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Vygen, J ; Byrka, J (Ed.)One of the most famous conjectures in combinatorial optimization is the fourthirds conjecture, which states that the integrality gap of the subtour LP relaxation of the TSP is equal to $\frac43$. For 40 years, the best known upper bound was $1.5$. Recently, Karlin, Klein, and Oveis Gharan \cite{KKO21b} showed that the max entropy algorithm for the TSP gives an improved bound of $1.5  10^{36}$. In this paper, we show that the approximation ratio of the max entropy algorithm is at least 1.375, even for graphic TSP. Thus the max entropy algorithm does not appear to be the algorithm that will ultimately resolve the fourthirds conjecture in the affirmative, should that be possible.more » « lessFree, publiclyaccessible full text available August 27, 2025

This paper considers the interplay between semidefinite programming, matrix rank, and graph coloring. Karger, Motwani, and Sudan give a vector program in which a coloring of a graph can be encoded as a semidefinite matrix of low rank. By complementary slackness conditions of semidefinite programming, if an optimal dual solution has high rank, any optimal primal solution must have low rank. We attempt to characterize graphs for which we can show that the corresponding dual optimal solution must have rank high enough that the primal solution encodes a coloring. In the case of the original Karger, Motwani, and Sudan vector program, we show that any graph which is a $k$tree has sufficiently high dual rank, and we can extract the coloring from the corresponding lowrank primal solution. We can also show that if a graph is not uniquely colorable, then no sufficiently high rank dual optimal solution can exist. This allows us to completely characterize the planar graphs for which dual optimal solutions have sufficiently high dual rank, since it is known that the uniquely colorable planar graphs are precisely the planar 3trees. We then modify the semidefinite program to have an objective function with costs, and explore when we can create an objective function such that the optimal dual solution has sufficiently high rank. We show that it is always possible to construct such an objective function given the graph coloring. The construction of the objective function gives rise to heuristics for 4coloring planar graphs. We enumerated all maximal planar graphs with an induced $K_4$ of up to 14 vertices; the heuristics successfully found a 4coloring for 99.75\% of them. Our research was motivated by trying to use semidefinite programming to prove the fourcolor theorem, which states that every planar graph can be colored with four colors. There is an intriguing connection of the KargerMotwaniSudan semidefinite program with the Colin de Verdi\`ere graph invariant (and a corresponding conjecture of Colin de Verdi\`ere), in which matrices that have some similarities to the dual feasible matrices of the semidefinite program must have high rank in the case that graphs are of a certain type; for instance, planar graphs have rank that would imply that the primal solution of the semidefinite program encodes a 4coloring.more » « lessFree, publiclyaccessible full text available July 1, 2025

This paper considers the relationship between semidefinite programs (SDPs), matrix rank, and maximum cuts of graphs. Utilizing complementary slackness conditions for SDPs, we investigate when the rank 1 feasible solution corresponding to a max cut is the unique optimal solution to the GoemansWilliamson max cut SDP by showing the existence of an optimal dual solution with rank n1 . Our results consider connected bipartite graphs and graphs with multiple max cuts. We conclude with a conjecture for general graphs.more » « lessFree, publiclyaccessible full text available March 1, 2025

We experimentally evaluate the performance of several Max Cut approximation algorithms. In particular, we compare the results of the Goemans and Williamson algorithm using semidefinite programming with Trevisan’s algorithm using spectral partitioning. The former algorithm has a known .878 approximation guarantee whereas the latter has a .614 approximation guarantee. We investigate whether this gap in approximation guarantees is evident in practice or whether the spectral algorithm performs as well as the SDP. We also compare the performances to the standard greedy Max Cut algorithm which has a .5 approximation guarantee, two additional spectral algorithms, and a heuristic from Burer, Monteiro, and Zhang (BMZ). The algorithms are tested on Erdős–Renyi random graphs, complete graphs from TSPLIB, and realworld graphs from the Network Repository. We find, unsurprisingly, that the spectral algorithms provide a significant speed advantage over the SDP. In our experiments, the spectral algorithms and BMZ heuristic return cuts with values which are competitive with those of the SDP.more » « lessFree, publiclyaccessible full text available December 31, 2024

