spanner upper bound applies in this setting. Our data structure lower bound extends to the case k=2k=2 (and hence we close the EFT problem for 33-approximations), but it falls to Ω(f1/21/(2k)n1+1/k)\Omega(f^{1/2 - 1/(2k)} \cdot n^{1 + 1/k}) for k3k \ge 3. We leave it as an open problem to close this gap.Comment: To appear in SODA 201

    Subgraph complementation and minimum rank

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    Any finite simple graph G=(V,E)G = (V,E) can be represented by a collection C\mathscr{C} of subsets of VV such that uvEuv\in E if and only if uu and vv appear together in an odd number of sets in C\mathscr{C}. Let c2(G)c_2(G) denote the minimum cardinality of such a collection. This invariant is equivalent to the minimum dimension of a faithful orthogonal representation of GG over F2\mathbb{F}_2 and is closely connected to the minimum rank of GG. We show that c2(G)=mr(G,F2)c_2(G) = \operatorname{mr}(G,\mathbb{F}_2) when mr(G,F2)\operatorname{mr}(G,\mathbb{F}_2) is odd, or when GG is a forest. Otherwise, mr(G,F2)c2(G)mr(G,F2)+1\operatorname{mr}(G,\mathbb{F}_2)\leq c_2(G)\leq \operatorname{mr}(G,\mathbb{F}_2)+1. Furthermore, we show that the following are equivalent for any graph GG with at least one edge: i. c2(G)=mr(G,F2)+1c_2(G)=\operatorname{mr}(G,\mathbb{F}_2)+1; ii. the adjacency matrix of GG is the unique matrix of rank mr(G,F2)\operatorname{mr}(G,\mathbb{F}_2) which fits GG over F2\mathbb{F}_2; iii. there is a minimum collection C\mathscr{C} as described in which every vertex appears an even number of times; and iv. for every component GG' of GG, c2(G)=mr(G,F2)+1c_2(G') = \operatorname{mr}(G',\mathbb{F}_2) + 1. We also show that, for these graphs, mr(G,F2)\operatorname{mr}(G,\mathbb{F}_2) is twice the minimum number of tricliques whose symmetric difference of edge sets is EE. Additionally, we provide a set of upper bounds on c2(G)c_2(G) in terms of the order, size, and vertex cover number of GG. Finally, we show that the class of graphs with c2(G)kc_2(G)\leq k is hereditary and finitely defined. For odd kk, the sets of minimal forbidden induced subgraphs are the same as those for the property mr(G,F2)k\operatorname{mr}(G,\mathbb{F}_2)\leq k, and we exhibit this set for c2(G)2c_2(G)\leq2

    How unproportional must a graph be?

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    Let uk(G,p)u_k(G,p) be the maximum over all kk-vertex graphs FF of by how much the number of induced copies of FF in GG differs from its expectation in the binomial random graph with the same number of vertices as GG and with edge probability pp. This may be viewed as a measure of how close GG is to being pp-quasirandom. For a positive integer nn and 0<p<10<p<1, let D(n,p)D(n,p) be the distance from p(n2)p\binom{n}{2} to the nearest integer. Our main result is that, for fixed k4k\ge 4 and for nn large, the minimum of uk(G,p)u_k(G,p) over nn-vertex graphs has order of magnitude Θ(max{D(n,p),p(1p)}nk2)\Theta\big(\max\{D(n,p), p(1-p)\} n^{k-2}\big) provided that p(1p)n1/2p(1-p)n^{1/2} \to \infty

