Stimulated Raman adiabatic passage-like protocols for amplitude transfer generalize to many bipartite graphs

Adiabatic passage techniques, used to drive a system from one quantum state into another, find widespread application in physics and chemistry. We focus on techniques to spatially transport a quantum amplitude over a strongly coupled system, such as STImulated Raman Adiabatic Passage (STIRAP) and Coherent Tunnelling by Adiabatic Passage (CTAP). Previous results were shown to work on certain graphs, such as linear chains, square and triangular lattices, and branched chains. We prove that similar protocols work much more generally, in a large class of (semi-)bipartite graphs. In particular, under random couplings, adiabatic transfer is possible on graphs that admit a perfect matching both when the sender is removed and when the receiver is removed. Many of the favorable stability properties of STIRAP/CTAP are inherited, and our results readily apply to transfer between multiple potential senders and receivers. We numerically test transfer between the leaves of a tree, and find surprisingly accurate transfer, especially when straddling is used. Our results may find applications in short-distance communication between multiple quantum computers, and open up a new question in graph theory about the spectral gap around the value 0.Comment: 17 pages, 3 figures. v2 is made more mathematical and precise than v

Optimal distance query reconstruction for graphs without long induced cycles

Let $G=(V,E)$ be an $n$-vertex connected graph of maximum degree $\Delta$. Given access to $V$ and an oracle that given two vertices $u,v\in V$, returns the shortest path distance between $u$ and $v$, how many queries are needed to reconstruct $E$? We give a simple deterministic algorithm to reconstruct trees using $\Delta n\log_\Delta n+(\Delta+2)n$ distance queries and show that even randomised algorithms need to use at least $\frac1{100} \Delta n\log_\Delta n$ queries in expectation. The best previous lower bound was an information-theoretic lower bound of $\Omega(n\log n/\log \log n)$. Our lower bound also extends to related query models including distance queries for phylogenetic trees, membership queries for learning partitions and path queries in directed trees. We extend our deterministic algorithm to reconstruct graphs without induced cycles of length at least $k$ using $O_{\Delta,k}(n\log n)$ queries, which includes various graph classes of interest such as chordal graphs, permutation graphs and AT-free graphs. Since the previously best known randomised algorithm for chordal graphs uses $O_{\Delta}(n\log^2 n)$ queries in expectation, we both get rid off the randomness and get the optimal dependency in $n$ for chordal graphs and various other graph classes. Finally, we build on an algorithm of Kannan, Mathieu, and Zhou [ICALP, 2015] to give a randomised algorithm for reconstructing graphs of treelength $k$ using $O_{\Delta,k}(n\log^2n)$ queries in expectation.Comment: 35 page

Parameterized Complexities of Dominating and Independent Set Reconfiguration

We settle the parameterized complexities of several variants of independent set reconfiguration and dominating set reconfiguration, parameterized by the number of tokens. We show that both problems are XL-complete when there is no limit on the number of moves and XNL-complete when a maximum length ? for the sequence is given in binary in the input. The problems are known to be XNLP-complete when ? is given in unary instead, and W[1]- and W[2]-hard respectively when ? is also a parameter. We complete the picture by showing membership in those classes. Moreover, we show that for all the variants that we consider, token sliding and token jumping are equivalent under pl-reductions. We introduce partitioned variants of token jumping and token sliding, and give pl-reductions between the four variants that have precise control over the number of tokens and the length of the reconfiguration sequence

Tight Bounds for Counting Colorings and Connected Edge Sets Parameterized by Cutwidth

We study the fine-grained complexity of counting the number of colorings and connected spanning edge sets parameterized by the cutwidth and treewidth of the graph. While decompositions of small treewidth decompose the graph with small vertex separators, decompositions with small cutwidth decompose the graph with small edge separators. Let p,q ? ? such that p is a prime and q ? 3. We show: - If p divides q-1, there is a (q-1)^{ctw}n^{O(1)} time algorithm for counting list q-colorings modulo p of n-vertex graphs of cutwidth ctw. Furthermore, there is no ? > 0 for which there is a (q-1-?)^{ctw} n^{O(1)} time algorithm that counts the number of list q-colorings modulo p of n-vertex graphs of cutwidth ctw, assuming the Strong Exponential Time Hypothesis (SETH). - If p does not divide q-1, there is no ? > 0 for which there exists a (q-?)^{ctw} n^{O(1)} time algorithm that counts the number of list q-colorings modulo p of n-vertex graphs of cutwidth ctw, assuming SETH. The lower bounds are in stark contrast with the existing 2^{ctw}n^{O(1)} time algorithm to compute the chromatic number of a graph by Jansen and Nederlof [Theor. Comput. Sci.\u2718]. Furthermore, by building upon the above lower bounds, we obtain the following lower bound for counting connected spanning edge sets: there is no ? > 0 for which there is an algorithm that, given a graph G and a cutwidth ordering of cutwidth ctw, counts the number of spanning connected edge sets of G modulo p in time (p - ?)^{ctw} n^{O(1)}, assuming SETH. We also give an algorithm with matching running time for this problem. Before our work, even for the treewidth parameterization, the best conditional lower bound by Dell et al. [ACM Trans. Algorithms\u2714] only excluded 2^{o(tw)}n^{O(1)} time algorithms for this problem. Both our algorithms and lower bounds employ use of the matrix rank method, by relating the complexity of the problem to the rank of a certain "compatibility matrix" in a non-trivial way