Over the last two decades, a significant line of work in theoretical algorithms has made progress in solving linear systems of the form $\mathbf{L}\mathbf{x} = \mathbf{b}$, where $\mathbf{L}$ is the Laplacian matrix of a weighted graph with weights $w(i,j)>0$ on the edges. The solution $\mathbf{x}$ of the linear system can be interpreted as the potentials of an electrical flow in which the resistance on edge $(i,j)$ is $1/w(i,j)$. Kelner, Orrechia, Sidford, and Zhu \cite{KOSZ13} give a combinatorial, nearlinear time algorithm that maintains the Kirchoff Current Law, and gradually enforces the Kirchoff Potential Law by updating flows around cycles ({\it cycle toggling}). In this paper, we consider a dual version of the algorithm that maintains the Kirchoff Potential Law, and gradually enforces the Kirchoff Current Law by {\it cut toggling}: each iteration updates all potentials on one side of a fundamental cut of a spanning tree by the same amount. We prove that this dual algorithm also runs in a nearlinear number of iterations. We show, however, that if we abstract cut toggling as a natural data structure problem, this problem can be reduced to the online vectormatrixvector problem (OMv), which has been conjectured to be difficult for dynamic algorithms \cite{HKNS15}. The conjecture implies that the data structure does not have an $O(n^{1\epsilon})$ time algorithm for any $\epsilon > 0$, and thus a straightforward implementation of the cuttoggling algorithm requires essentially linear time per iteration. To circumvent the lower bound, we batch update steps, and perform them simultaneously instead of sequentially. An appropriate choice of batching leads to an $\widetilde{O}(m^{1.5})$ time cuttoggling algorithm for solving Laplacian systems. Furthermore, we show that if we sparsify the graph and call our algorithm recursively on the Laplacian system implied by batching and sparsifying, we can reduce the running time to $O(m^{1 + \epsilon})$ for any $\epsilon > 0$. Thus, the dual cuttoggling algorithm can achieve (almost) the same running time as its primal cycletoggling counterpart.more » « lessFree, publiclyaccessible full text available December 1, 2024

Del Pia, Alberto ; Kaibel, Volker (Ed.)A longstanding conjecture for the traveling salesman problem (TSP) states that the integrality gap of the standard linear programming relaxation of the TSP (sometimes called the Subtour LP or the HeldKarp bound) is at most 4/3 for symmetric instances of the TSP obeying the triangle inequality. In this paper we consider the halfintegral case, in which a feasible solution to the LP has solution values in {0,1/2,1} . Karlin, Klein, and Oveis Gharan [9], in a breakthrough result, were able to show that in the halfintegral case, the integrality gap is at most 1.49993; Gupta et al. [6] showed a slight improvement of this result to 1.4983. Both of these papers consider a hierarchy of critical tight sets in the support graph of the LP solution, in which some of the sets correspond to cycle cuts and the others to degree cuts. Here we show that if all the sets in the hierarchy correspond to cycle cuts, then we can find a distribution of tours whose expected cost is at most 4/3 times the value of the halfintegral LP solution; sampling from the distribution gives us a randomized 4/3approximation algorithm. We note that known bad cases for the integrality gap have a gap of 4/3 and have a halfintegral LP solution in which all the critical tight sets in the hierarchy are cycle cuts; thus our result is tight.more » « less

Facetdefining inequalities of the symmetric traveling salesman problem (TSP) polytope play a prominent role in both polyhedral TSP research and stateoftheart TSP solvers. In this paper, we introduce a new class of facetdefining inequalities, the circlet inequalities. These inequalities were first conjectured in Gutekunst and Williamson [Gutekunst SC, Williamson DP (2019) Characterizing the integrality gap of the subtour LP for the circulant traveling salesman problem. SIAM J. Discrete Math. 33(4):2452–2478] when studying the circulant TSP, and they provide a bridge between polyhedral TSP research and numbertheoretic investigations of Hamiltonian cycles stemming from a conjecture from Marco Buratti in 2007. The circlet inequalities exhibit circulant symmetry by placing the same weight on all edges of a given length; our main proof exploits this symmetry to prove the validity of the circlet inequalities. We then show that the circlet inequalities are facetdefining and compute their strength following Goemans [Goemans MX (1995) Worstcase comparison of valid inequalities for the TSP. Math. Programming 69:335–349]; they achieve the same worst case strength as the similarly circulant crown inequalities of Naddef and Rinaldi [Naddef D, Rinaldi G (1992) The crown inequalities for the symmetric traveling salesman polytope. Math. Oper. Res. 17(2):308–326] but are generally stronger.more » « less
Funding: This material is based upon work supported by the National Science Foundation Graduate Research Fellowship Program [Grant DGE1650441] and by the National Science Foundation Division of Computing and Communications Foundations [Grant CCF1908517].