    On Minrank and Forbidden Subgraphs

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    The minrank over a field F\mathbb{F} of a graph GG on the vertex set {1,2,,n}\{1,2,\ldots,n\} is the minimum possible rank of a matrix MFn×nM \in \mathbb{F}^{n \times n} such that Mi,i0M_{i,i} \neq 0 for every ii, and Mi,j=0M_{i,j}=0 for every distinct non-adjacent vertices ii and jj in GG. For an integer nn, a graph HH, and a field F\mathbb{F}, let g(n,H,F)g(n,H,\mathbb{F}) denote the maximum possible minrank over F\mathbb{F} of an nn-vertex graph whose complement contains no copy of HH. In this paper we study this quantity for various graphs HH and fields F\mathbb{F}. For finite fields, we prove by a probabilistic argument a general lower bound on g(n,H,F)g(n,H,\mathbb{F}), which yields a nearly tight bound of Ω(n/logn)\Omega(\sqrt{n}/\log n) for the triangle H=K3H=K_3. For the real field, we prove by an explicit construction that for every non-bipartite graph HH, g(n,H,R)nδg(n,H,\mathbb{R}) \geq n^\delta for some δ=δ(H)>0\delta = \delta(H)>0. As a by-product of this construction, we disprove a conjecture of Codenotti, Pudl\'ak, and Resta. The results are motivated by questions in information theory, circuit complexity, and geometry.Comment: 15 page

    Finding Large H-Colorable Subgraphs in Hereditary Graph Classes

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    We study the \textsc{Max Partial HH-Coloring} problem: given a graph GG, find the largest induced subgraph of GG that admits a homomorphism into HH, where HH is a fixed pattern graph without loops. Note that when HH is a complete graph on kk vertices, the problem reduces to finding the largest induced kk-colorable subgraph, which for k=2k=2 is equivalent (by complementation) to \textsc{Odd Cycle Transversal}. We prove that for every fixed pattern graph HH without loops, \textsc{Max Partial HH-Coloring} can be solved: \bullet in {P5,F}\{P_5,F\}-free graphs in polynomial time, whenever FF is a threshold graph; \bullet in {P5,bull}\{P_5,\textrm{bull}\}-free graphs in polynomial time; \bullet in P5P_5-free graphs in time nO(ω(G))n^{\mathcal{O}(\omega(G))}; \bullet in {P6,1-subdivided claw}\{P_6,\textrm{1-subdivided claw}\}-free graphs in time nO(ω(G)3)n^{\mathcal{O}(\omega(G)^3)}. Here, nn is the number of vertices of the input graph GG and ω(G)\omega(G) is the maximum size of a clique in~GG. Furthermore, combining the mentioned algorithms for P5P_5-free and for {P6,1-subdivided claw}\{P_6,\textrm{1-subdivided claw}\}-free graphs with a simple branching procedure, we obtain subexponential-time algorithms for \textsc{Max Partial HH-Coloring} in these classes of graphs. Finally, we show that even a restricted variant of \textsc{Max Partial HH-Coloring} is NP\mathsf{NP}-hard in the considered subclasses of P5P_5-free graphs, if we allow loops on HH

    On the decomposition threshold of a given graph

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    We study the FF-decomposition threshold δF\delta_F for a given graph FF. Here an FF-decomposition of a graph GG is a collection of edge-disjoint copies of FF in GG which together cover every edge of GG. (Such an FF-decomposition can only exist if GG is FF-divisible, i.e. if e(F)e(G)e(F)\mid e(G) and each vertex degree of GG can be expressed as a linear combination of the vertex degrees of FF.) The FF-decomposition threshold δF\delta_F is the smallest value ensuring that an FF-divisible graph GG on nn vertices with δ(G)(δF+o(1))n\delta(G)\ge(\delta_F+o(1))n has an FF-decomposition. Our main results imply the following for a given graph FF, where δF\delta_F^\ast is the fractional version of δF\delta_F and χ:=χ(F)\chi:=\chi(F): (i) δFmax{δF,11/(χ+1)}\delta_F\le \max\{\delta_F^\ast,1-1/(\chi+1)\}; (ii) if χ5\chi\ge 5, then δF{δF,11/χ,11/(χ+1)}\delta_F\in\{\delta_F^{\ast},1-1/\chi,1-1/(\chi+1)\}; (iii) we determine δF\delta_F if FF is bipartite. In particular, (i) implies that δKr=δKr\delta_{K_r}=\delta^\ast_{K_r}. Our proof involves further developments of the recent `iterative' absorbing approach.Comment: Final version, to appear in the Journal of Combinatorial Theory, Series
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