We investigate the problem @math of counting all induced subgraphs of size @math in a graph @math that satisfy a given property @math . This continues the work of Jerrum and Meeks who proved the problem to be @math -hard for some families of properties which include, among others, (dis)connectedness [JCSS 15] and even- or oddness of the number of edges [Combinatorica 17]. Using the recent framework of graph motif parameters due to Curticapean, Dell and Marx [STOC 17], we discover that for monotone properties @math , the problem @math is hard for @math if the reduced Euler characteristic of the associated simplicial (graph) complex of @math is non-zero. This observation links @math to Karp's famous Evasiveness Conjecture, as every graph complex with non-vanishing reduced Euler characteristic is known to be evasive. Applying tools from the "topological approach to evasiveness" which was introduced in the seminal paper of Khan, Saks and Sturtevant [FOCS 83], we prove that @math is @math -hard for every monotone property @math that does not hold on the Hamilton cycle as well as for some monotone properties that hold on the Hamilton cycle such as being triangle-free or not @math -edge-connected for @math . Moreover, we show that for those properties @math can not be solved in time @math for any computable function @math unless the Exponential Time Hypothesis (ETH) fails. In the final part of the paper, we investigate non-monotone properties and prove that @math is @math -hard if @math is any non-trivial modularity constraint on the number of edges with respect to some prime @math or if @math enforces the presence of a fixed isolated subgraph.
Those latter results are independent from ours in the sense that ours do not imply theirs and vice versa. One example of a property whose hardness does not follow from the results of Jerrum and Meeks is bipartiteness: The edge-densities of both, the properties of being bipartite and not bipartite grow asymptotically as fast as @math and the edge-minimal non-bipartite graphs are odd cycles, hence having treewidth @math . However hardness for the property of being bipartite follows from the first condition of Theorem as odd cycles are not bipartite. Moreover, we remark that Meek's reduction in @cite_24 uses the Excluded Grid Theorem and hence does not imply a tight lower bound under ETH.
@cite_24
2963702702 2155295958 2144865572 2170546552
We prove tight upper bounds on the logarithmic derivative of the independence and matching polynomials of d-regular graphs. For independent sets, this theorem is a strengthening of Kahn's result that a disjoint union of copies of Kd;d maximizes the number of independent sets of a bipartite d-regular graph, Galvin and Tet ali's result that the independence polynomial is maximized by the same, and Zhao's extension of both results to all d-regular graphs. For matchings, this shows that the matching polynomial and the total number of matchings of a d-regular graph are maximized by a union of copies of Kd;d. Using this we prove the asymptotic upper matching conjecture of Friedland, Krop, Lundow, and Markstrom. In probabilistic language, our main theorems state that for all d-regular graphs and all �, the occupancy fraction of the hard-core model and the edge occupancy fraction of the monomer-dimer model with fugacity � are maximized by Kd;d. Our method involves constrained optimization problems over distributions of random variables and applies to all d-regular graphs directly, without a reduction to the bipartite case. Using a variant of the method we prove a lower bound on the occupancy fraction of the hard-core model on any d-regular, vertex-transitive, bipartite graph: the occupancy fraction of such a graph is strictly greater than the occupancy fraction of the unique translationinvariant hard-core measure on the infinite d-regular tree Suppose we are given a bipartite graph with vertex set X, Y, |X| = n , |Y| = N , each point in X (Y) has degree D(d) fixed, respectively, moreover, each pair of points x, x' ∈ X has at most D (log n) 3 (common) neighbours. Let t (X, Y) denote the minimum number of vertices of Y needed to cover all vertices of X. We prove (Theorem 1.1) that t (X, Y) d n tends to 1 as n tends to infinity. This result has many applications: Theorem [5]. Suppose k > r > 1 are fixed, n → ∞. Then there exists a collection of (1 + o (1)) × ( n r ) ( k r ) k -subsets of an n -set so that each r -subset is contained in at least one member of the collection. Analogues and strengthenings of this result are deduced. E.g. for vector spaces, orthogonal or simplectic geometries, random collections of k -sets with constant probabilities, etc. Theorem 3.3. Suppose G is a graph on v vertices and e edges and ℛ is a random graph on n vertices and edge probability e ( v 2 ). Then there exists a collection of (1 + o (1))( n 2 ) ( v 2 ) induced subgraphs of ℛ on v vertices, isomorphic to G and such that each edge (non-edge) of ℛ is covered by an edge (non-edge) of a graph in the collection. We give a sharp lower bound on the number of matchings of a given size in a bipartite graph. When specialized to regular bipartite graphs, our results imply Friedland's Lower Matching Conjecture and Schrijver's theorem proven by Gurvits and Csikvari. Indeed, our work extends the recent work of Csikvari done for regular and bi-regular bipartite graphs. Moreover, our lower bounds are order optimal as they are attained for a sequence of @math -lifts of the original graph as well as for random @math -lifts of the original graph when @math tends to infinity. We then extend our results to permanents and subpermanents sums. For permanents, we are able to recover the lower bound of Schrijver recently proved by Gurvits using stable polynomials. Our proof is algorithmic and borrows ideas from the theory of local weak convergence of graphs, statistical physics and covers of graphs. We provide new lower bounds for subpermanents sums and obtain new results on the number of matching in random @math -lifts with some implications for the matching measure and the spectral measure of random @math -lifts as well as for the spectral measure of infinite trees. Mulmuley [Mul12a] recently gave an explicit version of Noether’s Normalization lemma for ring of invariants of matrices under simultaneous conjugation, under the conjecture that there are deterministic black-box algorithms for polynomial identity testing (PIT). He argued that this gives evidence that constructing such algorithms for PIT is beyond current techniques. In this work, we show this is not the case. That is, we improve Mulmuley’s reduction and correspondingly weaken the conjecture regarding PIT needed to give explicit Noether Normalization. We then observe that the weaker conjecture has recently been nearly settled by the authors ([FS12]), who gave quasipolynomial size hitting sets for the class of read-once oblivious algebraic branching programs (ROABPs). This gives the desired explicit Noether Normalization unconditionally, up to quasipolynomial factors. As a consequence of our proof we give a deterministic parallel polynomial-time algorithm for deciding if two matrix tuples have intersecting orbit closures, under simultaneous conjugation. We also study the strength of conjectures that Mulmuley requires to obtain similar results as ours. We prove that his conjectures are stronger, in the sense that the computational model he needs PIT algorithms for is equivalent to the well-known algebraic branching program (ABP) model, which is provably stronger than the ROABP model. Finally, we consider the depth-3 diagonal circuit model as defined by Saxena [Sax08], as PIT algorithms for this model also have implications in Mulmuley’s work. Previous work (such as [ASS12] and [FS12]) have given quasipolynomial size hitting sets for this model. In this work, we give a much simpler construction of such hitting sets, using techniques of Shpilka and Volkovich [SV09].