Tauman Kalai, Yael (Ed.)Over the last two decades, a significant line of work in theoretical algorithms has made progress in solving linear systems of the form 𝐋𝐱 = 𝐛, where 𝐋 is the Laplacian matrix of a weighted graph with weights w(i,j) > 0 on the edges. The solution 𝐱 of the linear system can be interpreted as the potentials of an electrical flow in which the resistance on edge (i,j) is 1/w(i,j). Kelner, Orrechia, Sidford, and Zhu [Kelner et al., 2013] give a combinatorial, nearlinear time algorithm that maintains the Kirchoff Current Law, and gradually enforces the Kirchoff Potential Law by updating flows around cycles (cycle toggling). In this paper, we consider a dual version of the algorithm that maintains the Kirchoff Potential Law, and gradually enforces the Kirchoff Current Law by cut toggling: each iteration updates all potentials on one side of a fundamental cut of a spanning tree by the same amount. We prove that this dual algorithm also runs in a nearlinear number of iterations. We show, however, that if we abstract cut toggling as a natural data structure problem, this problem can be reduced to the online vectormatrixvector problem (OMv), which has been conjectured to be difficult for dynamic algorithms [Henzinger et al., 2015]. The conjecture implies that the data structure does not have an O(n^{1ε}) time algorithm for any ε > 0, and thus a straightforward implementation of the cuttoggling algorithm requires essentially linear time per iteration. To circumvent the lower bound, we batch update steps, and perform them simultaneously instead of sequentially. An appropriate choice of batching leads to an Õ(m^{1.5}) time cuttoggling algorithm for solving Laplacian systems. Furthermore, we show that if we sparsify the graph and call our algorithm recursively on the Laplacian system implied by batching and sparsifying, we can reduce the running time to O(m^{1 + ε}) for any ε > 0. Thus, the dual cuttoggling algorithm can achieve (almost) the same running time as its primal cycletoggling counterpart.more » « less

Kavitha, Telikepalli ; Mehlhorn, Kurt (Ed.)This paper revisits the 2approximation algorithm for kMST presented by Garg [9] in light of a recent paper of Paul et al. [14]. In the kMST problem, the goal is to return a tree spanning k vertices of minimum total edge cost. Paul et al. [14] extend Garg's primaldual subroutine to improve the approximation ratios for the budgeted prizecollecting traveling salesman and minimum spanning tree problems. We follow their algorithm and analysis to provide a cleaner version of Garg's result. Additionally, we introduce the novel concept of a kernel which allows an easier visualization of the stages of the algorithm and a clearer understanding of the pruning phase. Other notable updates include presenting a linear programming formulation of the kMST problem, including pseudocode, replacing the coloring scheme used by Garg with the simpler concept of neutral sets, and providing an explicit potential function.more » « less

Schulz, Christian ; Ucar, Bora (Ed.)We experimentally evaluate the performance of several Max Cut approximation algorithms. In particular, we compare the results of the Goemans and Williamson algorithm using semidefinite programming with Trevisan’s algorithm using spectral partitioning. The former algorithm has a known .878 approximation guarantee whereas the latter has a .614 approximation guarantee. We investigate whether this gap in approximation guarantees is evident in practice or whether the spectral algorithm performs as well as the SDP. We also compare the performances to the standard greedy Max Cut algorithm which has a .5 approximation guarantee and two additional spectral algorithms. The algorithms are tested on ErdősRenyi random graphs, complete graphs from TSPLIB, and realworld graphs from the Network Repository. We find, unsurprisingly, that the spectral algorithms provide a significant speed advantage over the SDP. In our experiments, the spectral algorithms return cuts with values which are competitive with those of the SDP.more » « less