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Register a new userSmall-Set Expansion in Shortcode Graph and the 2-to-2 Conjecture
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Added by
Pravesh K Kothari
Publication date
2018
fields
Informatics Engineering
and research's language is
English
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Dinur, Khot, Kindler, Minzer and Safra (2016) recently showed that the (imperfect completeness variant of) Khots 2 to 2 games conjecture follows from a combinatorial hypothesis about the soundness of a certain Grassmanian agreement tester. In this work, we show that the hypothesis of Dinur et. al. follows from a conjecture we call the Inverse Shortcode Hypothesis characterizing the non-expanding sets of the degree-two shortcode graph. We also show the latter conjecture is equivalent to a characterization of the non-expanding sets in the Grassman graph, as hypothesized by a follow-up paper of Dinur et. al. (2017). Following our work, Khot, Minzer and Safra (2018) proved the Inverse Shortcode Hypothesis. Combining their proof with our result and the reduction of Dinur et. al. (2016), completes the proof of the 2 to 2 conjecture with imperfect completeness. Moreover, we believe that the shortcode graph provides a useful view of both the hypothesis and the reduction, and might be useful in extending it further.
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We study the NP-hard textsc{$k$-Sparsest Cut} problem ($k$SC) in which, given an undirected graph $G = (V, E)$ and a parameter $k$, the objective is to partition vertex set into $k$ subsets whose maximum edge expansion is minimized. Herein, the edge expansion of a subset $S subseteq V$ is defined as the sum of the weights of edges exiting $S$ divided by the number of vertices in $S$. Another problem that has been investigated is textsc{$k$-Small-Set Expansion} problem ($k$SSE), which aims to find a subset with minimum edge expansion with a restriction on the size of the subset. We extend previous studies on $k$SC and $k$SSE by inspecting their parameterized complexity. On the positive side, we present two FPT algorithms for both $k$SSE and 2SC problems where in the first algorithm we consider the parameter treewidth of the input graph and uses exponential space, and in the second we consider the parameter vertex cover number of the input graph and uses polynomial space. Moreover, we consider the unweighted version of the $k$SC problem where $k geq 2$ is fixed and proposed two FPT algorithms with parameters treewidth and vertex cover number of the input graph. We also propose a randomized FPT algorithm for $k$SSE when parameterized by $k$ and the maximum degree of the input graph combined. Its derandomization is done efficiently. oindent On the negative side, first we prove that for every fixed integer $k,taugeq 3$, the problem $k$SC is NP-hard for graphs with vertex cover number at most $tau$. We also show that $k$SC is W[1]-hard when parameterized by the treewidth of the input graph and the number~$k$ of components combined using a reduction from textsc{Unary Bin Packing}. Furthermore, we prove that $k$SC remains NP-hard for graphs with maximum degree three and also graphs with degeneracy two. Finally, we prove that the unweighted $k$SSE is W[1]-hard for the parameter $k$.
In this work, we achieve gap amplification for the Small-Set Expansion problem. Specifically, we show that an instance of the Small-Set Expansion Problem with completeness $epsilon$ and soundness $frac{1}{2}$ is at least as difficult as Small-Set Expansion with completeness $epsilon$ and soundness $f(epsilon)$, for any function $f(epsilon)$ which grows faster than $sqrt{epsilon}$. We achieve this amplification via random walks -- our gadget is the graph with adjacency matrix corresponding to a random walk on the original graph. An interesting feature of our reduction is that unlike gap amplification via parallel repetition, the size of the instances (number of vertices) produced by the reduction remains the same.
We give an algorithm for solving unique games (UG) instances whenever low-degree sum-of-squares proofs certify good bounds on the small-set-expansion of the underlying constraint graph via a hypercontractive inequality. Our algorithm is in fact more versatile, and succeeds even when the constraint graph is not a small-set expander as long as the structure of non-expanding small sets is (informally speaking) characterized by a low-degree sum-of-squares proof. Our results are obtained by rounding emph{low-entropy} solutions -- measured via a new global potential function -- to sum-of-squares (SoS) semidefinite programs. This technique adds to the (currently short) list of general tools for analyzing SoS relaxations for emph{worst-case} optimization problems. As corollaries, we obtain the first polynomial-time algorithms for solving any UG instance where the constraint graph is either the emph{noisy hypercube}, the emph{short code} or the emph{Johnson} graph. The prior best algorithm for such instances was the eigenvalue enumeration algorithm of Arora, Barak, and Steurer (2010) which requires quasi-polynomial time for the noisy hypercube and nearly-exponential time for the short code and Johnson graphs. All of our results achieve an approximation of $1-epsilon$ vs $delta$ for UG instances, where $epsilon>0$ and $delta > 0$ depend on the expansion parameters of the graph but are independent of the alphabet size.
We give improved separations for the query complexity analogue of the log-approximate-rank conjecture i.e. we show that there are a plethora of total Boolean functions on $n$ input bits, each of which has approximate Fourier sparsity at most $O(n^3)$ and randomized parity decision tree complexity $Theta(n)$. This improves upon the recent work of Chattopadhyay, Mande and Sherif (JACM 20) both qualitatively (in terms of designing a large number of examples) and quantitatively (improving the gap from quartic to cubic). We leave open the problem of proving a randomized communication complexity lower bound for XOR compositions of our examples. A linear lower bound would lead to new and improved refutations of the log-approximate-rank conjecture. Moreover, if any of these compositions had even a sub-linear cost randomized communication protocol, it would demonstrate that randomized parity decision tree complexity does not lift to randomized communication complexity in general (with the XOR gadget).
The degree-$4$ Sum-of-Squares (SoS) SDP relaxation is a powerful algorithm that captures the best known polynomial time algorithms for a broad range of problems including MaxCut, Sparsest Cut, all MaxCSPs and tensor PCA. Despite being an explicit algorithm with relatively low computational complexity, the limits of degree-$4$ SoS SDP are not well understood. For example, existing integrality gaps do not rule out a $(2-varepsilon)$-algorithm for Vertex Cover or a $(0.878+varepsilon)$-algorithm for MaxCut via degree-$4$ SoS SDPs, each of which would refute the notorious Unique Games Conjecture. We exhibit an explicit mapping from solutions for degree-$2$ Sum-of-Squares SDP (Goemans-Williamson SDP) to solutions for the degree-$4$ Sum-of-Squares SDP relaxation on boolean variables. By virtue of this mapping, one can lift lower bounds for degree-$2$ SoS SDP relaxation to corresponding lower bounds for degree-$4$ SoS SDPs. We use this approach to obtain degree-$4$ SoS SDP lower bounds for MaxCut on random $d$-regular graphs, Sherington-Kirkpatrick model from statistical physics and PSD Grothendieck problem. Our constructions use the idea of pseudocalibration towards candidate SDP vectors, while it was previously only used to produce the candidate matrix which one would show is PSD using much technical work. In addition, we develop a different technique to bound the spectral norms of _graphical matrices_ that arise in the context of SoS SDPs. The technique is much simpler and yields better bounds in many cases than the _trace method_ -- which was the sole technique for this purpose.
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