The curse of ties in congestion games with limited lookahead

We introduce a novel framework to model limited lookahead in congestion games. Intuitively, the players enter the game sequentially and choose an optimal action under the assumption that the k - 1 subsequent players play subgame-perfectly. Our model naturally interpolates between outcomes of greedy best-response (k = 1) and subgame-perfect outcomes (k = n, the number of players). We study the impact of limited lookahead (parameterized by k) on the stability and inefficiency of the resulting outcomes. As our results reveal, increased lookahead does not necessarily lead to better outcomes; in fact, its effect crucially depends on the existence of ties and the type of game under consideration

On the Parameterized Complexity of Computing Tree-Partitions

We study the parameterized complexity of computing the tree-partition-width, a graph parameter equivalent to treewidth on graphs of bounded maximum degree. On one hand, we can obtain approximations of the tree-partition-width efficiently: we show that there is an algorithm that, given an $n$-vertex graph $G$ and an integer $k$, constructs a tree-partition of width $O(k^7)$ for $G$ or reports that $G$ has tree-partition width more than $k$, in time $k^{O(1)}n^2$. We can improve on the approximation factor or the dependence on $n$ by sacrificing the dependence on $k$. On the other hand, we show the problem of computing tree-partition-width exactly is XALP-complete, which implies that it is $W[t]$-hard for all $t$. We deduce XALP-completeness of the problem of computing the domino treewidth. Finally, we adapt some known results on the parameter tree-partition-width and the topological minor relation, and use them to compare tree-partition-width to tree-cut width

On the parameterized complexity of computing tree-partitions

We study the parameterized complexity of computing the tree-partition-width, a graph parameter equivalent to treewidth on graphs of bounded maximum degree. On one hand, we can obtain approximations of the tree-partition-width efficiently: we show that there is an algorithm that, given an $n$-vertex graph $G$ and an integer $k$, constructs a tree-partition of width $O(k^7)$ for $G$ or reports that $G$ has tree-partition width more than $k$, in time $k^{O(1)}n^2$. We can improve on the approximation factor or the dependence on $n$ by sacrificing the dependence on $k$. On the other hand, we show the problem of computing tree-partition-width exactly is XALP-complete, which implies that it is $W[t]$-hard for all $t$. We deduce XALP-completeness of the problem of computing the domino treewidth. Finally, we adapt some known results on the parameter tree-partition-width and the topological minor relation, and use them to compare tree-partition-width to tree-cut width

Parameterized Complexity of Binary CSP: Vertex Cover, Treedepth, and Related Parameters

We investigate the parameterized complexity of Binary CSP parameterized by the vertex cover number and the treedepth of the constraint graph, as well as by a selection of related modulator-based parameters. The main findings are as follows: - Binary CSP parameterized by the vertex cover number is W[3]-complete. More generally, for every positive integer d, Binary CSP parameterized by the size of a modulator to a treedepth-d graph is W[2d+1]-complete. This provides a new family of natural problems that are complete for odd levels of the W-hierarchy. - We introduce a new complexity class XSLP, defined so that Binary CSP parameterized by treedepth is complete for this class. We provide two equivalent characterizations of XSLP: the first one relates XSLP to a model of an alternating Turing machine with certain restrictions on conondeterminism and space complexity, while the second one links XSLP to the problem of model-checking first-order logic with suitably restricted universal quantification. Interestingly, the proof of the machine characterization of XSLP uses the concept of universal trees, which are prominently featured in the recent work on parity games. - We describe a new complexity hierarchy sandwiched between the W-hierarchy and the A-hierarchy: For every odd t, we introduce a parameterized complexity class S[t] with W[t] ? S[t] ? A[t], defined using a parameter that interpolates between the vertex cover number and the treedepth. We expect that many of the studied classes will be useful in the future for pinpointing the complexity of various structural parameterizations of graph problems